Derivations and the first Hochschild cohomology group of the quantum grassmannian

Stéphane Launois; Thomas H. Lenagan

doi:10.4171/jncg/662

JournalsjncgVol. 20, No. 3pp. 845–870

Derivations and the first Hochschild cohomology group of the quantum grassmannian

Stéphane Launois
Université Caen Normandie, France
Thomas H. Lenagan
The University of Edinburgh, UK
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Abstract

We calculate the derivations and the first Hochschild cohomology group of the quantum grassmannian over a field of characteristic zero in the generic case when the deformation parameter is not a root of unity. Using graded techniques and two special homogeneous normal elements of the quantum grassmannian, we reduce the problem to computing derivations of the quantum grassmannian that act trivially on these two normal elements. We then use the dehomogenisation equality which shows that a localisation of the quantum grassmannian is equal to a skew Laurent extension of quantum matrices. This equality is used to connect derivations of the quantum grassmannian with those of quantum matrices. More precisely, again using graded techniques, we show that derivations of the quantum grassmannian that act trivially on our two normal elements restrict to homogeneous derivations of quantum matrices. The derivations of quantum matrices are known in the square case, and technical details needed to deal with the general case are given in the appendix. This allows us to explicitly describe the first Hochschild cohomology group of the quantum grassmannian.

1. Introduction

Let

K

denote a field of characteristic zero and

q \in K

be a nonzero element which is not a root of unity. The quantum grassmannian

𝒪_{q} (G (k, n))

is a noncommutative algebra that is a deformation of the homogeneous coordinate ring of the classical grassmannian of

k -

planes in

n -

space. In this paper, we calculate the derivations and the first Hochschild cohomology group of the quantum grassmannian.

One of the motivations for this paper is [7

M. Movshev and A. Schwarz, Quantum deformation of planar amplitudes. J. High Energy Phys. 2018 (2018), no. 4, article no. 121, 19 pp. Zbl 1390.81613 MR 3801153

] where a connection between the totally nonnegative grassmannian – or rather its cell decomposition into the union of so-called positroid cells – and Hochschild (co)homology of

𝒪_{q} (G (k, n))

was established. More precisely, a link between the volume form of positroids and Hochschild (co)homology of the quantum grassmannian

𝒪_{q} (G (k, n))

is established, allowing the contribution of a positroid cell to the scattering amplitude (in the

N = 4

supersymmetric Yang–Mills theory) to be

q -

deformed. The present paper is the first one in a project to compute the Hochschild (co)homology of quantum positroid varieties. More precisely, the quantum grassmannian can be viewed as a “quantum positroid variety” (corresponding to the totally positive grassmannian) thanks to [5

S. Launois, T. H. Lenagan, and B. M. Nolan, Total positivity is a quantum phenomenon: the grassmannian case. Mem. Amer. Math. Soc. 291 (2023), no. 1448, 109 pp. Zbl 1548.16002 MR 4665212

], and we compute the first Hochschild cohomology group of the quantum grassmannian. We will come back to the general case in future work.

Assume

k \leq n .

Recall that

𝒪_{q} (G (k, n))

is the subalgebra of the quantum matrix algebra

𝒪_{q} (M (k, n))

that is generated by the

k \times k

quantum minors of

𝒪_{q} (M (k, n)) .

We call these

k \times k

quantum minors quantum Plücker coordinates of

𝒪_{q} (G (k, n)) .

They are indexed by

k -

subsets of

{1, \dots, n}

and denoted by

[I]

for each

k -

subset

I .

The algebra

𝒪_{q} (G (k, n))

is graded with all quantum Plücker coordinates homogeneous of degree one.

Quantum Plücker coordinates formed on consecutive columns play a special role: they are normal, and each one generates a completely prime ideal of

𝒪_{q} (G (k, n)) .

They are referred to as prime quantum Plücker coordinates. Among the prime quantum Plücker coordinates, two are special:

[u] = [1, \dots, k]

and

[w] = [n - k + 1, \dots, n] .

The reason for this is that

[u]

(respectively,

[w]

)

q -

commutes with every quantum Plücker coordinate

[I]

with a nonnegative (respectively, nonpositive) power of

q,

that is,

[u] [I] = q^{\geq 0} [I] [u] and [w] [I] = q^{\leq 0} [I] [w] .

We exploit this observation and other graded techniques in order to reduce the problem of computing derivations of

𝒪_{q} (G (k, n))

to computing derivations

D

of

𝒪_{q} (G (k, n))

that act trivially on both

[u]

and

[w],

that is, derivations

D

such that

D ([u]) = D ([w]) = 0 .

We then employ the dehomogenisation equality which shows that a localisation of the quantum grassmannian is equal to a skew Laurent extension of quantum matrices. This equality is used to connect the set of derivations of the quantum grassmannian acting trivially on

[u]

and

[w]

with that of quantum matrices. More precisely, again using graded techniques, we show that derivations of the quantum grassmannian that act trivially on

[u]

and

[w]

restrict to homogeneous derivations of quantum matrices. To conclude, we use our knowledge of derivations of quantum matrices: the set of derivations is known in the case of square quantum matrices; see [3

S. Launois and T. H. Lenagan, The first Hochschild cohomology group of quantum matrices and the quantum special linear group. J. Noncommut. Geom. 1 (2007), no. 3, 281–309 Zbl 1137.16015 MR 2314098

]. Some technical details that are needed to deal with the fact that the non-square case has not yet been covered are given in the appendix.

This allows us to show that the first Hochschild cohomology group of

𝒪_{q} (G (k, n))

is a

K -

vector space of dimension

n,

with basis given by the cosets of the derivations

D_{1}, \dots, D_{n},

where

D_{i}

is the derivation of

𝒪_{q} (G (k, n))

defined by

D_{i} ([I]) = {\begin{matrix} [I] & if i \in I; \\ 0 & otherwise . \end{matrix}

2. Basic definitions

Throughout the paper,

K

denotes a field of characteristic zero and

q \in K

is a nonzero element which is not a root of unity.

The algebra of

m \times n

quantum matrices over

K,

denoted by

𝒪_{q} (M (m, n)),

is the algebra generated over

K

by

m n

indeterminates

x_{i j},

with

1 \leq i \leq m

and

1 \leq j \leq n,

which commute with the elements of

K

and are subject to the relations

\begin{matrix} x_{i j} x_{i l} & = q x_{i l} x_{i j} & for 1 \leq i \leq m and 1 \leq j < l \leq n; \\ x_{i j} x_{k j} & = q x_{k j} x_{i j} & for 1 \leq i < k \leq m and 1 \leq j \leq n; \\ x_{i j} x_{k l} & = x_{k l} x_{i j} & for 1 \leq k < i \leq m and 1 \leq j < l \leq n; \\ x_{i j} x_{k l} - x_{k l} x_{i j} & = (q - q^{- 1}) x_{i l} x_{k j} & for 1 \leq i < k \leq m and 1 \leq j < l \leq n . \end{matrix}

It is well known that

𝒪_{q} (M (m, n))

is an iterated Ore extension over

K

with the

x_{i j}

added in lexicographic order. An immediate consequence is that

𝒪_{q} (M (m, n))

is a noetherian domain.

When

m = n,

the quantum determinant

D_{q}

is defined by

D_{q} := \sum {(- q)}^{l (σ)} x_{1 σ (1)} \dots x_{n σ (n)},

where the sum is over all permutations

σ

of

{1, \dots, n}

and where

l (σ)

denotes the length of the permutation

σ .

The quantum determinant is a central element in the algebra of quantum matrices

𝒪_{q} (M (n, n)) .

If

I

and

J

are

t -

element subsets of

{1, \dots, m}

and

{1, \dots, n},

respectively, then the quantum minor

[I | J]

is defined to be the quantum determinant of the

t \times t

quantum matrix subalgebra generated by the variables

x_{i j}

with

i \in I

and

j \in J .

The quantum matrix algebra

𝒪_{q} (M (m, n))

is a connected

ℕ -

graded algebra with each generator

x_{i j}

given degree one. Note that each

t \times t

quantum minor has degree

t .

Definition 2.1.

Let

k \leq n .

The homogeneous coordinate ring of the $k \times n$ quantum grassmannian,

𝒪_{q} (G (k, n))

(known informally as the quantum grassmannian), is the subalgebra of the quantum matrix algebra

𝒪_{q} (M (k, n))

that is generated by the

k \times k

quantum minors of

𝒪_{q} (M (k, n));

see, for example, [2

A. C. Kelly, T. H. Lenagan, and L. Rigal, Ring theoretic properties of quantum grassmannians. J. Algebra Appl. 3 (2004), 9–30 Zbl 1081.16048 MR 2047633

].

A

k \times k

quantum minor of

𝒪_{q} (M (k, n))

must use all of the

k

rows, and so we can specify the quantum minor by specifying the columns that define it. With this in mind, we will write

[J]

for the quantum minor

[1, \dots, k | J]

for any

k -

element subset

J

of

{1, \dots, n} .

Quantum minors of this type are called quantum Plücker coordinates. The quantum grassmannian

𝒪_{q} (G (k, n))

is a connected

ℕ -

graded algebra with each quantum Plücker coordinate given degree one. The set of quantum Plücker coordinates in

𝒪_{q} (G (k, n))

is denoted by

Π .

There is a natural partial order on

Π

defined in the following way: if

I = [i_{1} < \dots < i_{k}]

and

J = [j_{1} < \dots < j_{k}],

then

[I] < [J]

if and only if

i_{l} \leq j_{l}

for each

l = 1, \dots, k .

A standard monomial in the quantum Plücker coordinates is an expression of the form

[I_{1}] [I_{2}] \dots [I_{t}],

where

I_{1} \leq I_{2} \leq \dots \leq I_{t}

in this partial order. The set of all standard monomials forms a vector space basis of

𝒪_{q} (G (k, n))

over

K;

see, for example, [2

A. C. Kelly, T. H. Lenagan, and L. Rigal, Ring theoretic properties of quantum grassmannians. J. Algebra Appl. 3 (2004), 9–30 Zbl 1081.16048 MR 2047633

, Corollary 2.1]. We will refer to the process of rewriting a monomial in terms of standard monomials as straightening.

The quantum grassmannian

𝒪_{q} (G (1, n))

is a quantum affine space, and, as such, its derivations are known; see [1

J. Alev and M. Chamarie, Dérivations et automorphismes de quelques algèbres quantiques. Comm. Algebra 20 (1992), no. 6, 1787–1802 Zbl 0760.17003 MR 1162608

, Corollaire 1.3.3]; so we will assume throughout this paper that

k > 1 .

As

𝒪_{q} (G (n - 1, n)) ≅ 𝒪_{q} (G (1, n)),

by [4

S. Launois and T. H. Lenagan, The automorphism group of the quantum grassmannian. Bull. Lond. Math. Soc. 56 (2024), no. 4, 1310–1327 Zbl 1554.16033 MR 4736773

, Proposition 3.4], we exclude the case

k = n - 1

as well. Thus, we will assume throughout the paper that

2 \leq k \leq n - 2,

and so

n \geq 4 .

Also, it is shown in [4

S. Launois and T. H. Lenagan, The automorphism group of the quantum grassmannian. Bull. Lond. Math. Soc. 56 (2024), no. 4, 1310–1327 Zbl 1554.16033 MR 4736773

, Proposition 3.4] that

𝒪_{q} (G (k, n)) ≅ 𝒪_{q} (G (n - k, n)),

when

2 k \leq n .

In Section 7, we need to assume that

2 k \leq n,

and the general result is obtained from this case in Subsection 8.2 by using the isomorphism just mentioned.

3. Derivations for $𝒪_{q} (G (k, n))$ inherited from $𝒪_{q} (M (k, n))$

We use the following notation for the delta truth function: if

P

is a proposition, then

δ (P) = 1

if

P

is true, and

δ (P) = 0

if

P

is false. As is traditional, we write

δ_{i j}

for

δ (i = j) .

Recall from [3

S. Launois and T. H. Lenagan, The first Hochschild cohomology group of quantum matrices and the quantum special linear group. J. Noncommut. Geom. 1 (2007), no. 3, 281–309 Zbl 1137.16015 MR 2314098

] that for each column of

𝒪_{q} (M (k, n))

there is a derivation that is the identity on any quantum minor that contains that column and is zero on the other quantum minors. These derivations restrict to the subalgebra

𝒪_{q} (G (k, n))

to give us

n

derivations which we denote by

D_{i}

for

i = 1, \dots, n .

We refer to these derivations as the column derivations of

𝒪_{q} (G (k, n)) .

Lemma 3.1.

For each $i = 1, \dots, n,$ there is a derivation $D_{i}$ whose action on quantum Plücker coordinates is given by $D_{i} ([I]) = δ (i \in I) [I] .$

Proof.

This is immediate from the definition of

D_{i}

as the restriction to

𝒪_{q} (G (k, n))

of a column derivation of

𝒪_{q} (M (k, n)) .

Corollary 3.2.

We have

\frac{1}{k} (\sum_{i = 1}^{n} D_{i}) ([I]) = [I]

for each quantum Plücker coordinate $[I]$ in $𝒪_{q} (G (k, n)) .$

Proof.

This follows immediately from the previous lemma, as there are

k

occurrences of

D_{i}

for which

D_{i} ([I]) = [I],

and otherwise

D_{i} ([I]) = 0 .

Our main aim in this paper is to prove the following conjecture.

Conjecture 3.3.

Let $K$ be a field of characteristic zero, and let $q$ be a nonzero element of $K$ that is not a root of unity. Then every derivation of $𝒪_{q} (G (k, n))$ can be written as a linear combination of inner derivations and column derivations. Furthermore, the column derivations are linearly independent modulo the space generated by the inner derivations.

As a consequence of the truth of this conjecture, we identify the first Hochschild cohomology group of the quantum grassmannian.

4. The dehomogenisation equality for $𝒪_{q} (G (k, n))$

In this section, we recall results concerning the dehomogenisation equality that occur in [4

S. Launois and T. H. Lenagan, The automorphism group of the quantum grassmannian. Bull. Lond. Math. Soc. 56 (2024), no. 4, 1310–1327 Zbl 1554.16033 MR 4736773

, Section 3]. We refer the reader to that paper for detailed definitions and proofs.

Set

u = {1, \dots, k} .

Then

[u]

commutes with all other quantum Plücker coordinates up to a power of

q,

by [4

S. Launois and T. H. Lenagan, The automorphism group of the quantum grassmannian. Bull. Lond. Math. Soc. 56 (2024), no. 4, 1310–1327 Zbl 1554.16033 MR 4736773

, Lemma 3.1]. As

𝒪_{q} (G (k, n))

is generated by the quantum Plücker coordinates, it follows that the element

[u]

is a normal element, and so we may invert

[u]

to obtain the overring

T := 𝒪_{q} (G (k, n)) [{[u]}^{- 1}] .

For

1 \leq i \leq k

and

1 \leq j \leq n - k,

set

x_{i j} := [1, \dots, \hat{k + 1 - i}, \dots, k, j + k] {[u]}^{- 1} \in T .

Set

p := n - k .

The elements

x_{i j}

generate a subalgebra of

T

that is a quantum matrix algebra

𝒪_{q} (M (k, p)) .

In fact,

T

is generated over

𝒪_{q} (M (k, p))

by

{[u]}^{\pm 1},

so

T = 𝒪_{q} (G (k, n)) [{[u]}^{- 1}] = 𝒪_{q} (M (k, p)) [{[u]}^{\pm 1}; σ],

where

σ

is the automorphism of

𝒪_{q} (M (k, p))

defined by

σ (x_{i j}) := q x_{i j}

(since

[u] x_{i j} = q x_{i j} [u]

for all

i, j

).

When we operate on the right-hand side of this equality, we will write

y

for

[u] .

Thus,

y x_{i j} = σ (x_{i j}) y

and

(4.1)

T = 𝒪_{q} (G (k, n)) [{[u]}^{- 1}] = 𝒪_{q} (M (k, p)) [y, y^{- 1}; σ] .

We refer to this equality as the dehomogenisation equality. We will make extensive use of the dehomogenisation equality to transfer derivations from

𝒪_{q} (M (k, p))

to

𝒪_{q} (G (k, n))

and vice versa.

Note that [4

S. Launois and T. H. Lenagan, The automorphism group of the quantum grassmannian. Bull. Lond. Math. Soc. 56 (2024), no. 4, 1310–1327 Zbl 1554.16033 MR 4736773

, Lemma 3.2] gives the formula for general quantum minors of the algebra

𝒪_{q} (M (k, p))

when viewed as elements in

𝒪_{q} (G (k, n)) [{[u]}^{- 1}],

and each quantum Plücker coordinate, multiplied by

{[u]}^{- 1},

occurs in the formula. The formula is

[I | J] = [{1, \dots, k} \ (k + 1 - I) ⊔ (k + J)] {[u]}^{- 1} .

In the reverse direction, for a quantum Plücker coordinate

[L]

of

𝒪_{q} (G (k, n)),

we have

[L] = [L_{\leq k} ⊔ L_{> k}] = [I | J] [u],

where

I = {(k + 1) - ({1, \dots, k} \ L_{\leq k})}

and

J = L_{> k} - k

with

L_{\leq k} := L \cap {1, \dots, k}

and

L_{> k} := L \cap {k + 1, \dots, n} .

5. Derivations arising via dehomogenisation

The

n

column derivations

D_{i}

of

𝒪_{q} (G (k, n))

that we have defined in Section 3 satisfy

D_{i} ([u]) = [u]

for

1 \leq i \leq k

and

D_{i} ([u]) = 0

for

k < i \leq n,

so they extend to

n

derivations

\tilde{D_{i}}

of

T = R [y, y^{- 1}; σ] = 𝒪_{q} (G (k, n)) {[u]}^{- 1}

with

\tilde{D_{i}} (y) = y

for

1 \leq i \leq k

and

\tilde{D_{i}} (y) = 0

for

k < i \leq n .

We will show that these derivations

\tilde{D_{i}}

of

T

coincide with extensions of known derivations of

𝒪_{q} (M (k, p))

which we now recall from [3

S. Launois and T. H. Lenagan, The first Hochschild cohomology group of quantum matrices and the quantum special linear group. J. Noncommut. Geom. 1 (2007), no. 3, 281–309 Zbl 1137.16015 MR 2314098

].

For

1 \leq i \leq k,

there are derivations

D_{i, *}

of

𝒪_{q} (M (k, p))

defined by

D_{i, *} (x_{r s}) := δ_{i r} x_{r s},

and for

1 \leq j \leq p,

there are derivations

D_{* j}

of

𝒪_{q} (M (k, p))

defined by

D_{* j} (x_{r s}) := δ_{j s} x_{r s} .

In other words,

D_{i, *}

fixes row

i

and kills all the other rows of

𝒪_{q} (M (k, p)),

while

D_{* j}

fixes column

j

and kills all other columns of

𝒪_{q} (M (k, p)) .

For these observations, see the comment after [3

S. Launois and T. H. Lenagan, The first Hochschild cohomology group of quantum matrices and the quantum special linear group. J. Noncommut. Geom. 1 (2007), no. 3, 281–309 Zbl 1137.16015 MR 2314098

, Corollary 2.8].

Note that the Leibniz formula for derivations shows that

D_{i, *} ([I | J]) = δ (i \in I) [I | J]

and that

D_{* j} ([I | J]) = δ (j \in J) [I | J]

for any quantum minor

[I | J]

of

𝒪_{q} (M (k, p)) .

The derivations that we have just defined are not linearly independent: it is easy to check that

\sum_{i = 1}^{k} D_{i, *} = \sum_{j = 1}^{p} D_{* j} .

We will extend these derivations of

𝒪_{q} (M (k, p))

to derivations

\tilde{D_{i, *}}

and

\tilde{D_{* j}}

of

T = 𝒪_{q} (M (k, p)) [y^{\pm 1}; σ] = 𝒪_{q} (G (k, n)) {[u]}^{- 1}

as follows. For each

i = 1, \dots, k,

set

\tilde{D_{i, *}} (y) = y

and note that by applying the Leibniz formula for derivatives to

1 = y y^{- 1}

we obtain

\tilde{D_{i, *}} (y^{- 1}) = - y^{- 1} .

For each

j = 1, \dots, p,

set

\tilde{D_{* j}} (y) = \tilde{D_{* j}} (y^{- 1}) = 0 .

These choices for the action on

y

are necessary to obtain Corollary 5.2.

In the next proposition, we calculate the effect of the derivations

\tilde{D_{i, *}}

and

\tilde{D_{* j}}

on quantum Plücker coordinates in

𝒪_{q} (G (k, n)) .

Proposition 5.1.

Let $[I]$ be a quantum Plücker coordinate in $𝒪_{q} (G (k, n)),$ and let $1 \leq i \leq k$ and $1 \leq j \leq n - k .$ Then

(i)

$\tilde{D_{i, *}} ([I]) = - δ (k + 1 - i \in I) [I],$
(ii)

$\tilde{D_{* j}} ([I]) = δ (k + j \in I) [I] .$

Proof.

Write

I = I_{\leq k} ⊔ I_{> k} .

Then, with

A = (k + 1) - ({1, \dots, k} \ I_{\leq k})

and

B = I_{> k} - k,

we see that

[I] = [A | B] y .

In case (i),

\begin{matrix} \tilde{D_{i, *}} ([I]) & = \tilde{D_{i, *}} ([A | B] y) \\ = D_{i, *} ([A | B]) y + [A | B] D_{i, *} (y) \\ = δ (i \in A) [A | B] y - [A | B] y \\ = (δ (i \in A) - 1) [I] . \end{matrix}

Notice that

A = (k + 1) - ({1, \dots, k} \ I_{\leq k}) = (k + 1) - ({1, \dots, k} \ I) .

Hence,

i \in A

if and only if

k + 1 - i \notin I .

It follows that

δ (i \in A) + δ (k + 1 - i \in I) = 1 .

Hence,

δ (i \in A) - 1 = - δ (k + 1 - i \in I),

and the result follows.

In case (ii),

\begin{matrix} \tilde{D_{* j}} ([I]) & = \tilde{D_{* j}} ([A | B] y) \\ = \tilde{D_{* j}} ([A | B]) y + [A | B] \tilde{D_{* j}} (y) \\ = D_{* j} ([A | B]) y + 0 \\ = δ (j \in B) [I] . \end{matrix}

Note that

B = I_{> k} - k .

Hence,

j \in B = I_{> k} - k

if and only if

k + j \in I,

and the result follows.

Corollary 5.2.

We have

(i)

$\tilde{D_{i, *}} = - \tilde{D_{k + 1 - i}}$ for $1 \leq i \leq k,$
(ii)

$\tilde{D_{* j}} = \tilde{D_{k + j}}$ for $1 \leq j \leq n - k .$

Example 5.3.

For

𝒪_{q} (G (2, 4)),

we have

\tilde{D_{1, *}} = - \tilde{D_{2}},

\tilde{D_{2, *}} = - \tilde{D_{1}}

and

\tilde{D_{* 1}} = \tilde{D_{3}},

\tilde{D_{* 2}} = \tilde{D_{4}} .

Remark 5.4.

Note, for later use, that

\begin{matrix} {(\tilde{D_{1}} + \dots + \tilde{D_{k}} + \tilde{D_{k + 1}} + \dots + \tilde{D_{k + p}}) |}_{𝒪_{q} (M (k, p))} \\ = D_{k + 1} + \dots + D_{k + p} - (D_{k} + \dots + D_{1}) = 0, \end{matrix}

while

(\tilde{D_{1}} + \dots + \tilde{D_{k}} + \tilde{D_{k + 1}} + \dots + \tilde{D_{k + p}}) (y) = k y .

6. Adjusting derivatives

In cohomology calculations, we can adjust our original derivation by adding or subtracting derivations that arise as inner derivations. In this section, we consider what can happen when we do this or when adjusting by column derivations.

Let

[u] = [1, \dots, k]

and

[w] = [n - k + 1, \dots, n]

denote the leftmost and rightmost quantum Plücker coordinates of

𝒪_{q} (G (k, n)),

respectively. For any quantum Plücker coordinate

[I],

set

d (I) := # (I \ (I \cap u))

and

e (I) := # (I \ (I \cap w)) .

This notation is fixed for the rest of the paper. We will use the commutation relations for

[u]

and

[w]

with other quantum Plücker coordinates that are described in the following lemma without comment throughout the paper.

Lemma 6.1.

We have

(i)

$[u] [I] = q^{d (I)} [I] [u],$
(ii)

$[w] [I] = q^{- e (I)} [I] [w] .$

Proof.

(i) is proved in [4

S. Launois and T. H. Lenagan, The automorphism group of the quantum grassmannian. Bull. Lond. Math. Soc. 56 (2024), no. 4, 1310–1327 Zbl 1554.16033 MR 4736773

, Lemma 3.1] and (ii) is proved in [2

A. C. Kelly, T. H. Lenagan, and L. Rigal, Ring theoretic properties of quantum grassmannians. J. Algebra Appl. 3 (2004), 9–30 Zbl 1081.16048 MR 2047633

, Lemma 1.5].

If

S = [I_{1}] \dots [I_{m}]

is a standard monomial, then set

d (S) := \sum d (I_{i})

and note that each

d (I_{i}) \geq 0

with

d (I_{i}) = 0

if and only if

[I_{i}] = [u] .

Then,

S [u] = q^{d (S)} [u] S,

and so

S u = u S

if and only if

d (S) = 0

(in which case

S = {[u]}^{m}

).

In any case, note that

[u] S

is a standard monomial, as

[u]

is the unique minimal quantum Plücker coordinate.

Recall from Section 2 that

𝒪_{q} (G (k, n))

is a graded algebra with each quantum Plücker coordinate having degree one.

Lemma 6.2.

Let $D$ be a derivation of $𝒪_{q} (G (k, n)) .$ Suppose that $D ([I]) = b_{0} + \dots + b_{t}$ is the homogeneous decomposition of $D ([I]) .$ Then $b_{0} = 0 .$

Proof.

Suppose that

[I] \neq [u],

and suppose that

D ([u]) = a_{0} + \dots + a_{s}

is the homogeneous decomposition of

D ([u]) .

Let

d = d ([I])

and note that

d > 0

so that

q^{d} \neq 1 .

By applying

D

to the equation

[u] [I] = q^{d} [I] [u],

we obtain

D ([u]) [I] + [u] D ([I]) = q^{d} D ([I]) [u] + q^{d} [I] D ([u]) .

Examination of the terms in degree one in this equation reveals that

a_{0} [I] + b_{0} [u] = q^{d} b_{0} [u] + q^{d} a_{0} [I],

from which it follows that

a_{0} = q^{d} a_{0}

and

b_{0} = q^{d} b_{0} .

Then

a_{0} = b_{0} = 0,

as

q^{d} \neq 1 .

The following lemma is deduced from results in [5

S. Launois, T. H. Lenagan, and B. M. Nolan, Total positivity is a quantum phenomenon: the grassmannian case. Mem. Amer. Math. Soc. 291 (2023), no. 1448, 109 pp. Zbl 1548.16002 MR 4665212

]. In the proof of the lemma, we use the notation developed in that paper without further explanation.

Lemma 6.3.

Let $v = {1, \dots, k - 1, k + 1},$ and let $[α] > [v]$ in the standard order on quantum Plücker coordinates. Set $t := | α \ (v \cap α) |$ and note that $t > 0 .$ Then $[v] [α] = q^{t} [α] [v]$ modulo $⟨ u ⟩ .$

Proof.

There is an isomorphism

Ψ : \frac{𝒪_{q} (G (k, n))}{⟨ u ⟩} [{[v]}^{- 1}] ⟶ 𝒪_{q^{- 1}} (Y_{λ}) [Y^{\pm 1}; σ],

where

σ (x_{i j}) = q x_{i j}

for the generators

x_{i j}

of the partition subalgebra

Y_{λ};

see [5

S. Launois, T. H. Lenagan, and B. M. Nolan, Total positivity is a quantum phenomenon: the grassmannian case. Mem. Amer. Math. Soc. 291 (2023), no. 1448, 109 pp. Zbl 1548.16002 MR 4665212

, Theorem 9.17]. If

{[I | J]}_{q^{- 1}}

is a

t \times t

pseudo-quantum minor in

Y_{λ},

then we have the following commutation relation:

Y {[I | J]}_{q^{- 1}} = q^{t} {[I | J]}_{q^{- 1}} Y .

Let

α \in Π \ {u, v},

and set

t := | α \ v | .

Then

Ψ (\bar{α} {\bar{v}}^{- 1}) = β {[I | J]}_{q^{- 1}}

for some

β \in K,

and

t \times t

pseudo-quantum minor

{[I | J]}_{q^{- 1}},

by [5

S. Launois, T. H. Lenagan, and B. M. Nolan, Total positivity is a quantum phenomenon: the grassmannian case. Mem. Amer. Math. Soc. 291 (2023), no. 1448, 109 pp. Zbl 1548.16002 MR 4665212

, Theorem 9.17].

Hence,

\begin{matrix} Ψ (\bar{v} \bar{α}) & = Ψ (\bar{v}) Ψ (\bar{α} {\bar{v}}^{- 1}) Ψ (\bar{v}) \\ = Y β {[I | J]}_{q^{- 1}} Y \\ = q^{t} β {[I | J]}_{q^{- 1}} Y^{2} \\ = q^{t} Ψ (\bar{α} {\bar{v}}^{- 1}) Ψ ({\bar{v}}^{2}) \\ = q^{t} Ψ (\bar{α} \bar{v}), \end{matrix}

and so

\bar{v} \bar{α} = q^{t} \bar{α} \bar{v},

as required.

Definition 6.4.

Suppose that

a = \sum_{i = 1}^{s} a_{i} S_{i}

is an expression for an element

a \in 𝒪_{q} (G (k, n))

in terms of standard monomials

S_{i}

and that each

a_{i}

is nonzero. Then the support of

a,

written

Supp (a),

is defined to be

{S_{i} | i = 1, \dots, s} .

Lemma 6.5.

Let $D$ be a derivation on $𝒪_{q} (G (k, n)) .$ Then

(i)

there are no standard monomials of the form ${[I_{1}]}^{a_{1}} {[I_{2}]}^{a_{2}} \dots {[I_{m}]}^{a_{m}}$ , with $I_{1} \neq u$ and $\sum a_{i} > 0$ , occurring in $Supp (D [u]),$
(ii)

there are no standard monomials of the form ${[I_{1}]}^{a_{1}} {[I_{2}]}^{a_{2}} \dots {[I_{m}]}^{a_{m}}$ , with $I_{m} \neq w$ and $\sum a_{i} > 0$ , occurring in $Supp (D [w]) .$

Proof.

(i) The proof is by contradiction, so suppose that such a standard monomial exists. Without loss of generality, we may assume that

I_{1} = v = {1, \dots, k - 1, k + 1}

as we are allowing the possibility that some

a_{i} = 0 .

As

u

and

v

differ only in one value, we know that

[u] [v] = q [v] [u] .

Apply

D

to the equation

[u] [v] = q [v] [u]

to obtain

D ([u]) [v] + [u] D ([v]) = q D ([v]) [u] + q [v] D ([u]) .

Suppose that

D ([u]) = \sum_{i} α_{i} S_{i} + \sum_{i} α_{i}^{'} S_{i}^{'}

where

α_{i} \in K

and the

S_{i}, S_{i}^{'}

are standard monomials such that

[u]

does not occur in the

S_{i},

but does occur in the

S_{i}^{'} .

Also, suppose that

D ([v]) = \sum_{j} β_{j} T_{j}

where

β_{j} \in K

and the

T_{i}

are standard monomials. Then

\begin{matrix} \sum_{i} α_{i} S_{i} [v] + \sum_{i} α_{i}^{'} S_{i}^{'} [v] + \sum_{j} β_{j} [u] T_{j} \\ = \sum_{j} q β_{j} T_{j} [u] + \sum_{i} q α_{i} [v] S_{i} + \sum_{i} q α_{i}^{'} [v] S_{i}^{'} . \end{matrix}

Now,

S_{i} [v] = q^{- t_{i}} [v] S_{i}

modulo

⟨ u ⟩

for some

t_{i} \geq 0,

by Lemma 6.3 (we allow zero as it might be that

S_{i}

is a power of

[v]

).

The image of the above equation in

𝒪_{q} (G (k, n)) / ⟨ [u] ⟩

gives

\sum_{i} q^{- t_{i}} α_{i} \bar{[v]} \bar{S_{i}} = \sum_{i} q α_{i} \bar{[v]} \bar{S_{i}}

or

\sum_{i} (q^{- t_{i}} - q) α_{i} \bar{[v]} \bar{S_{i}} = 0 .

Now,

𝒪_{q} (G (k, n)) / ⟨ [u] ⟩

is the quantum Schubert variety determined by the quantum Plücker coordinate

[v]

(see [6

T. H. Lenagan and L. Rigal, Quantum graded algebras with a straightening law and the AS–Cohen–Macaulay property for quantum determinantal rings and quantum grassmannians. J. Algebra 301 (2006), 670–702 Zbl 1108.16026 MR 2236763

] for the definition of quantum Schubert varieties of the quantum grassmannian). As such,

𝒪_{q} (G (k, n)) / ⟨ [u] ⟩

has a basis consisting of the images of the standard monomials in

𝒪_{q} (G (k, n))

that do not involve

[u];

see [6

T. H. Lenagan and L. Rigal, Quantum graded algebras with a straightening law and the AS–Cohen–Macaulay property for quantum determinantal rings and quantum grassmannians. J. Algebra 301 (2006), 670–702 Zbl 1108.16026 MR 2236763

, Example 2.1.3]. Consequently, each

(q^{- t_{i}} - q) α_{i}

in the equation above is equal to zero. As at least one

α_{i}

is nonzero, this gives

q^{- t_{i}} - q = 0,

for that

i,

which is a contradiction as

q

is not a root of unity and

t_{i} \geq 0 .

(ii) Follows in a similar fashion.

The next corollary follows immediately from the previous lemma.

Corollary 6.6.

Let $D$ be a derivation of $𝒪_{q} (G (k, n)) .$ Suppose that the homogeneous decomposition of $D ([u])$ is given by $D ([u]) = a_{1} + \dots + a_{s}$ and that $D ([w]) = b_{1} + \dots + b_{t}$ is the homogeneous decomposition of $D ([w]) .$ Then $a_{1}$ is a scalar multiple of $[u]$ and $b_{1}$ is a scalar multiple of $[w] .$

In order to prove Conjecture 3.3, we will start by showing that we can adjust an arbitrary derivation

D

by adding or subtracting derivations coming from the derivations mentioned in the conjecture (column derivations and inner derivations), so that the adjusted derivation, which we will continue to denote by

D,

satisfies

D ([u]) = D ([w]) = 0 .

This will enable us to transfer the study of

D

into a quantum matrix problem by using the dehomogenisation equality. We show that we can make this adjustment in a number of steps that demonstrate how to remove standard monomials that occur in

D ([u])

(and

D ([w])

).

As a result of Lemma 6.5, any standard monomial that occurs in the support of

D ([u])

must start with

[u] .

Similarly, any standard monomial that occurs in the support of

D ([w])

must finish with

[w] .

The next lemma shows that, by adjusting

D

by suitable inner derivations, we can remove terms of

D ([u])

that are not of the form

α {[u]}^{a}

for values of

a \geq 1 .

Lemma 6.7.

Suppose that $S = {[u]}^{a} {[I_{1}]}^{b_{1}} {[I_{2}]}^{b_{2}} \dots {[I_{m}]}^{b_{m}}$ is a standard monomial with $[v] \leq [I_{1}]$ and $\sum_{i} b_{i} > 0,$ while $a \geq 1 .$ Let $D$ be a derivation of $𝒪_{q} (G (k, n)),$ and suppose that $S$ occurs in the support of $D ([u])$ with nonzero scalar coefficient $α .$ Set $d := d (S) = b_{1} d (I_{1}) + \dots + b_{m} d (I_{m}),$ and set $z := \frac{α}{q^{- d} - 1} {[u]}^{a - 1} {[I_{1}]}^{b_{1}} {[I_{2}]}^{b_{2}} \dots {[I_{m}]}^{b_{m}} .$ Also, set $D^{'} := D - {ad}_{z} .$ Then

Supp (D^{'} ([u])) = Supp (D ([u])) \ {S} .

Proof.

Note that

d > 0,

so

q^{- d} - 1 \neq 0 .

We calculate

\begin{matrix} {ad}_{z} ([u]) & = z u - u z \\ = \frac{α}{q^{- d} - 1} ({[u]}^{a - 1} {[I_{1}]}^{b_{1}} {[I_{2}]}^{b_{2}} \dots {[I_{m}]}^{b_{m}} [u] - {[u]}^{a} {[I_{1}]}^{b_{1}} {[I_{2}]}^{b_{2}} \dots {[I_{m}]}^{b_{m}}) \\ = \frac{α}{q^{- d} - 1} (q^{- d} {[u]}^{a} {[I_{1}]}^{b_{1}} {[I_{2}]}^{b_{2}} \dots {[I_{m}]}^{b_{m}} - {[u]}^{a} {[I_{1}]}^{b_{1}} {[I_{2}]}^{b_{2}} \dots {[I_{m}]}^{b_{m}}) \\ = \frac{α}{q^{- d} - 1} ((q^{- d} - 1) {[u]}^{a} {[I_{1}]}^{b_{1}} {[I_{2}]}^{b_{2}} \dots {[I_{m}]}^{b_{m}}) \\ = α S . \end{matrix}

It follows that

Supp (D^{'} ([u])) = Supp (D ([u])) \ {S},

as required.

Corollary 6.8.

Let $D$ be a derivation of $𝒪_{q} (G (k, n)) .$ Then there is a derivation $D^{'}$ such that $D^{'} - D$ is a sum of inner derivations and such that the homogeneous terms of $D^{'} ([u])$ are of the form $λ_{a} {[u]}^{a}$ for $a \geq 1$ and $λ_{a} \in K .$

Proof.

Lemma 6.5 shows that

Supp (D ([u]))

has no terms whose standard monomials do not begin with

[u] .

By using Lemma 6.7 an appropriate number of times, we can remove terms in

Supp (D ([u]))

that involve

[u]

and at least one other quantum Plücker coordinate by adjusting with suitable inner derivations. What remains is a derivation

D^{'}

whose support only involves terms of the form

{[u]}^{a} .

Our next task is to show that we can adjust further, if necessary, to see that we can reduce to

a = 1

being the only possibility.

Lemma 6.9.

Suppose that $D$ is a derivation of $𝒪_{q} (G (k, n))$ such that ${[u]}^{a - 1} [w]$ occurs in $Supp (D ([w]))$ with $a > 1 .$ Then there is a derivation $D^{'}$ of $𝒪_{q} (G (k, n))$ such that $D^{'} - D$ is inner and $D^{'} ([u]) = D ([u])$ while $Supp (D^{'} ([w])) = Supp (D ([w])) \ {[u]}^{a - 1} [w] .$

Proof.

Suppose that

{[u]}^{a - 1} [w]

occurs in

Supp (D ([w])),

say with nonzero coefficient

β .

Now,

{ad}_{{[u]}^{a - 1}} ([w]) = {[u]}^{a - 1} [w] - [w] {[u]}^{a - 1} = (1 - q^{- d (w) (a - 1)}) {[u]}^{a - 1} [w],

and note that

1 - q^{- d (w) (a - 1)} \neq 0,

as both

d (w)

and

a - 1

are nonzero and

q

is not a root of unity. Set

D^{'} := D - β {(1 - q^{- d (w) (a - 1)})}^{- 1} {ad}_{{[u]}^{a - 1}} .

Then

D^{'} ([u]) = D ([u]),

as

{ad}_{{[u]}^{a - 1}} ([u]) = 0 .

Also,

D^{'} ([w]) = D ([w]) - β {(1 - q^{- d (w) (a - 1)})}^{- 1} {ad}_{{[u]}^{a - 1}} ([w]) = D ([w]) - β {[u]}^{a - 1} [w],

so that

Supp (D^{'} ([w])) = Supp (D ([w])) \ {[u]}^{a - 1} [w],

as required.

The following corollary now follows by applying the previous lemma an appropriate number of times.

Corollary 6.10.

Let $D$ be a derivation of $𝒪_{q} (G (k, n)) .$ Then there is a derivation $D^{'}$ of $𝒪_{q} (G (k, n))$ such that $D^{'} - D$ is a sum of inner derivations, $D^{'} ([u]) = D ([u])$ and such that there are no terms of the form ${[u]}^{a - 1} [w]$ with $a > 1$ occurring in $Supp (D ([w])) .$

Lemma 6.11.

Let $D$ be a derivation of $𝒪_{q} (G (k, n))$ such that $D ([u]) = \sum λ_{i} {[u]}^{i}$ for some $λ_{i} \in K,$ and suppose that no terms of the form ${[u]}^{a - 1} [w]$ with $a > 1$ occur in $Supp (D ([w])) .$ Then $D ([u]) = λ_{1} [u] .$

Proof.

Suppose that

D ([w]) = \sum β_{i} S_{i}

for some standard monomials

S_{i}

and

0 \neq β_{i} \in K,

and note that there is no

S_{i}

such that

S_{i} = {[u]}^{a - 1} [w]

for any

a > 1,

by assumption. Apply

D

to the equation

[u] [w] = q^{d (w)} [w] [u]

to obtain

D ([u]) [w] + [u] D ([w]) = q^{d (w)} D ([w]) [u] + q^{d (w)} [w] D ([u]) .

Hence,

\sum λ_{i} {[u]}^{i} [w] + \sum β_{i} [u] S_{i} = \sum q^{d (w)} β_{i} S_{i} [u] + \sum q^{d (w)} λ_{i} [w] {[u]}^{i}

and so

\sum λ_{i} {[u]}^{i} [w] + \sum β_{i} [u] S_{i} = \sum q^{d (w) - d (S_{i})} β_{i} [u] S_{i} + \sum q^{d (w) - i \cdot d (w)} λ_{i} {[u]}^{i} [w] .

The terms in this equation are all scalar multiples of standard monomials. Consider the occurrences of the standard monomial

{[u]}^{a} [w]

for a given

a > 1 .

If

[u] S_{i} = {[u]}^{a} [w],

then

S_{i} = {[u]}^{a - 1} [w],

which does not occur, by assumption. Hence, the second term on the left side of this equation and the first term on the right side do not contain

{[u]}^{a} [w] .

It follows that

λ_{a} {[u]}^{a} [w] = q^{(1 - a) d (w)} λ_{a} {[u]}^{a} [w],

and this forces

λ_{a} (1 - q^{(1 - a) d (w)}) = 0 .

Now,

(1 - a) d (w) \neq 0,

as

a \neq 1

so

(1 - q^{(1 - a) d (w)}) \neq 0 .

Hence,

λ_{a} = 0 .

As this is true for all

a > 1,

we obtain the required result.

Recall from Lemma 3.1 that the column derivation

D_{1}

is acting on a quantum Plücker coordinate

[I]

by

D_{1} ([I]) = δ (1 \in I) [I] .

Corollary 6.12.

Let $D$ be a derivation of $𝒪_{q} (G (k, n))$ such that $D ([u]) = λ [u]$ for some $λ \in K,$ and suppose that no terms of the form ${[u]}^{a - 1} [w]$ with $a > 1$ occur in $Supp (D ([w])) .$ Then there is a derivation $D^{'}$ of $D$ such that the following hold:

(i)

$D^{'} ([u]) = 0 .$
(ii)

$D^{'} - D = λ D_{1} .$
(iii)

There are no terms of the form ${[u]}^{a - 1} [w]$ with $a > 1$ that occur in $Supp (D^{'} ([w])) .$

Proof.

We start by observing that

D_{1} ([u]) = D_{1} ([1, \dots, k]) = [u]

and

D_{1} ([w]) = D_{1} ([n - k + 1, \dots, n]) = 0,

as

1 < n - k + 1 .

Set

D^{'} = D - λ D_{1},

so that

D^{'} ([u]) = D ([u]) - λ D_{1} ([u]) = λ [u] - λ [u] = 0 .

Also,

D^{'} ([w]) = D ([w]) - λ D_{1} ([w]) = D ([w]),

so no terms of the form

{[u]}^{a - 1} [w]

with

a > 1

occur in

Supp (D ([w])) .

Lemma 6.13.

Let $D$ be a derivation of $𝒪_{q} (G (k, n))$ with the following properties:

(i)

$D ([u]) = 0 .$
(ii)

$Supp (D ([w]))$ contains no term of the form ${[u]}^{a - 1} [w]$ for $a > 1 .$

Then $D ([w]) = α [w]$ for some $α \in K .$

Proof.

Recall, from Lemma 6.5, that

[w]

must occur in any standard monomial contained in

Supp (D ([w])) .

Suppose that

S = {[u]}^{a} {[I_{1}]}^{a_{1}} \dots {[I_{m}]}^{a_{m}} {[w]}^{b} \in Supp (D ([w]))

with

u < I_{1}

and

I_{m} < w,

while

a, a_{i} \geq 0, b > 0 .

Suppose that

S

occurs in

D ([w])

with nonzero coefficient

α \in K .

Note that

d (I_{i}) > 0

for each

i .

Apply

D

to the equation

[u] [w] = q^{d (w)} [w] [u],

remembering that

D ([u]) = 0,

to obtain

[u] D ([w]) = q^{d (w)} D ([w]) [u] .

Examination of the standard monomials of the form

[u] S

in this equation reveals that

\begin{matrix} α {[u]}^{a + 1} {[I_{1}]}^{a_{1}} \dots {[I_{m}]}^{a_{m}} {[w]}^{b} \\ = α q^{d (w)} {[u]}^{a} {[I_{1}]}^{a_{1}} \dots {[I_{m}]}^{a_{m}} {[w]}^{b} [u] \\ = α q^{d (w) - (d (w) b + a_{1} d (I_{1}) + \dots + a_{m} d (I_{m}))} {[u]}^{a + 1} {[I_{1}]}^{a_{1}} \dots {[I_{m}]}^{a_{m}} {[w]}^{b} . \end{matrix}

As

q

is not a root of unity, the only possibility is that the power of

q

on the right-hand side is

q^{0},

and this is only possible for

b = 1

and

a_{1} = \dots = a_{m} = 0 .

Thus, the only possible terms in

Supp (D [w])

are of the form

{[u]}^{a} [w] .

Taking into account condition (ii) in the statement of the lemma, we see that

a > 0

is not allowed, so

D [w] = α [w],

as required.

Recall from Lemma 3.1 that there are column derivations

D_{i}

for

i = 1, \dots, n

such that

D_{i} ([I]) = δ (i \in I) [I]

for each quantum Plücker coordinate

[I] .

The results of this section are summarised in the following proposition.

Proposition 6.14.

Let $D$ be a derivation of $𝒪_{q} (G (k, n)) .$ Then there is a derivation $D^{'}$ of $𝒪_{q} (G (k, n))$ with $D^{'} ([u]) = D^{'} ([w]) = 0$ and such that $(D^{'} - D)$ is a linear combination of derivations of the form ${ad}_{z},$ with $z \in 𝒪_{q} (G (k, n)),$ and column derivations $D_{i}$ for $i = 1, \dots, n .$

Proof.

We know the following facts hold for any derivation

D

of

𝒪_{q} (G (k, n))

and so will hold for the any derivation that occurs when we adjust a given derivation by adding or subtracting inner derivations and scalar multiples of the

D_{i}

: (i) the degree zero parts of

D ([u])

and

D ([w])

are both zero (see Lemma 6.2); (ii) the degree one part of

D ([u])

is a scalar multiple of

[u]

and, similarly, the degree one part of

D ([w])

is a scalar multiple of

[w]

(see Corollary 6.6); (iii) any standard monomial occurring in the support of

D ([u])

must start with at least one occurrence of

[u]

and, similarly, any standard monomial occurring in the support of

D ([w])

must end with at least one occurrence of

[w]

(see Lemma 6.5).

Let

D

be an arbitrary derivation of

𝒪_{q} (G (k, n)) .

By Corollary 6.8, there is a derivation

D^{(1)}

of

𝒪_{q} (G (k, n))

such that

D^{(1)} - D

is a sum of inner derivations and such that the homogeneous terms of

D^{(1)} ([u])

are of the form

λ_{a} {[u]}^{a}

for

a \geq 1

and

α_{a} \in K .

By Corollary 6.10, there is a derivation

D^{(2)}

of

𝒪_{q} (G (k, n))

such that

D^{(2)} - D^{(1)}

is a sum of inner derivations; the homogeneous terms of

D^{(2)} ([u]) = D^{(1)} ([u])

are of the form

λ_{a} {[u]}^{a}

for

a \geq 1

and

α_{a} \in K,

and there are no terms of the form

{[u]}^{a - 1} [w]

with

a > 1

occurring in

Supp (D^{(2)} ([w])) .

By Lemma 6.11,

D^{(2)} ([u]) = λ [u]

for some

λ \in K .

Set

D^{(3)} := D^{(2)} - λ D_{1} .

Then

D^{(3)} ([u]) = 0,

while

D^{(3)} ([w]) = D^{(2)} ([w]) - λ D_{1} ([w]) = D^{(2)} ([w]),

as

D_{1} ([w]) = 0 .

Hence, there are no terms of the form

{[u]}^{a - 1} [w]

with

a > 1

in

D^{(3)} ([w]) .

It follows from Lemma 6.13 that

D^{(3)} ([w]) = α [w]

for some

α \in K .

Finally, set

D^{'} := D^{(3)} - α D_{n} .

Then

D^{'} ([u]) = D^{(3)} [u] = 0,

as

D_{n} ([u]) = 0

and

D^{'} ([w]) = D^{(3)} ([w]) - α D_{n} ([w]) = 0 .

The passage from

D

to

D^{'}

via

D^{(1)}, D^{(2)}, D^{(3)}

only involves adjustments by adding or subtracting derivations of the form

{ad}_{z},

with

z \in 𝒪_{q} (G (k, n)),

and

D_{i}

for

i = 1, \dots, n

at each stage, so the required result follows.

7. Transferring derivations of $𝒪_{q} (G (k, n))$ to $𝒪_{q} (M (k, p))$

Throughout this section, we assume that

2 k \leq n .

Recall the dehomogenisation equality from Section 4

T = 𝒪_{q} (G (k, n)) [{[u]}^{- 1}] = 𝒪_{q} (M (k, p)) [y, y^{- 1}; σ] .

Given a derivation

D

of

𝒪_{q} (G (k, n))

with

D ([u]) = D ([w]) = 0,

we may extend

D

to

T

by setting

D ({[u]}^{- 1}) = 0

and then transfer to

𝒪_{q} (M (k, p)) [y, y^{- 1}; σ]

via the dehomogenisation equality. We then know that

D (y) = D ([u]) = 0 .

We retain the notation

D

for this extension to

T .

Recall that in Section 4 we set

R := 𝒪_{q} (M (k, p))

where

p = n - k .

The quantum matrix generators

x_{i j}

of

R

are defined in Section 4

x_{i j} := [1, \dots, \hat{k + 1 - i}, \dots, k, j + k] {[u]}^{- 1} \in T .

Our aim in this section is to show that

D (R) \subseteq R

for such a derivation

D .

We will use a pair of gradings of

T

that were developed in [4

S. Launois and T. H. Lenagan, The automorphism group of the quantum grassmannian. Bull. Lond. Math. Soc. 56 (2024), no. 4, 1310–1327 Zbl 1554.16033 MR 4736773

, Section 6] to discuss a similar result for certain automorphisms of

𝒪_{q} (G (k, n)) .

As

2 k \leq n,

we know that

k \leq n - k = p,

and so

R

has at least

k

columns and the quantum minor

[I | J] := [1 \dots k | p + 1 - k, \dots, p] \in R

is defined (we are using all the rows of

R = 𝒪_{q} (M (k, p))

and the last

k

columns). As noted in [4

S. Launois and T. H. Lenagan, The automorphism group of the quantum grassmannian. Bull. Lond. Math. Soc. 56 (2024), no. 4, 1310–1327 Zbl 1554.16033 MR 4736773

, Lemma 6.1],

x_{i j} [I | J] = q [I | J] x_{i j}

when

j < p + 1 - k

while

x_{i j} [I | J] = [I | J] x_{i j}

when

j \geq p + 1 - k .

As a consequence,

[I | J]

is a normal element in

R

and also in

T .

Lemma 7.1.

Let $D$ be a derivation of $𝒪_{q} (G (k, n)),$ where $2 k \leq n,$ with $D ([u]) = D ([w]) = 0 .$ Let $[I | J]$ be defined as in the previous paragraph. Then $D ([I | J]) = 0 .$

Proof.

The discussion at the end of Section 4 shows that

[I | J] = [p + 1 \dots n] {[1 \dots k]}^{- 1} = [w] {[u]}^{- 1},

and so

D ([I | J]) = D ([w] {[u]}^{- 1}) = 0 .

Also, we can calculate how

[I | J]

commutes with

[u] = [1 \dots k] .

Note that

k < n - k + 1 = p + 1,

as

2 k \leq n .

Thus, the index sets

{1, \dots, k}

and

{p + 1, \dots, n}

do not overlap, and

[u] [I | J] = [u] [w] {[u]}^{- 1} = q^{k} [w] [u] {[u]}^{- 1} = q^{k} [w] {[u]}^{- 1} [u] = q^{k} [I | J] [u],

where the second equality comes from Lemma 6.1.

The two gradings that were used in [4

S. Launois and T. H. Lenagan, The automorphism group of the quantum grassmannian. Bull. Lond. Math. Soc. 56 (2024), no. 4, 1310–1327 Zbl 1554.16033 MR 4736773

, Section 6] are defined by considering how elements of

T

commute with

y = [u]

and with

[I | J] .

We set

T_{i} := {a \in T | y a y^{- 1} = q^{i} a}

and

T^{(i)} := {a \in T | [I | J] a {[I | J]}^{- 1} = q^{- i} a} .

Lemma 7.2.

We have

(i)

$T = ⨁_{i = 1}^{\infty} T_{i},$
(ii)

$T = ⨁_{i = ℤ} T^{(i)},$
(iii)

$(T^{(0)} \cup T^{(1)}) \cap T_{1} \subseteq 𝒪_{q} (M (k, p)) = R .$

Proof.

These results are established in [4

S. Launois and T. H. Lenagan, The automorphism group of the quantum grassmannian. Bull. Lond. Math. Soc. 56 (2024), no. 4, 1310–1327 Zbl 1554.16033 MR 4736773

, Lemmas 6.2 (i), 6.3 (i) and 6.4].

Theorem 7.3.

Suppose that $2 k \leq n$ and that $D$ is a derivation of $𝒪_{q} (G (k, n))$ such that $D ([u]) = D ([w]) = 0 .$ Then $D (R) \subseteq R .$

Proof.

It is enough to show that

D (x_{i j}) \in R

for each generator

x_{i j}

of

R .

Note that

x_{i j} \in (T^{(0)} \cup T^{(1)}) \cap T_{1} .

This claim follows from the commutation rules given in [4

S. Launois and T. H. Lenagan, The automorphism group of the quantum grassmannian. Bull. Lond. Math. Soc. 56 (2024), no. 4, 1310–1327 Zbl 1554.16033 MR 4736773

, Lemma 6.1] and the fact that

y x_{i j} = q x_{i j} y .

Let

a \in T_{1} .

Then

y a = q a y .

Apply

D

to this equation, noting that

D (y) = 0,

to obtain

y D (a) = q D (a) y,

so that

D (T_{1}) \subseteq T_{1} .

Similar calculations, using the fact that

D ([I | J]) = 0

show that

D (T^{(0)}) \subseteq T^{(0)}

and

D (T^{(1)}) \subseteq T^{(1)} .

It follows that

D (x_{i j}) \in D ((T^{(0)} \cup T^{(1)}) \cap T_{1}) \subseteq (T^{(0)} \cup T^{(1)}) \cap T_{1} \subseteq R .

As

D

takes each generator

x_{i j}

of

R

into

R,

we see that

D (R) \subseteq R,

as required.

8. The main theorem

Recall that we are assuming that

K

is a field of characteristic zero and that

q \in K

is a nonzero element that is not a root of unity. We are also assuming that

k \neq 1,

as in this case the quantum grassmannian is a quantum affine space, where the results are known; see [1

J. Alev and M. Chamarie, Dérivations et automorphismes de quelques algèbres quantiques. Comm. Algebra 20 (1992), no. 6, 1787–1802 Zbl 0760.17003 MR 1162608

]. As

𝒪_{q} (G (n - 1, n)) ≅ 𝒪_{q} (G (1, n)),

we exclude this case as well. Thus, we are assuming that

2 \leq k \leq n - 2 .

In this section, we prove Conjecture 3.3. The proof proceeds by first analysing the case where

2 k \leq n .

The general case is then obtained by using the isomorphism

𝒪_{q} (G (k, n)) ≅ 𝒪_{q} (G (n - k, n)) .

In order to avoid breaking the flow of the main result, we relegate to the appendix a discussion concerning homogeneous derivations on non-square quantum matrices that we use in obtaining the truth of the conjecture in the case where

2 k < n .

In this section, in order to make reading easier, we will use

δ

to denote an arbitrary derivative.

8.1. The case where $2 k \leq n$

In this subsection, we consider

𝒪_{q} (G (k, n))

in the case that

2 k \leq n .

In this case, the dehomogenisation equality is

𝒪_{q} (G (k, n)) [{[u]}^{- 1}] = T = 𝒪_{q} (M (k, p)) [y, y^{- 1}; σ],

where

p = n - k .

We set

R := 𝒪_{q} (M (k, p)) .

When

2 k = n,

so that

R = 𝒪_{q} (M (k, k)),

it is well known that the centre of

R

is

K [D_{q}],

where

D_{q}

is the quantum determinant of

𝒪_{q} (M (k, k)) .

It is also well known that when

2 k < n,

so that

R

is non-square, the centre of

R

is

K .

We also recall that the centre of

𝒪_{q} (G (k, n))

is always reduced to scalars. This follows easily from the basis of standard monomials by first observing that an element is central in

𝒪_{q} (G (k, n))

if and only if all the standard monomials in its support are central, and next by noting that there are no non-trivial central standard monomials since the only standard monomial commuting with both

[u]

and

[w]

are scalars by Lemma 6.1.

Proposition 8.1.

Assume that $2 k \leq n .$ Then any derivation $δ$ of $𝒪_{q} (G (k, n))$ is equal, modulo inner derivations, to a linear combination of $D_{1}, \dots, D_{n} .$ Furthermore, these $n$ derivations are linearly independent modulo the inner derivations.

Proof.

We use the same notation

δ

for the extension to

T .

After possibly adjusting

δ

by inner derivations and linear combinations of

D_{1}, \dots, D_{n},

we may assume that

δ (y) = δ ([u]) = δ ([w]) = 0,

by Proposition 6.14, and then

δ (R) \subseteq R,

by Theorem 7.3. Apply

δ

to the equation

y x_{i j} = q x_{i j} y

to obtain

y δ (x_{i j}) = q δ (x_{i j}) y .

Given this equation and the fact that

δ (R) \subseteq R,

we conclude that

δ (x_{i j})

is homogeneous of degree one.

In order to prove the first claim, we consider the two cases (i)

2 k = n

and (ii)

2 k < n

separately in order to show the subclaim that

{δ |}_{R}

can be written as a linear combination of the row and column derivations

D_{i, *}, D_{* j}

of the quantum matrix algebra

R

introduced in Section 5.

Subclaim: Case (i)

First, suppose that

2 k = n,

so that

k = n - k .

In this case,

R = 𝒪_{q} (M (k, k)),

and so

R

is a square quantum matrix algebra and the centre of

R

is

K [D_{q}],

where

D_{q}

is the quantum determinant of

𝒪_{q} (M (k, k)) .

By [3

S. Launois and T. H. Lenagan, The first Hochschild cohomology group of quantum matrices and the quantum special linear group. J. Noncommut. Geom. 1 (2007), no. 3, 281–309 Zbl 1137.16015 MR 2314098

, Theorem 2.9], there are polynomials

P_{1}, \dots, P_{k}, Q_{1}, \dots, Q_{k} \in K [D_{q}]

and an element

z \in R

such that

{δ |}_{R} = {ad}_{z} + \sum_{i = 1}^{k} P_{i} D_{i, *} + \sum_{j = 1}^{n - k} Q_{j} D_{* j},

where

D_{i, *}, D_{* j}

are the row and column derivations of

R

introduced in Section 5.

Let

a_{i}

be the constant term in

P_{i}

and

b_{j}

be the constant term in

Q_{j} .

Then

\begin{matrix} δ (x_{r s}) - (\sum_{i = 1}^{k} a_{i} D_{i, *} (x_{r s}) + \sum_{j = 1}^{n - k} b_{j} D_{* j} (x_{r s})) \\ = z x_{r s} - x_{r s} z + \sum_{i = 1}^{k} (P_{i} - a_{i}) D_{i, *} (x_{r s}) + \sum_{j = 1}^{n - k} (Q_{j} - b_{j}) D_{* j} (x_{r s}) \end{matrix}

for all

r, s .

The terms on the left-hand side of this equation all have degree one, whereas the terms on the right-hand side have degree greater than one, because

z x_{r s} - x_{r s} z

has no degree zero or degree one terms.

It follows that both sides are zero, and so

{δ |}_{R} = \sum_{i = 1}^{k} a_{i} D_{i, *} + \sum_{j = 1}^{n - k} b_{j} D_{* j},

which establishes the subclaim in the case that

2 k = n .

Subclaim: Case (ii)

Next, suppose that

2 k < n .

In this case,

R = 𝒪_{q} (M (k, p)),

where

p = n - k > k

and so

R

is a non-square quantum matrix algebra with more columns than rows, and the centre of

R

is

K .

The proof of this case is substantially more complicated than that of case (i) due to the fact that [3

S. Launois and T. H. Lenagan, The first Hochschild cohomology group of quantum matrices and the quantum special linear group. J. Noncommut. Geom. 1 (2007), no. 3, 281–309 Zbl 1137.16015 MR 2314098

, Theorem 2.9] only covers derivations for square quantum matrices. To avoid disturbing the flow of the proof of this proposition, the proof of this subclaim is treated in the appendix and finally established in Proposition A.6.

Having established the subclaim, we revert to the condition that

2 k \leq n .

By using Corollary 5.2, we see that

{δ |}_{R} = \sum_{i = 1}^{k} a_{i} D_{i, *} + \sum_{j = 1}^{n - k} b_{j} D_{* j} = (- \sum_{i = 1}^{k} a_{i} \tilde{D_{i}} + \sum_{j = 1}^{n - k} b_{j} \tilde{D_{k + j}}) |_{R} .

We define

\tilde{δ} = - \sum_{i = 1}^{k} a_{i} \tilde{D_{i}} + \sum_{j = 1}^{n - k} b_{j} \tilde{D_{k + j}},

so that

{δ |}_{R} = {\tilde{δ} |}_{R} .

Note that

\tilde{δ} (y) = - (\sum_{i = 1}^{k} a_{i}) y,

as

\tilde{D_{i}} (y) = y

for

i = 1, \dots, k,

while

\tilde{D_{k + j}} (y) = 0

for

j = 1, \dots, n - k .

Recall from Remark 5.4 that

\sum_{i = 1}^{n} \tilde{D_{i}}

acts trivially on

R,

while

\sum_{i = 1}^{n} \tilde{D_{i}} (y) = k y .

Set

\hat{δ} := (\frac{1}{k} \sum_{i = 1}^{k} a_{i}) (\sum_{i = 1}^{n} \tilde{D_{i}}) .

Then, for all

r

and

s,

we have

(\tilde{δ} + \hat{δ}) (x_{r s}) = \tilde{δ} (x_{r s}) + \hat{δ} (x_{r s}) = \tilde{δ} (x_{r s}) + 0 = δ (x_{r s}),

while

(\tilde{δ} + \hat{δ}) (y) = \tilde{δ} (y) + \hat{δ} (y) = - (\sum a_{i}) y + (\frac{1}{k} \sum a_{i}) (k y) = 0 = δ (y) .

As

δ

and

\tilde{δ} + \hat{δ}

agree on the generating set

x_{i j}, y,

they are equal as derivatives.

For the proof of the second part, suppose that

{ad}_{z} + \sum_{i = 1}^{n} a_{i} D_{i} = 0

for some

z \in 𝒪_{q} (G (k, n))

and

a_{i} \in K .

Thus,

{ad}_{z} ([I]) + \sum_{i = 1}^{n} a_{i} δ (i \in I) [I] = 0

for each quantum Plücker coordinate

[I] \in 𝒪_{q} (G (k, n)) .

The first term has no components in degree one and the other terms are all in degree one, so we deduce that

{ad}_{z} ([I]) = 0

for each quantum Plücker coordinate

[I] \in 𝒪_{q} (G (k, n)),

so that

{ad}_{z} = 0 .

Thus, we obtain

\sum_{i = 1}^{n} a_{i} D_{i} = 0 .

For

r = 1, \dots, n + 1 - k,

set

[I_{r}] := [1, \dots, k - 1, k - 1 + r],

and observe that

0 = \sum_{i = 1}^{n} a_{i} D_{i} [I_{r}] = ((a_{1} + \dots + a_{k - 1}) + a_{k - 1 + r}) [I_{r}] .

Thus,

(a_{1} + \dots + a_{k - 1}) + a_{k - 1 + r} = 0

for each of these values of

r .

It follows that

a_{k} = a_{k + 1} = \dots = a_{n} .

In a similar manner, set

[J_{r}] := [n - k + 1 - r, n - k + 2, \dots, n]

for

r = 0, \dots, n - k

to observe that

a_{n - k + 1 - r} + (a_{n - k + 2} + \dots + a_{n}) = 0

for these values of

r .

It follows that

a_{1} = \dots = a_{n - k + 1} .

These two ranges of values must overlap, or else

n - k + 1 < k,

so that

n + 1 < 2 k \leq n,

a contradiction. Thus,

a_{1} = a_{2} = \dots = a_{n},

and from this and Corollary 3.2, it follows that each

a_{i} = 0 .

8.2. The general case

We have now proved our conjecture for

𝒪_{q} (G (k, n))

in the case where

2 k \leq n .

In order to remove this restriction, we use the fact that

𝒪_{q} (G (k, n)) ≅ 𝒪_{q} (G (n - k, n));

see, for example, [4

S. Launois and T. H. Lenagan, The automorphism group of the quantum grassmannian. Bull. Lond. Math. Soc. 56 (2024), no. 4, 1310–1327 Zbl 1554.16033 MR 4736773

, Proposition 3.1]. We will use

\bar{D}

to distinguish derivations of

𝒪_{q} (G (n - k, n))

from derivations of

𝒪_{q} (G (k, n)) .

Let

ψ : 𝒪_{q} (G (k, n)) \to 𝒪_{q} (G (n - k, n))

be the automorphism of [4

S. Launois and T. H. Lenagan, The automorphism group of the quantum grassmannian. Bull. Lond. Math. Soc. 56 (2024), no. 4, 1310–1327 Zbl 1554.16033 MR 4736773

, Proposition 3.1], so that

ψ ([I]) = [w_{0} (\hat{I})]

for each quantum Plücker coordinate

[I]

of

𝒪_{q} (G (k, n)),

where

\hat{I} := {1, \dots, n} \ I

and

w_{0}

is the longest element of the symmetric group

S_{n} .

Let

D

be a derivation of

𝒪_{q} (G (k, n)) .

It is easy to check that

ψ D ψ^{- 1}

is a derivation of

𝒪_{q} (G (n - k, n)) .

Similarly, if

\bar{D}

is a derivation of

𝒪_{q} (G (n - k, n)),

then

ψ^{- 1} \bar{D} ψ

is a derivation of

𝒪_{q} (G (k, n)) .

Recall that we have the derivations

D_{i}

of

𝒪_{q} (G (k, n))

for

i = 1, \dots, n,

with

D_{i} ([I]) = δ (i \in I) [I],

and similarly we have derivations

\bar{D_{j}}

of

𝒪_{q} (G (n - k, n))

for

j = 1, \dots, n .

For each

i,

we need to see how

ψ D_{i} ψ^{- 1}

acts as a derivation on

𝒪_{q} (G (n - k, n))

in terms of the

\bar{D_{j}} .

Note from Corollary 3.2 that

\frac{1}{n - k} (\sum_{j = 1}^{n} \bar{D_{j}}) ([J]) = [J]

for each quantum Plücker coordinate

[J] \in 𝒪_{q} (G (n - k, n)) .

Lemma 8.2.

We have

ψ D_{i} ψ^{- 1} = \frac{1}{n - k} (\sum_{j = 1}^{n} \bar{D_{j}}) - {\bar{D}}_{w_{0} (i)} .

Proof.

Let

[J]

be a quantum Plücker coordinate in

𝒪_{q} (G (n - k, n))

and suppose that

[J] = ψ ([I]) = [w_{0} (\hat{I})]

for a quantum Plücker coordinate

[I]

of

𝒪_{q} (G (k, n)) .

Before we do the calculation of

ψ D_{i} ψ^{- 1},

note the following evaluation of a truth function:

δ (i \in I) = 1 - δ (i \in \hat{I}) = 1 - δ (w_{0} (i) \in w_{0} (\hat{I})) = 1 - δ (w_{0} (i) \in J) .

We obtain

\begin{matrix} ψ D_{i} ψ^{- 1} ([J]) & = ψ D_{i} ([I]) = ψ (δ (i \in I) [I]) = δ (i \in I) [J] \\ = (1 - δ (w_{0} (i) \in J)) [J] = [J] - {\bar{D}}_{w_{0} (i)} ([J]) \\ = {\frac{1}{n - k} (\sum_{j = 1}^{n} \bar{D_{j}}) - {\bar{D}}_{w_{0} (i)}} ([J]), \end{matrix}

as required.

We can now obtain our main theorem without any restriction other than

1 < k < n - 1 .

Given that we have proved the conjecture in the case that

2 k \leq n,

it is enough to prove the result for

𝒪_{q} (G (n - k, n))

when

2 k \leq n .

Proposition 8.3.

Assume that $2 k \leq n .$ In $𝒪_{q} (G (n - k, n))$ any derivation is equal, modulo inner derivations, to a linear combination of $\bar{D_{1}}, \dots, \bar{D_{n}} .$ Furthermore, these $n$ derivations are linearly independent modulo the inner derivations.

Proof.

Let

\bar{D}

be a derivation on

𝒪_{q} (G (n - k, n)) .

Then

ψ^{- 1} \bar{D} ψ

is a derivation of the algebra

𝒪_{q} (G (k, n)) .

Hence,

ψ^{- 1} \bar{D} ψ = {ad}_{z} + \sum_{i = 1}^{n} a_{i} D_{i}

for some

z \in 𝒪_{q} (G (k, n))

and

a_{i} \in K,

by Proposition 8.1.

Therefore,

\bar{D} = {ad}_{ψ (z)} + \sum_{i = 1}^{n} a_{i} ψ D_{i} ψ^{- 1} = {ad}_{ψ (z)} + \sum_{i = 1}^{n} a_{i} {\frac{1}{n - k} (\sum_{j = 1}^{n} \bar{D_{j}}) - {\bar{D}}_{w_{0} (i)}},

and the first claim follows.

The proof of the second part follows in the same way as for the second part of Proposition 8.1.

We are now ready to state and prove our main result.

Theorem 8.4.

Let $2 \leq k \leq n - 2 .$ Then any derivation of $𝒪_{q} (G (k, n))$ is equal, modulo inner derivations, to a linear combination of $D_{1}, \dots, D_{n} .$ Furthermore, these $n$ derivations are linearly independent modulo the inner derivations.

Proof.

The result in the case that

2 k \leq n

has been established in Proposition 8.1, so suppose that

2 k > n .

Set

k^{'} := n - k,

then

2 k^{'} < n

and

n - k^{'} = k .

As

2 k^{'} < n,

the result holds for

𝒪_{q} (G (k^{'}, n)) .

It then follows that the result holds for

𝒪_{q} (G (k, n)) = 𝒪_{q} (G (n - k^{'}, n)),

by using Proposition 8.3.

Recall that the Hochschild cohomology group in degree one of a ring

R,

denoted by

{HH}^{1} (R),

is defined by

{HH}^{1} (R) := Der (R) / InnDer (R),

where

InnDer (R) := {{ad}_{z} | z \in R}

is the Lie algebra of inner derivations of

R .

It is well known that

{HH}^{1} (R)

is a module over

{HH}^{0} (R) := Z (R) .

The following corollary is immediate from the above theorem.

Corollary 8.5.

Let $2 \leq k \leq n - 2 .$ The first Hochschild cohomology group of the quantum grassmannian, ${HH}^{1} (𝒪_{q} (G (k, n))),$ is an $n -$ dimensional vector space over $K$ with basis (the cosets of) $D_{1}, \dots, D_{n} .$

A. Derivations on non-square quantum matrices

In this appendix, we prove case (ii) of the subclaim in the proof of Proposition 8.1. To be more specific, in case (ii) of the subclaim, we are dealing with a derivation of

𝒪_{q} (M (k, p)),

where

k < p,

that arises, via the dehomogenisation equality, from a derivation

D

of

𝒪_{q} (G (k, n))

that has the following properties: (i)

D ([u]) = D ([w]) = 0;

(ii)

D

is a homogeneous derivative of

𝒪_{q} (M (k, p)) .

As a consequence of Lemma 7.1, the first condition implies that the derivative

D

acts trivially on the rightmost quantum minor of

𝒪_{q} (M (k, p)) .

Hence, with a change of notation, throughout this appendix, we assume that we are considering derivations on

𝒪_{q} (M (m, n))

where

m < n

and that we have a derivation

D

acting on

𝒪_{q} (M (m, n))

with the following properties:

(i)

The derivation $D$ is homogeneous; that is, all terms appearing non-trivially in $D (x_{r s})$ have degree one.
(ii)

$D ([1, \dots, m | n + 1 - m, \dots, n]) = 0 .$

The aim in this appendix is to show that such a derivation can be written as a linear combination of the row and column derivations

D_{i, *}

and

D_{* j}

that were introduced in Section 5. With this in mind, we fix the following notation.

Notation A.1.

Throughout the appendix,

B

denotes the quantum matrix subalgebra of

𝒪_{q} (M (m, n))

generated by

x_{r s}

in the first

n - m

columns of

𝒪_{q} (M (m, n))

and

C

denotes the square quantum matrix subalgebra of

𝒪_{q} (M (m, n))

generated by the

x_{r s}

in the final

m

columns. The quantum determinant of

C

is

[I | J] := [1, \dots, m | n - m + 1, \dots, n]

and

[I | J]

is in the centre of

C .

A.1. Action of derivations on first $m$ columns of $𝒪_{q} (M (m, n))$

Lemma A.2.

Use Notation A.1. Let $D$ be a derivation on $𝒪_{q} (M (m, n))$ such that $D (x_{i j})$ is homogeneous of degree one for all $x_{i j},$ and suppose that $D ([I | J]) = 0 .$ Then $D (B) \subseteq B .$

Proof.

It is enough to show that

D (x_{i j}) \in B

for

j \leq m .

For such an

x_{i j},

suppose that

D (x_{i j}) = \sum_{\begin{matrix} r \leq m \\ s \leq n \end{matrix}} a_{r s} x_{r s}

with

a_{r s} \in K .

Now,

x_{i j} [I | J] = q [I | J] x_{i j},

by [4

S. Launois and T. H. Lenagan, The automorphism group of the quantum grassmannian. Bull. Lond. Math. Soc. 56 (2024), no. 4, 1310–1327 Zbl 1554.16033 MR 4736773

, Lemma 6.1 (ii)]. Apply

D

to this equation, noting that

D ([I | J]) = 0,

to obtain

\sum_{\begin{matrix} 1 \leq r \leq m \\ 1 \leq s \leq n \end{matrix}} a_{r s} x_{r s} [I | J] = q [I | J] \sum_{\begin{matrix} 1 \leq r \leq m \\ 1 \leq s \leq n \end{matrix}} a_{r s} x_{r s} .

As

x_{r s} [I | J] = q [I | J] x_{r s}

when

s \leq m,

and

x_{r s} [I | J] = [I | J] x_{r s}

for

s > m,

this gives

\sum_{\begin{matrix} r \leq m \\ s > m \end{matrix}} (1 - q) a_{r s} x_{r s} [I | J] = 0,

and it follows that

a_{r s} = 0

when

s > m,

so that

D (x_{i j}) \in B,

as required.

A.2. We can adjust $D,$ so that $D$ is trivial on $C$

Lemma A.3.

Use Notation A.1. Let $D$ be a derivation on $𝒪_{q} (M (m, n))$ such that $D (x_{i k})$ is homogeneous of degree one for all $x_{i k},$ and suppose that $D ([I | J]) = 0 .$ Then $D (C) \subseteq C .$

Proof.

This is proved in a similar manner to the proof of Lemma A.2, using the fact that

x_{i k}

commutes with

[I | J]

for

x_{i k}

in

C .

We now show that we can adjust

D

by row and column derivations, so that the adjusted derivation acts trivially on

C .

Lemma A.4.

Use Notation A.1. Let $D$ be a derivation on $𝒪_{q} (M (m, n))$ such that $D (x_{r s})$ is homogeneous of degree one for all $x_{r s},$ and suppose that $D ([I | J]) = 0 .$ There is a homogeneous derivation $D^{'}$ of $𝒪_{q} (M (m, n))$ with ${D^{'} |}_{C} = 0$ and $D^{'} (B) \subseteq B$ such that $D^{'} - D$ is a linear combination of $D_{i, *},$ for $i = 1, \dots, m,$ and $D_{* j},$ for $n - m + 1 \leq j \leq n,$ the row and column derivations of $𝒪_{q} (M (m, n))$ introduced in Section 5.

Proof.

Note that

D (C) \subseteq C,

by Lemma A.3. Set

Y := [I | J],

the quantum determinant of

C .

By [3

S. Launois and T. H. Lenagan, The first Hochschild cohomology group of quantum matrices and the quantum special linear group. J. Noncommut. Geom. 1 (2007), no. 3, 281–309 Zbl 1137.16015 MR 2314098

, Theorem 2.9], there are polynomials

P_{1}, \dots, P_{m}, Q_{n - m + 1}, \dots, Q_{n} \in K [Y]

and an element

z \in C

such that

{D |}_{C} = {ad}_{z} + \sum_{i = 1}^{m} P_{i} D_{i, *} |_{C} + \sum_{j = n - m + 1}^{n} Q_{j} D_{* j} |_{C} .

Let

s \geq n - m + 1,

so that

x_{r s} \in C .

Then

{D |}_{C} (x_{r s}) = {ad}_{z} (x_{r s}) + \sum_{i = 1}^{m} P_{i} D_{i, *} |_{C} (x_{r s}) + \sum_{j = n - m + 1}^{n} Q_{j} D_{* j} |_{C} (x_{r s}) .

Let

a_{i}

be the constant term in

P_{i}

and

b_{j}

be the constant term in

Q_{j},

so that

\begin{matrix} {D |}_{C} (x_{r s}) - \sum_{i = 1}^{m} a_{i} D_{i, *} |_{C} (x_{r s}) + \sum_{j = n - m + 1}^{n} b_{j} D_{* j} |_{C} (x_{r s}) \\ = z x_{r s} - x_{r s} z + \sum_{i = 1}^{m} (P_{i} - a_{i}) D_{i, *} |_{C} (x_{r s}) + \sum_{j = n - m + 1}^{n} (Q_{j} - b_{j}) D_{* j} |_{C} (x_{r s}) . \end{matrix}

As

D (x_{r s})

is homogeneous of degree one, the terms on the left-hand side of the equation all have degree one, whereas the nonzero terms on the right-hand side all have degree greater than one. It follows that both sides are zero, and so

{D |}_{C} (x_{r s}) = \sum_{i = 1}^{m} a_{i} D_{i, *} |_{C} (x_{r s}) + \sum_{i = n - m + 1}^{n} b_{j} D_{* j} |_{C} (x_{r s}) .

Set

D^{'} := D - \sum_{i = 1}^{m} a_{i} D_{i, *} + \sum_{i = n - m + 1}^{n} b_{j} D_{* j} .

Then

{D^{'} |}_{C} = 0 .

Note that

D^{'}

is a homogeneous derivation as

D,

D_{i, *}

and

D_{* j}

are all homogeneous. Lemma A.2 shows that

D^{'} (B) \subseteq B .

A.3. Derivatives column by column

Let

ε_{1}, \dots, ε_{m}

and

ε_{1}^{'}, \dots, ε_{n}^{'}

be the standard bases for

ℤ^{m}

and

ℤ^{n},

respectively. The quantum matrix algebra

𝒪_{q} (M (m, n))

has a natural

ℤ^{m} \times ℤ^{n}

bigrading defined by giving each

x_{r s}

bidegree, also called bicontent,

(ε_{r}, ε_{s}^{'}) .

Observe that any quantum minor

[U | V]

is homogeneous of bidegree

(χ_{U}, χ_{V}^{'}),

where

χ_{S}

(respectively,

χ_{S}^{'}

) stands for the characteristic function of a subset

S

of

{1, \dots, m}

(respectively, of

{1, \dots, n}

). If the bidegree of a homogeneous element is

(χ_{r}, χ_{c}),

we refer to

χ_{r}

as the row content of the element and

χ_{c}

as the column content of the element. The main use of the notion of content will be that in any equation of the form

P = Q,

the terms of the same bicontent on each side of the equation must be equal.

We consider

𝒪_{q} (M (m, n))

with

m < n

and assume that we have a homogeneous derivation

D

that acts trivially on

C

and that

D (B) \subseteq B .

Lemma A.5.

Use Notation A.1. Let $D$ be a derivation on $𝒪_{q} (M (m, n))$ such that $D (x_{i k})$ is homogeneous of degree one for all $x_{i k}$ and that $D (C) = 0 .$ Then $D$ is a linear combination of $D_{* k}$ for $k = 1, \dots, n - m .$

Proof.

Let

x_{i k} \in B,

so that

1 \leq i \leq m

and

1 \leq k \leq n - m .

As

D (B) \subseteq B,

by Lemma A.2, we can write

D (x_{i k}) = \sum_{\begin{matrix} 1 \leq r \leq m \\ 1 \leq s \leq n - m \end{matrix}} a_{r s}^{(i k)} x_{r s}

for some

a_{r s}^{(i k)} \in K .

Our first aim is to show that

a_{r s}^{(i k)} = 0

whenever

r \neq i,

so that

D (x_{i k}) = \sum_{1 \leq s \leq n - m} a_{i s}^{(i k)} x_{i s} .

Set

l := n - m + 1 .

Choose any

j < i .

Note that

x_{i k} x_{j l} = x_{j l} x_{i k} .

As

x_{j l} \in C,

we know that

D (x_{j l}) = 0,

so that when we apply the derivative

D

to this equation, we obtain

\sum_{\begin{matrix} 1 \leq r \leq m \\ 1 \leq s < l \end{matrix}} a_{r s}^{(i k)} x_{r s} x_{j l} = \sum_{\begin{matrix} 1 \leq r \leq m \\ 1 \leq s < l \end{matrix}} a_{r s}^{(i k)} x_{j l} x_{r s}

or

\sum_{\begin{matrix} 1 \leq r \leq m \\ 1 \leq s < l \end{matrix}} a_{r s}^{(i k)} (x_{r s} x_{j l} - x_{j l} x_{r s}) = 0 .

Consider terms in this equation with bicontent

(2 ε_{j}; ε_{s} + ε_{l})

(which from now on we also denote by

(j, j; s, l)

to ease notation) to see that

\sum_{1 \leq s < l} a_{j s}^{(i k)} (x_{j s} x_{j l} - x_{j l} x_{j s}) = 0 .

As

s < l,

this gives

\sum_{1 \leq s < l} a_{j s}^{(i k)} (q - 1) x_{j l} x_{j s} = 0 .

From which it follows that

a_{j s}^{(i k)} = 0

for all

s

when

j < i .

Next, choose any

j > i .

As

D (x_{j k}) \in B,

write

D (x_{j k}) = \sum_{\begin{matrix} 1 \leq r \leq m \\ 1 \leq s < l \end{matrix}} a_{r s}^{(j k)} x_{r s} .

Apply

D

to the equation

x_{i k} x_{j l} - x_{j l} x_{i k} = \hat{q} x_{i l} x_{j k}

(where

\hat{q} := q - q^{- 1}

), noting that

D (x_{* l}) = 0

as

x_{* l} \in C,

to obtain

(A.1)

\sum_{\begin{matrix} 1 \leq r \leq m \\ 1 \leq s < l \end{matrix}} a_{r s}^{(i k)} x_{r s} x_{j l} - \sum_{\begin{matrix} 1 \leq r \leq m \\ 1 \leq s < l \end{matrix}} a_{r s}^{(i k)} x_{j l} x_{r s} = \hat{q} (\sum_{\begin{matrix} 1 \leq r \leq m \\ 1 \leq s < l \end{matrix}} a_{r s}^{(j k)} x_{i l} x_{r s}) .

Look at terms with bicontent

(j, j; s, l)

in equation (A.1). There are no such terms on the right-hand side, as

i

occurs as a row index and

i \neq j,

while on the left-hand side, we get such terms when

r = j .

Hence,

a_{j s}^{(i k)} x_{j s} x_{j l} - a_{j s}^{(i k)} x_{j l} x_{j s} = 0

for each

s .

As

s < l,

this gives the equality

(q - 1) a_{j s}^{(i k)} x_{j s} x_{j l} = 0

and so

a_{j s}^{(i k)} = 0

for all

s

and each

j > i .

As we already know that

a_{j s}^{(i k)} = 0

for all

s

when

j < i,

this gives

D (x_{i k}) = \sum_{1 \leq s < l} a_{i s}^{(i k)} x_{i s}

for

1 \leq i \leq m

as required.

Fix

k,

with

1 \leq k < l

and

j > 1 .

Apply the derivative

D

to the equation

x_{1 k} x_{j l} - x_{j l} x_{1 k} = \hat{q} x_{1 l} x_{j k},

using the expressions we have just derived, to obtain

(A.2)

\sum_{1 \leq s < l} a_{1 s}^{(1 k)} x_{1 s} x_{j l} - \sum_{1 \leq s < l} a_{1 s}^{(1 k)} x_{j l} x_{1 s} = \hat{q} \sum_{1 \leq s < l} a_{j s}^{(j k)} x_{1 l} x_{j s} .

Look at terms with content

(1, j; s, l)

in equation (A.2) for fixed

s .

On the left-hand side, we have

a_{1 s}^{(1 k)} x_{1 s} x_{j l} - a_{1 s}^{(1 k)} x_{j l} x_{1 s},

and this is equal to

\hat{q} a_{1 s}^{(1 k)} x_{1 l} x_{j s} .

On the right-hand side, we have

\hat{q} a_{j s}^{(j k)} x_{1 l} x_{j s} .

It follows that

a_{j s}^{(j k)} = a_{1 s}^{(1 k)}

for each

s

with

1 \leq s < l,

so that

D (x_{j k}) = \sum_{1 \leq s < l} a_{1 s}^{(1 k)} x_{j s}

for each

j

with

1 \leq j \leq m .

Hence,

(A.3)

D (\begin{matrix} x_{1 k} \\ x_{2 k} \\ ⋮ \\ x_{m k} \end{matrix}) = a_{11}^{(1 k)} (\begin{matrix} x_{11} \\ x_{21} \\ ⋮ \\ x_{m 1} \end{matrix}) + a_{12}^{(1 k)} (\begin{matrix} x_{12} \\ x_{22} \\ ⋮ \\ x_{m 2} \end{matrix}) + \dots + a_{1, l - 1}^{(1 k)} (\begin{matrix} x_{1, l - 1} \\ x_{2, l - 1} \\ ⋮ \\ x_{m, l - 1} \end{matrix}) .

For each

k = 1, \dots, l - 1,

we want to show that

a_{1 r}^{(1 k)} = 0

when

r \neq k,

so that

D (\begin{matrix} x_{1 k} \\ x_{2 k} \\ ⋮ \\ x_{m k} \end{matrix}) = a_{1 k}^{(1 k)} (\begin{matrix} x_{1 k} \\ x_{2 k} \\ ⋮ \\ x_{m k} \end{matrix}) = a_{1 k}^{(1 k)} D_{* k} (\begin{matrix} x_{1 k} \\ x_{2 k} \\ ⋮ \\ x_{m k} \end{matrix}) .

Fix

k

with

1 \leq k \leq l - 1 .

First, we show that

a_{1 r}^{(1 k)} = 0

for each

r

such that

k < r \leq l - 1 .

If

k = l - 1,

there is no such

r

to consider, so assume that

k < l - 1,

in which case

k + 1 < l

and

x_{1, k + 1} \in B .

Apply

D

to the equation

x_{1 k} x_{1, k + 1} = q x_{1, k + 1} x_{1 k}

for

k + 1 < l

to obtain

D (x_{1 k}) x_{1, k + 1} + x_{1 k} D (x_{1, k + 1}) = q D (x_{1, k + 1}) x_{1 k} + q x_{1, k + 1} D (x_{1 k}),

which gives

(A.4)

\begin{matrix} \sum_{1 \leq s < l} a_{1 s}^{(1 k)} x_{1 s} x_{1, k + 1} + \sum_{1 \leq s < l} a_{1 s}^{(1, k + 1)} x_{1 k} x_{1 s} \\ = \sum_{1 \leq s < l} q a_{1 s}^{(1, k + 1)} x_{1 s} x_{1 k} + \sum_{1 \leq s < l} q a_{1 s}^{(1 k)} x_{1, k + 1} x_{1 s} . \end{matrix}

Consider terms in this equation with content

(1, 1; k + 1, r)

for

r > k .

These occur in the first and fourth sums when

s = r

and do not occur in the second and third sums because

k

is in the column content of the terms in these sums. Hence,

a_{1 r}^{(1 k)} x_{1 r} x_{1, k + 1} = q a_{1 r}^{(1 k)} x_{1, k + 1} x_{1 r} .

If

r = k + 1,

then we get

a_{1, k + 1}^{(1 k)} x_{1, k + 1} x_{1, k + 1} = q a_{1, k + 1}^{(1 k)} x_{1, k + 1} x_{1, k + 1}

and so

a_{1, k + 1}^{(1 k)} = 0 .

If

r > k + 1,

then

x_{1 r} x_{1, k + 1} = q^{- 1} x_{1, k + 1} x_{1 r} .

So we see that

q^{- 1} a_{1 r}^{(1 k)} x_{1, k + 1} x_{1 r} = q a_{1 r}^{(1 k)} x_{1, k + 1} x_{1 r}

and so

a_{1 r}^{(1 k)} = 0 .

Hence,

a_{1, r}^{(1, k)} = 0

for all

r > k .

Next, we show

a_{1 r}^{(1 k)} = 0

when

r < k .

If

k = 1,

then there is no such

r

to consider, so assume that

k > 1,

in which case

k - 1 \geq 1,

so that

x_{1, k - 1}

exists and is in

B .

Apply

D

to the equation

x_{1, k - 1} x_{1, k} = q x_{1, k} x_{1, k - 1}

to obtain

D (x_{1, k - 1}) x_{1, k} + x_{1, k - 1} D (x_{1, k}) = q D (x_{1, k}) x_{1, k - 1} + q x_{1, k} D (x_{1, k - 1}),

which gives

(A.5)

\begin{matrix} \sum_{1 \leq s < l} a_{1 s}^{(1, k - 1)} x_{1 s} x_{1, k} + \sum_{1 \leq s < l} a_{1 s}^{(1, k)} x_{1, k - 1} x_{1 s} \\ = \sum_{1 \leq s < l} q a_{1 s}^{(1 k)} x_{1 s} x_{1, k - 1} + \sum_{1 \leq s < l} q a_{1 s}^{(1, k - 1)} x_{1 k} x_{1 s} . \end{matrix}

Consider terms in this equation with content

(1, 1; r, k - 1)

with

r < k .

There are no such terms in the first and fourth terms, as

k

is in the column content of the terms in these sums, and such terms occur in the second and third terms when

s = r .

Hence,

a_{1 r}^{(1, k)} x_{1, k - 1} x_{1 r} = q a_{1 r}^{(1, k)} x_{1 r} x_{1, k - 1} .

When

r = k - 1,

we see that

a_{1, k - 1}^{(1, k)} x_{1, k - 1} x_{1, k - 1} = q a_{1, k - 1}^{(1, k)} x_{1, k - 1} x_{1, k - 1},

so that

a_{1, k - 1}^{(1, k)} = 0 .

When

r < k - 1,

then

x_{1, k - 1} x_{1 r} = q^{- 1} x_{1 r} x_{1, k - 1},

so we see that

q^{- 1} a_{1, r}^{(1, k)} x_{1 r} x_{1, k - 1} = q a_{1, r}^{(1, k)} x_{1, r} x_{1, k - 1}

and so

a_{1, r}^{(1, k)} = 0 .

Hence,

a_{1, r}^{(1, k)} = 0

for all

r < k .

Thus,

a_{1 r}^{(1 k)} = 0

whenever

r \neq k,

so that

D (x_{i k}) = a_{i k}^{(1 k)} x_{i k}

for each

1 \leq i \leq m,

as required to show that

D (\begin{matrix} x_{1 k} \\ x_{2 k} \\ ⋮ \\ x_{m k} \end{matrix}) = a_{1 k}^{(1 k)} (\begin{matrix} x_{1 k} \\ x_{2 k} \\ ⋮ \\ x_{m k} \end{matrix})

for each

1 \leq k < l .

As we already know that

D

acts trivially on columns

n + 1 - m

up to

n,

this gives

D = a_{11}^{(1, 1)} D_{* 1} + a_{12}^{(1, 2)} D_{* 2} + \dots + a_{1, l - 1}^{(1, l - 1)} D_{*, l - 1},

as required.

A.4. Conclusion of appendix

The preceding analysis proves the following proposition.

Proposition A.6.

Suppose that $m < n$ and that we have a derivation $D$ of $𝒪_{q} (M (m, n))$ such that

(i)

the derivation $D$ is homogeneous; that is, all terms appearing non-trivially in $D (x_{i j})$ have degree one,
(ii)

$D ([1, \dots, m | n + 1 - m, \dots, n]) = 0 .$

Then

D = \sum_{i = 1}^{m} a_{i} D_{i, *} + \sum_{k = 1}^{n} b_{k} D_{* k}

for some $a_{i}, b_{k} \in K .$

Proof.

By Lemma A.4, there is a homogeneous derivation

D^{(1)}

such that

D^{(1)} - D

is a linear combination of the

D_{i, *},

with

1 \leq i \leq m,

and

D_{* j},

with

n - m + 1 \leq j \leq n,

and such that

{D^{(1)} |}_{C} = 0 .

By Lemma A.5,

D^{(1)}

is a linear combination of the

D_{* k}

with

1 \leq k \leq n - m .

Writing

D = D^{(1)} - (D^{(1)} - D)

gives the required result.

Funding

This research was partly supported by EPSRC grant EP/R009279/1.

References

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Cite this article

Stéphane Launois, Thomas H. Lenagan, Derivations and the first Hochschild cohomology group of the quantum grassmannian. J. Noncommut. Geom. 20 (2026), no. 3, pp. 845–870

DOI 10.4171/JNCG/662

Abstract

1. Introduction

2. Basic definitions

Definition 2.1.

3. Derivations for 𝒪q(G(k,n)) inherited from 𝒪q(M(k,n))

Lemma 3.1.

Proof.

Corollary 3.2.

Proof.

Conjecture 3.3.

4. The dehomogenisation equality for 𝒪q(G(k,n))

5. Derivations arising via dehomogenisation

Proposition 5.1.

Proof.

Corollary 5.2.

Example 5.3.

Remark 5.4.

6. Adjusting derivatives

Lemma 6.1.

Proof.

Lemma 6.2.

Proof.

Lemma 6.3.

Proof.

Definition 6.4.

Lemma 6.5.

Proof.

Corollary 6.6.

Lemma 6.7.

Proof.

Corollary 6.8.

Proof.

Lemma 6.9.

Proof.

Corollary 6.10.

Lemma 6.11.

Proof.

Corollary 6.12.

Proof.

Lemma 6.13.

Proof.

Proposition 6.14.

Proof.

7. Transferring derivations of 𝒪q(G(k,n)) to 𝒪q(M(k,p))

Lemma 7.1.

Proof.

Lemma 7.2.

Proof.

Theorem 7.3.

Proof.

8. The main theorem

8.1. The case where 2k≤n

Proposition 8.1.

Proof.

Subclaim: Case (i)

Subclaim: Case (ii)

8.2. The general case

Lemma 8.2.

Proof.

Proposition 8.3.

Proof.

Theorem 8.4.

Proof.

Corollary 8.5.

A. Derivations on non-square quantum matrices

Notation A.1.

A.1. Action of derivations on first m columns of 𝒪q(M(m,n))

Lemma A.2.

Proof.

A.2. We can adjust D, so that D is trivial on C

Lemma A.3.

Proof.

Lemma A.4.

Proof.

A.3. Derivatives column by column

Lemma A.5.

Proof.

A.4. Conclusion of appendix

Proposition A.6.

Proof.

3. Derivations for $𝒪_{q} (G (k, n))$ inherited from $𝒪_{q} (M (k, n))$

4. The dehomogenisation equality for $𝒪_{q} (G (k, n))$

7. Transferring derivations of $𝒪_{q} (G (k, n))$ to $𝒪_{q} (M (k, p))$

8.1. The case where $2 k \leq n$

A.1. Action of derivations on first $m$ columns of $𝒪_{q} (M (m, n))$

A.2. We can adjust $D,$ so that $D$ is trivial on $C$