Let K denote a field of characteristic zero and q∈K be a nonzero element which is not a root of unity. The quantum grassmannian 𝒪q(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] 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 𝒪q(G(k,n)) was established. More precisely, a link between the volume form of positroids and Hochschild (co)homology of the quantum grassmannian 𝒪q(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], 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≤n. Recall that 𝒪q(G(k,n)) is the subalgebra of the quantum matrix algebra 𝒪q(M(k,n)) that is generated by the k×k quantum minors of 𝒪q(M(k,n)). We call these k×k quantum minors quantum Plücker coordinates of 𝒪q(G(k,n)). They are indexed by k-subsets of {1,…,n} and denoted by [I] for each k-subset I. The algebra 𝒪q(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 𝒪q(G(k,n)). They are referred to as prime quantum Plücker coordinates. Among the prime quantum Plücker coordinates, two are special: [u]=[1,…,k] and [w]=[n−k+1,…,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≥0[I][u] and [w][I]=q≤0[I][w].

We exploit this observation and other graded techniques in order to reduce the problem of computing derivations of 𝒪q(G(k,n)) to computing derivations D of 𝒪q(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]. 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 𝒪q(G(k,n)) is a K-vector space of dimension n, with basis given by the cosets of the derivations D1,…,Dn, where Di is the derivation of 𝒪q(G(k,n)) defined by

Di([I])={[I] if i∈I;0 otherwise.

Throughout the paper, K denotes a field of characteristic zero and q∈K is a nonzero element which is not a root of unity.

The algebra of m×n quantum matrices over K, denoted by 𝒪q(M(m,n)), is the algebra generated over K by mn indeterminates xij, with 1≤i≤m and 1≤j≤n, which commute with the elements of K and are subject to the relations

xijxil =qxilxij  for 1≤i≤m and 1≤j<l≤n; xijxkj =qxkjxij  for 1≤i<k≤m and 1≤j≤n; xijxkl =xklxij  for 1≤k<i≤m and 1≤j<l≤n; xijxkl−xklxij =(q−q−1)xilxkj  for 1≤i<k≤m and 1≤j<l≤n.

It is well known that 𝒪q(M(m,n)) is an iterated Ore extension over K with the xij added in lexicographic order. An immediate consequence is that 𝒪q(M(m,n)) is a noetherian domain.

When m=n, the quantum determinant Dq is defined by

Dq:=∑(−q)l(σ)x1σ(1)⋯xnσ(n),

where the sum is over all permutations σ of {1,…,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 𝒪q(M(n,n)).

If I and J are t-element subsets of {1,…,m} and {1,…,n}, respectively, then the quantum minor [I|J] is defined to be the quantum determinant of the t×t quantum matrix subalgebra generated by the variables xij with i∈I and j∈J.

The quantum matrix algebra 𝒪q(M(m,n)) is a connected ℕ-graded algebra with each generator xij given degree one. Note that each t×t quantum minor has degree t.

A k×k quantum minor of 𝒪q(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,…,k|J] for any k-element subset J of {1,…,n}. Quantum minors of this type are called quantum Plücker coordinates. The quantum grassmannian 𝒪q(G(k,n)) is a connected ℕ-graded algebra with each quantum Plücker coordinate given degree one. The set of quantum Plücker coordinates in 𝒪q(G(k,n)) is denoted by Π. There is a natural partial order on Π defined in the following way: if I=[i1<⋯<ik] and J=[j1<⋯<jk], then [I]<[J] if and only if il≤jl for each l=1,…,k. A standard monomial in the quantum Plücker coordinates is an expression of the form [I1][I2]⋯[It], where I1≤I2≤⋯≤It in this partial order. The set of all standard monomials forms a vector space basis of 𝒪q(G(k,n)) over K; see, for example, [2, Corollary 2.1]. We will refer to the process of rewriting a monomial in terms of standard monomials as straightening.

The quantum grassmannian 𝒪q(G(1,n)) is a quantum affine space, and, as such, its derivations are known; see [1, Corollaire 1.3.3]; so we will assume throughout this paper that k>1 1\)]]>. As 𝒪q(G(n−1,n))≅𝒪q(G(1,n)), by [4, Proposition 3.4], we exclude the case k=n−1 as well. Thus, we will assume throughout the paper that 2≤k≤n−2, and so n≥4.

Also, it is shown in [4, Proposition 3.4] that 𝒪q(G(k,n))≅𝒪q(G(n−k,n)), when 2k≤n. In Section 7, we need to assume that 2k≤n, and the general result is obtained from this case in Subsection 8.2 by using the isomorphism just mentioned.

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 δij for δ(i=j).

Recall from [3] that for each column of 𝒪q(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 𝒪q(G(k,n)) to give us n derivations which we denote by Di for i=1,…,n. We refer to these derivations as the column derivations of 𝒪q(G(k,n)).

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

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

In this section, we recall results concerning the dehomogenisation equality that occur in [4, Section 3]. We refer the reader to that paper for detailed definitions and proofs.

Set u={1,…,k}. Then [u] commutes with all other quantum Plücker coordinates up to a power of q, by [4, Lemma 3.1]. As 𝒪q(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:=𝒪q(G(k,n))[[u]−1].

For 1≤i≤k and 1≤j≤n−k, set

xij:=[1,…,k+1−i^,…,k,j+k][u]−1∈T.

Set p:=n−k. The elements xij generate a subalgebra of T that is a quantum matrix algebra 𝒪q(M(k,p)). In fact, T is generated over 𝒪q(M(k,p)) by [u]±1, so

T=𝒪q(G(k,n))[[u]−1]=𝒪q(M(k,p))[[u]±1;σ],

where σ is the automorphism of 𝒪q(M(k,p)) defined by σ(xij):=qxij (since [u]xij=qxij[u] for all i,j).

When we operate on the right-hand side of this equality, we will write y for [u]. Thus, yxij=σ(xij)y and

(4.1) T=𝒪q(G(k,n))[[u]−1]=𝒪q(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 𝒪q(M(k,p)) to 𝒪q(G(k,n)) and vice versa.

Note that [4, Lemma 3.2] gives the formula for general quantum minors of the algebra 𝒪q(M(k,p)) when viewed as elements in 𝒪q(G(k,n))[[u]−1], and each quantum Plücker coordinate, multiplied by [u]−1, occurs in the formula. The formula is

[I|J]=[{1,…,k}\(k+1−I)⊔(k+J)][u]−1.

In the reverse direction, for a quantum Plücker coordinate [L] of 𝒪q(G(k,n)), we have

[L]=[L≤k⊔L>k]=[I|J][u], k}]=[I\mid J][u],$$]]>

where I={(k+1)−({1,…,k}\L≤k)} and J=L>k−k k}-k\)]]> with L≤k:=L∩{1,…,k} and L>k:=L∩{k+1,…,n} k}:=L\cap\{k+1,\ldots,n\}\)]]>.

The n column derivations Di of 𝒪q(G(k,n)) that we have defined in Section 3 satisfy Di([u])=[u] for 1≤i≤k and Di([u])=0 for k<i≤n, so they extend to n derivations Di~ of T=R[y,y−1;σ]=𝒪q(G(k,n))[u]−1 with Di~(y)=y for 1≤i≤k and Di~(y)=0 for k<i≤n.

We will show that these derivations Di~ of T coincide with extensions of known derivations of 𝒪q(M(k,p)) which we now recall from [3].

For 1≤i≤k, there are derivations Di⁣∗ of 𝒪q(M(k,p)) defined by Di⁣∗(xrs):=δirxrs, and for 1≤j≤p, there are derivations D∗j of 𝒪q(M(k,p)) defined by D∗j(xrs):=δjsxrs. In other words, Di⁣∗ fixes row i and kills all the other rows of 𝒪q(M(k,p)), while D∗j fixes column j and kills all other columns of 𝒪q(M(k,p)). For these observations, see the comment after [3, Corollary 2.8].

Note that the Leibniz formula for derivations shows that Di⁣∗([I|J])=δ(i∈I)[I|J] and that D∗j([I|J])=δ(j∈J)[I|J] for any quantum minor [I|J] of 𝒪q(M(k,p)). The derivations that we have just defined are not linearly independent: it is easy to check that ∑i=1kDi⁣∗=∑j=1pD∗j.

We will extend these derivations of 𝒪q(M(k,p)) to derivations Di⁣∗~ and D∗j~ of T=𝒪q(M(k,p))[y±1;σ]=𝒪q(G(k,n))[u]−1 as follows. For each i=1,…,k, set Di⁣∗~(y)=y and note that by applying the Leibniz formula for derivatives to 1=yy−1 we obtain Di⁣∗~(y−1)=−y−1. For each j=1,…,p, set D∗j~(y)=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 Di⁣∗~ and D∗j~ on quantum Plücker coordinates in 𝒪q(G(k,n)).

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,…,k] and [w]=[n−k+1,…,n] denote the leftmost and rightmost quantum Plücker coordinates of 𝒪q(G(k,n)), respectively. For any quantum Plücker coordinate [I], set d(I):=#(I\(I∩u)) and e(I):=#(I\(I∩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.

If S=[I1]⋯[Im] is a standard monomial, then set d(S):=∑d(Ii) and note that each d(Ii)≥0 with d(Ii)=0 if and only if [Ii]=[u]. Then, S[u]=qd(S)[u]S, and so Su=uS 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 𝒪q(G(k,n)) is a graded algebra with each quantum Plücker coordinate having degree one.

The following lemma is deduced from results in [5]. In the proof of the lemma, we use the notation developed in that paper without further explanation.

The next corollary follows immediately from the previous lemma.

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≥1.