JournalsaihpdVol. 1, No. 1pp. 1–46

Noncommutative determinants, Cauchy–Binet formulae, and Capelli-type identities II. Grassmann and quantum oscillator algebra representation

  • Sergio Caracciolo

    Università degli Studi di Milano, Italy
  • Andrea Sportiello

    Université Paris-Nord, Villetaneuse, France
Noncommutative determinants, Cauchy–Binet formulae, and Capelli-type  identities II. Grassmann and quantum oscillator algebra representation cover
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Abstract

We prove that, for XX, YY, AA and BB matrices with entries in a non-commutative ring such that

[Xij,Yk]=AiBkj,\hbox{$[X_{ij},Y_{k\ell}]=-A_{i\ell} B_{kj}$},

satisfying suitable commutation relations (in particular, XX is a Manin matrix), row-pseudo-commutative matrix (a Manin matrix), the following identity holds:

coldet X coldet Y =0coldet (aA+X(IaB)1Y)0\mathrm {col-det } \ X \ \mathrm { col-det } \ Y \ = \langle 0\mid \mathrm { col-det } \ (aA + X (I-a^{\dagger} B)^{-1} Y)\mid 0\rangle

Furthermore, if also YY is a Manin matrix, [Yij,Ykl]=0[Y_{ij},Y_{kl}]=0 for iki\neq k, jlj\neq l

coldet X coldet Y=D(ψ,ψˉ)exp(k0(ψˉAψ)kk+1(ψˉXBkYψ))\mathrm {col-det } \ X \ \mathrm { col-det } \ Y =\int \mathcal{D}(\psi, \bar{\psi}) \exp \big(\sum_{k \geq 0}\frac{(\bar{\psi} A \psi)^{k}}{k+1}(\bar{\psi} X B^k Y \psi)\big)

Here 0\langle 0 \mid and 0\mid 0\rangle, are respectively the bra and the ket of the ground state, aa^{\dagger} and aa the creation and annihilation operators of a quantum harmonic oscillator, while ψˉi\bar{\psi}_i and ψi\psi_i are Grassmann variables in a Berezin integral. These results should be seen as a generalization of the classical Cauchy–Binet formula, in which AA and BB are null matrices, and of the non-commutative generalization, the Capelli identity, in which AA and BB are identity matrices and [Xij,Xk]=[Yij,Yk]=0[X_{ij},X_{k\ell}]=[Y_{ij},Y_{k\ell}]=0.

Cite this article

Sergio Caracciolo, Andrea Sportiello, Noncommutative determinants, Cauchy–Binet formulae, and Capelli-type identities II. Grassmann and quantum oscillator algebra representation. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 1 (2014), no. 1, pp. 1–46

DOI 10.4171/AIHPD/1