# 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

## Abstract

We prove that, for $X$, $Y$, $A$ and $B$ matrices with entries in a non-commutative ring such that

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

Furthermore, if also $Y$ is a Manin matrix, $[Y_{ij},Y_{kl}]=0$ for $i=k$, $j=l$

Here $⟨0∣$ and $∣0⟩$, are respectively the bra and the ket of the ground state, $a_{†}$ and $a$ the creation and annihilation operators of a quantum harmonic oscillator, while $ψˉ _{i}$ and $ψ_{i}$ are Grassmann variables in a Berezin integral. These results should be seen as a generalization of the classical Cauchy–Binet formula, in which $A$ and $B$ are null matrices, and of the non-commutative generalization, the Capelli identity, in which $A$ and $B$ are identity matrices and $[X_{ij},X_{kℓ}]=[Y_{ij},Y_{kℓ}]=0$.

## 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