Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities II. Grassmann and quantum oscillator algebra representation
Articolo
Data di Pubblicazione:
2014
Citazione:
Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities II. Grassmann and quantum oscillator algebra representation / S. Caracciolo, A. Sportiello. - In: ANNALES DE L'INSTITUT HENRI POINCARE D: COMBINATORICS, PHYSICS AND THEIR INTERACTIONS. - ISSN 2308-5827. - 1:1(2014), pp. 1-46. [10.4171/AIHPD/1]
Abstract:
We prove that, for X, Y, A and B matrices with entries in a non-commutative ring such that [Xij,Ykℓ]=−AiℓBkj, satisfying suitable commutation relations (in particular, X is a Manin matrix), the following identity holds: coldetXcoldetY=<0|coldet(aA+X(I−a†B)−1Y)|0>. Furthermore, if also Y is a Manin matrix, coldetXcoldetY=∫D(ψ,ψ†)exp[∑k≥01k+1(ψ†Aψ)k(ψ†XBkYψ)]. Notations: <0|, |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 ψ†i and ψi 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 [Xij,Xkℓ]=[Yij,Ykℓ]=0.
Tipologia IRIS:
01 - Articolo su periodico
Keywords:
invariant theory ; Capelli identity ; non-commutative determinant ; row- and column- determinants ; Cauchy–Binet theorem ; Weyl algebra ; right-quantum matrix ; Cartier–Foata matrix ; Manin matrix ; quantum oscillator algebra ; Grassmann algebra ; Campbell–Baker– Hausdorff formula ; Łukasiewicz paths
Elenco autori:
S. Caracciolo, A. Sportiello
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