Multiparameter Quantum Cauchy-Binet Formulas
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Multiparameter Quantum Cauchy-Binet Formulas Matthias Flore´ 1 Received: 13 April 2019 / Accepted: 21 May 2020 / © Springer Nature B.V. 2020
Abstract The quantum Cayley-Hamilton theorem for the generator of the reflection equation algebra has been proven by Pyatov and Saponov, with explicit formulas for the coefficients in the Cayley-Hamilton formula. However, these formulas do not give an easy way to compute these coefficients. Jordan and White provided an elegant formula for the coefficients given with respect to the generators of the reflection equation algebra. In this paper, we provide R (M (C)), Cauchy-Binet formulas for these coefficients with respect to generators of Oq,Q N ∗ the multiparameter quantized -algebra of functions on MN (C) as a real variety, which contains the reflection equation algebra as a subalgebra. We also prove a Cauchy-Binet formula for the inverse of a matrix involving these generators. Keywords Quantum matrices · Quantum minor · Cayley-Hamilton · Reflection equation algebra · Cauchy-Binet · Multiparameter quantum deformation Mathematics Subject Classification (2010) 17B37 · 20G42 · 81R50
1 Introduction The Cauchy-Binet formula is a classical formula in linear algebra and states (in the case of square matrices) that if A, X, Y ∈ MN (C) with A = XY then the minors of A can be computed by [A]I,J = [X]I,K [Y ]K,J (1.1) K∈([N ]) [N] i [N] for all I, J ∈ i . Here, we write i for the set of i-element subsets of [N ], with [N ] = {1, . . . , N } for N ∈ N0 . Thus [N] i = {J = (j1 , . . . , ji ) | 1 ≤ j1 < · · · < ji ≤ N } [N] for i ∈ [N ] and 0 = {∅}. For J, K ∈ [N] we denote by [A]J,K the i × i-minor of the i N × N -matrix A with rows j1 , . . . , ji and columns k1 , . . . , ki .
Presented by: Michel Van den Bergh Matthias Flor´e
[email protected] 1
Department of Mathematics, Vrije Universiteit Brussel (VUB), B-1050 Brussels, Pleinlaan 2, Belgium
M. Flor´e
In Theorem 1.1 and Theorem 6.7, we prove analogues over the deformation of MN (C) (see Section 2) of 2 particular cases where the Cauchy-Binet formula (1.1) is used in the classical case, which can thus be seen as multiparameter quantum Cauchy-Binet formulas. In e.g. [11] the Cayley-Hamilton theorem is proved for the generating matrix A = XY of the reflection equation algebra: N
σq,Q (i)(−XY )N−i = 0.
(1.2)
i=0 R (M (C)) (see Theorem 3.3). The σq,Q (i), 0 ≤ i ≤ N are self-adjoint and central in Oq,Q N Hence they are also central in the reflection equation algebra and in fact, they generate the center of the reflection equation algebra, see [8] and [11]. We will prove that they also satisfy an invariance property, see Theorem 4.6. The formulas [11, (3.2) and (3.3)] ((3.8) and (3.10)) for the coefficients σq,Q (i) in the Cayley-Hamilton theorem (1.2) do not give a concrete description of them. In [12, Theorem 1.3] ((3.12)), an elegant formula for the coefficients in the case that qij = q, i = j is given with respect to the generators of the reflection equation algebra. In particular:
It is a q-deformati
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