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On “quantum” deformations of irreducible finite-dimensional representations of \({\mathfrak gl}_ N\). (English. Russian original) Zbl 0623.20030

Sov. Math., Dokl. 33, 507-510 (1986); translation from Dokl. Akad. Nauk SSSR 287, 1076-1079 (1986).
For \(R=\sum_{\alpha,\beta}w_{\alpha \beta}(u)I_{\alpha}\otimes I_{\beta}\), where \(\{I_{\alpha}\}\) is a base of a matrix algebra \(M_ n\) of order n, and \(\{w_{\alpha \beta}\}\) is a family of functions of a parameter \(u\in {\mathbb{C}}\), the corresponding matrix \(^{ij}R\) is defined as \[ ^{ij}R=\sum_{\alpha,\beta}w_{\alpha \beta}(u_ i-u_ j)^ iI_{\alpha}^ jI_{\beta} \] where \[ ^ kI_{\alpha}=I_ 0\otimes...\otimes I_ 0\otimes I_{\alpha}\otimes I_ 0\otimes...\otimes I_ 0\;(k-1\text{ times \(I_ 0\) in the first place)} \] with \(I_ 0\) denoting the identity matrix, and it is assumed that \(^{12}R^{13}R^{23}R=^{23}R^{13}R^{12}R\) for arbitrary \(u_ 1,u_ 2,u_ 3\in {\mathbb{C}}.\)
For fixed R, a quantum analogue of a d-dimensional representation of \(gl_ N\) is defined to be a family of matrices \(\{J_{\alpha}\}\subset M_ d\) for which the Yang-Baxter-Faddeev relation \(^{12}R^ 1L^ 2L=^ 2L^ 1L^{12}R\) holds, where \(^ iL=\sum_{\alpha,\beta}w_{\alpha \beta}(u_ i)^ iI_{\alpha}J_{\beta}.\)
In this note the author proposes a method for constructing quantum representations of \(gl_ N\) corresponding to any Young schemes for arbitrary R-matrices with rational \(w_{\alpha\beta}(u)\), which have only one (modulo symmetries) pole of the first order for u. Particular attention is given to Baxter-Belavin elliptic R-matrices.
Reviewer: W.Guz

MSC:

20G05 Representation theory for linear algebraic groups
81T08 Constructive quantum field theory
20G45 Applications of linear algebraic groups to the sciences
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