Gauss decomposition of connection matrices and application to Yang-Baxter equation. II
[For part I see the authors' paper in ibid. 69, No. 7, 238-242 (1993; Zbl 0797.33011).] It is the second part of the paper consisting of two parts. Here the authors study matrix functions introduced in the first part. Basic properties of the principal connection matrix \(G= G(x\mid \alpha_ 1)\), \(x= (x_ 1, x_ 2)\), are given. This matrix function depends only on \(x_ 2/ x_ 1\). The formula is presented relating \(G= G(x\mid \alpha_ 1)\) with \(G= G(x^{-1} \mid \alpha_ 1^ \prime)\). The special cycles \(\text{reg } Y^ -_ i \otimes Y^ +_{ n-i}\) are defined which are connected with \(G\). It is given the asymptotics characterizing them. The cycle \(\text{reg } Y^ -_ i \otimes Y^ +_{n -i}\) is expressed as a linear combination of \(\text{reg } Y^ -_{j, n-j}\) and \(\text{reg } Y^ +_{n-i, n-j}\). Explicit expressions for the coefficients are given in terms of theta monomials. Matrix elements of the matrix \(W_{\tau_ r}\), introduced in the first part of the paper, is expressed in terms of the connection matrix for Jackson integrals of the function depending only on \(x_ r\) and \(x_{r+1}\). The authors obtain the one cocycle condition for \(\{ W^{(m) }_{\tau_ r} \}_{r=1}^{m -1}\) which coincides with the Yang-Baxter equation.
scientific article; zbMATH DE number 549047