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Connection coefficients for \(A\)-type Jackson integral and Yang-Baxter equation. (English) Zbl 0813.33010
Sally, Paul J. jun. (ed.) et al., Mathematical aspects of conformal and topological field theories and quantum groups. AMS-IMS-SIAM summer research conference, June 13-19, 1992, Mount Holyoke College, South Hadley, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 175, 1-26 (1994).
The characteristic cycles for Jackson integrals are constructed as a class of special \(q\)-cycles (\(q= e^{2\pi i\tau}, \text{Im}(\tau) >0)\). The \(n\)-dimensional Jackson integral is defined as the integral of a \(q\)- multiplicative function \(\Phi\) over a suitable cycle. Then, the regularized Jackson integrals are introduced by using the action of the permutation group \(\sigma_ n\) on the functions \(\Phi\). Special cycles in the \(n\)-dimensional de Rham cohomology over which Jackson integrals have monomial asymptotic expansions are named characteristic. It is shown that the canonical action of \(\sigma^ n\) induces a linear representation satisfying the Yang-Baxter equation. Some particular cases (2-dimensional representations and one-dimensional Jackson integrals) are considered and explicit formulas are obtained for the associated characteristic cycles.
For the entire collection see [Zbl 0801.00049].
Reviewer: G.Zet (Iaşi)

33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
33E05 Elliptic functions and integrals
39A10 Additive difference equations