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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)

##### MSC:
 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
Jackson integral