Yangian Algebras and Classical Riemann Problems

We investigate different Hopf algebras associated to Yang's solution of quantum Yang-Baxter equation. It is shown that for the precise definition of the algebra one needs the commutation relations for the deformed algebra of formal currents and the specialization of the Riemann problem for the currents. Two different Riemann problems are considered. They lead to the central extended Yangian double associated with ${sl}_2$ and to the degeneration of scaling limit of elliptic affine algebra. Unless the defining relations for the generating functions of the both algebras coincide their properties and the theory of infinite-dimensional representations are quite different. We discuss also the Riemann problem for twisted algebras and for scaled elliptic algebra.Comment: 36 pages, 3 figures under bezier.sty, corrected some typo

Baxter operator and Archimedean Hecke algebra

In this paper we introduce Baxter integral Q-operators for finite-dimensional Lie algebras gl(n+1) and so(2n+1). Whittaker functions corresponding to these algebras are eigenfunctions of the Q-operators with the eigenvalues expressed in terms of Gamma-functions. The appearance of the Gamma-functions is one of the manifestations of an interesting connection between Mellin-Barnes and Givental integral representations of Whittaker functions, which are in a sense dual to each other. We define a dual Baxter operator and derive a family of mixed Mellin-Barnes-Givental integral representations. Givental and Mellin-Barnes integral representations are used to provide a short proof of the Friedberg-Bump and Bump conjectures for G=GL(n+1) proved earlier by Stade. We also identify eigenvalues of the Baxter Q-operator acting on Whittaker functions with local Archimedean L-factors. The Baxter Q-operator introduced in this paper is then described as a particular realization of the explicitly defined universal Baxter operator in the spherical Hecke algebra H(G(R),K), K being a maximal compact subgroup of G. Finally we stress an analogy between Q-operators and certain elements of the non-Archimedean Hecke algebra H(G(Q_p),G(Z_p)).Comment: 32 pages, typos corrected