Algebras of functions on compact quantum groups, Schubert cells and quantum tori

Abstract

The structures of Poisson Lie groups on a simple compact group are parametrized by pairs (α,w), where aeR, ueA 2 \) R , and ί) R is a real Cartan subalgebra of complexifϊcation of Lie algebra of the group in question. In the present article the description of the symplectic leaves for all pairs (α, u) is given. Also, the corresponding quantized algebras of functions are constructed and their irreducible representations are described. In the course of investigation Schubert cells and quantum tori appear. At the end of the article the quantum analog of the Weyl group is constructed and some of its applications, among them the formula for the universal ^-matrix, are given.

Quantum Symmetry Groups of Noncommutative Spheres

“…and similarly for the other terms, this equation is satisfied when r ∈ h ⊕ h for an abelian Lie subalgebra h of g. Although there are other solutions (see [18] for an exhaustive treatment of Poisson Lie group structures on simple compact Lie groups and the several algebraic quantizations of the Hopf algebra of representative functions), we shall focus on the case r ∈ h ⊕h. On using our previous identification of h with R l , we can write r as a skewsymmetric l × l matrix Q.…”

“…At the algebraic level, there are other deformations of flag manifolds [18] which go beyond those considered here, in that more general solutions of the classical Yang-Baxter equation are used for the deformation directions. These could yield further examples of quantum homogeneous spaces.…”

Admissible diagrams in quantum nilpotent algebras and combinatoric properties of Weyl groups

“…The reader will observe a little difference between those formulas and the original Levendorskii-Soibelman one ( [16], prop. 5.5.2.…”

Lacunary Fourier Series for Compact Quantum Groups

“…Let G be a simply connected semisimple compact Lie group. It follows from the work of Levendorskii and Soibelman [LS91,Soi90] that given any q > 0 one can define a compact quantum group G q , called the Drinfeld-Jimbo qdeformation of G such that the fusion rules, the classical dimension function and the coamenablity do not depend on q. More precisely, we may state the following property (see for example [NT13, Theorem 2.4.7], [Ban99] and references therein).…”