Quantum symmetric spaces

Abstract

Let $G$ be a semisimple Lie group, $\mathfrak{g}$ its Lie algebra. For any symmetric space $M$ over $G$ we construct a new (deformed) multiplication in the space $A$ of smooth functions on $M$. This multiplication is invariant under the action of the Drinfeld--Jimbo quantum group $U_h\mathfrak{g}$ and is commutative with respect to an involutive operator $\tilde{S}:A\otimes A\to A\otimes A$. Such a multiplication is unique. Let $M$ be a kählerian symmetric space with the canonical Poisson structure. Then we construct a $U_h\mathfrak{g}$-invariant multiplication in $A$ which depends on two parameters and is a quantization of that structure.

16 pp, LaTeX

Cited by in corpus (1)

• Quantization of Poisson pencils and generalized Lie algebras