Chevalley restriction theorem for vector-valued functions on quantum groups

TL;DR

This paper extends Chevalley's restriction theorem to quantum groups, demonstrating injectivity and describing the image of the restriction map for vector-valued functions, thus broadening the classical invariant theory to quantum algebra settings.

Contribution

It generalizes Chevalley's theorem to quantum groups with vector-valued functions, providing new insights into invariant theory in quantum algebra.

Findings

Restriction map is injective for quantum groups.

Explicit description of the image of the restriction map.

Extension of classical invariant theory to quantum group context.

Abstract

We generalize Chevalley's theorem about restriction of \mathfrak{g}-invariant polynomial functions \mathfrak{g}->C to W-invariant functions on the Cartan \mathfrak{h}->C. We consider the case when \mathfrak{g} is replaced by a quantum group and the target space of the polynomial maps is replaced by a finite dimensional representation V of this quantum group. We prove that the restriction map Res:(O_q(G)\otimes V)^{U_q(\mathfrak{g})}-> O(H)\otimes V is injective and describe the image.