Quantum Affine Schubert Cells and FRT-Bialgebras: The E_6^{(1)} Case

TL;DR

This paper explores the structure of quantum Schubert cell algebras related to E_6 and E_6^{(1)} types, revealing their isomorphic relationship to subalgebras of FRT-bialgebras via cocycle twists.

Contribution

It demonstrates an isomorphism between quotients of quantum Schubert cell algebras and subalgebras of FRT-bialgebras for E_6 cases, extending understanding of their algebraic structure.

Findings

Identified isomorphic subalgebras within FRT-bialgebras for E_6^{(1)}

Mapped quantum Schubert cell algebras to FRT-bialgebras with cocycle twists

Characterized quotients by specific submodules under adjoint action

Abstract

De Concini, Kac, and Procesi defined a family of subalgebras Uq[w] of the quantized enveloping algebra Uq(g) associated to elements w in the Weyl group of a simple Lie algebra g. These algebras are called quantum Schubert cell algebras. We show that, up to a mild cocycle twist, quotients of certain quantum Schubert cell algebras of types E_6 and E_6^{(1)} map isomorphically onto distinguished subalgebras of the Faddeev-Reshetikhin-Takhtajan universal bialgebra associated to the braiding on the quantum half-spin representation of Uq(so_{10}). We identify the quotients as those obtained by factoring out the quantum Schubert cell algebras by ideals generated by certain submodules with respect to the adjoint action of Uq(so_{10}).