A new realization of the i-quantum group U^j(n)

TL;DR

This paper introduces a new realization of the i-quantum groups U^j(n) of type B, providing a basis, multiplication formulas, and connections to integral q-Schur algebras and finite orthogonal groups.

Contribution

It develops a new realization for U^j(n) of type B, including basis and multiplication formulas, and links representations to finite orthogonal groups, extending type A theory.

Findings

Established a basis and multiplication formulas for U^j(n).

Constructed a surjective algebra homomorphism to q-Schur algebras.

Connected representations of i-quantum hyperalgebras to finite orthogonal groups.

Abstract

We follow the approach developed by Beilinson-Lusztig-MacPherson and modified by Fu and the first author to investigate a new realization for the i-quantum groups U^j(n) of type B, building on the multiplication formulas discovered in [BKLW,Lem.~3.2]. This allows us to present U^j(n) via a basis and multiplication formulas by generators. We also establish a surjective algebra homomorphism from a Lusztig type form of U^j(n) to integral q-Schur algebras of type B. Thus, base changes allow us to relate representations of the i-quantum hyperalgebras of U^j(n) to representations of finite orthogonal groups of odd degree in non-defining characteristics. This generalizes part of Dipper--James' type A theory to the type B case.