Equivariant quantizations of the positive nilradical and covariant differential calculi

TL;DR

This paper develops a method to quantize the positive nilradical of complex semisimple Lie algebras respecting their module structure, and constructs compatible covariant differential calculi on quantum flag manifolds.

Contribution

It introduces a new quantization of the positive nilradical as a module under a Levi factor and constructs covariant differential calculi on quantum flag manifolds.

Findings

Quantization of the positive nilradical as a module under Levi factors.

Construction of covariant differential calculi on quantum flag manifolds.

Identification of conditions under which the quantized structures form coideals.

Abstract

Consider a decomposition $\mathfrak{n} = \mathfrak{n}_1 \oplus \cdots \oplus \mathfrak{n}_r$ of the positive nilradical of a complex semisimple Lie algebra of rank $r$, where each $\mathfrak{n}_k$ is a module under an appropriate Levi factor. We show that this can be quantized as a finite-dimensional subspace $\mathfrak{n}^q_k = \mathfrak{n}^q_1 \oplus \cdots \oplus \mathfrak{n}^q_r$ of the positive part of the quantized enveloping algebra, where each $\mathfrak{n}^q_k$ is a module under the left adjoint action of a quantized Levi factor. Furthermore, we show that $\mathbb{C} \oplus \mathfrak{n}^q$ is a left coideal, with the possible exception of components corresponding to some exceptional Lie algebras. Finally we use these quantizations to construct covariant first-order differential calculi on quantum flag manifolds, compatible in a certain sense with the decomposition above, which coincide with those introduced by Heckenberger-Kolb in the irreducible case.