Equivariant quantizations of the positive nilradical and covariant differential calculi

TL;DR

The paper constructs finite-dimensional, equivariant quantizations of the positive nilradical $\mathfrak{n}$ of a complex semisimple Lie algebra $\mathfrak{g}$ with a fixed decomposition, embedding each graded piece as a module under the corresponding quantized Levi factor and ensuring $\mathbb{C}\oplus\mathfrak{n}^q$ acts as a left coideal under $U_q(\mathfrak{b})$ (with mild exclusions for exceptional types). By relating these quantizations to twisted quantum Schubert cells, it provides explicit degree-one building blocks generated by quantum root vectors and shows the higher-degree pieces arise via braided shuffle products; the construction yields left-covariant first-order differential calculi on quantum flag manifolds compatible with the decomposition, aligning with the Heckenberger-Kolb calculi in the irreducible case. The results rely on quasitriangular Hopf algebra techniques, transmutation, and a detailed analysis of coideal properties up to degree three, with a concrete G2 example illustrating the method. The framework organizes covariant calculi on chains of quantum flag manifolds, and the authors discuss extensions to right coideals via the quantum Chevalley involution, suggesting a broad, symmetry-aware approach to quantum geometry.

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.

Figures (1)

• Figure A.1: The exceptional simple Lie algebras and the numbering we adopt.

Theorems & Definitions (83)

• Proposition 1.3
• Theorem 1.4
• Conjecture 1.5
• Theorem 1.6
• Example 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Proposition 3.1