Holomorphic Relative Hopf Modules over the Irreducible Quantum Flag Manifolds

Holomorphic deformation of Hopf algebras and applications to quantum groups

In this article we propose a new and so-called holomorphic deformation scheme for locally convex algebras and Hopf algebras. Essentially we regard converging power series expansion of a deformed product on a locally convex algebra, thus giving the means to actually insert complex values for the deformation parameter. Moreover we establish a topological duality theory for locally convex Hopf algebras. Examples coming from the theory of quantum groups are reconsidered within our holomorphic deformation scheme and topological duality theory. It is shown that all the standard quantum groups compri…

全纯有限生成 Hopf 代数与量子李群

Holomorphically finitely generated Hopf algebras and quantum Lie groups

本文研究 HFG 拓扑 Hopf 代数，刻画交换情形与 Stein 群的范畴等价，分析 A_exp(G) 的 HFG 性质及连通 G 下依赖大尺度几何的解析结构，并给出 hbar-adic 量子群的复解析例子。

A Dolbeault-Dirac Spectral Triple for the $B_2$-Irreducible Quantum Flag Manifold

The quantum version of the Bernstein-Gelfand-Gelfand resolution is used to construct a Dolbeault-Dirac operator on the anti-holomorphic forms of the Heckenberger-Kolb calculus for the $B_2$-irreducible quantum flag manifold. The spectrum and the multiplicities of the eigenvalues of the Dolbeault-Dirac operator are computed. It is shown that this construction yields an equivariant, even, $0^+$-summable spectral triple.

Differential and holomorphic differential operators on noncommutative algebras

This paper deals with sheaves of differential operators on noncommutative algebras. The sheaves are defined by quotienting a the tensor algebra of vector fields (suitably deformed by a covariant derivative) to ensure zero curvature. As an example we can obtain enveloping algebra like relations for Hopf algebras with differential structures which are not bicovariant. Symbols of differential operators are defined, but not studied. These sheaves are shown to be in the center of as category of bimodules with flat bimodule covariant derivatives. Also holomorphic differential operators are considere…

Bimodule Connections for Relative Line Modules over the Irreducible Quantum Flag Manifolds

It was recently shown (by the second author and Díaz García, Krutov, Somberg, and Strung) that every relative line module over an irreducible quantum flag manifold $\mathcal{O}_q(G/L_S)$ admits a unique $\mathcal{O}_q(G)$-covariant connection with respect to the Heckenberger-Kolb differential calculus $Ω^1_q(G/L_S)$. In this paper we show that these connections are bimodule connections with an invertible associated bimodule map. This is proved by applying general results of Beggs and Majid, on principal connections for quantum principal bundles, to the quantum principal bundle presentation of …

Lusztig 量子根向量与 A 系列全量子旗流形的 Dolbeault 复形

Lusztig's Quantum Root Vectors and a Dolbeault Complex for the A-Series Full Quantum Flag Manifolds

本文证明 Lusztig 正根向量在好约化分解下构成 A 系列全量子旗流形的量子切空间，其微分演算为经典反全纯 Dolbeault 复形的 q-形变，并由此建立量子 Borel-Weil 定理，实现所有有限维 type-1 不可约表示。