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A remark on the Hochschild dimension of liberated quantum groups

Tomasz Brzeziński, Ulrich Krähmer, Réamonn Ó Buachalla, Karen R. Strung

Abstract

Let $A$ be a Hopf algebra equipped with a projection onto the coordinate Hopf algebra $\mathcal{O}(G)$ of a semisimple algebraic group $G$. It is shown that if $A$ admits a suitably non-degenerate comodule $V$ and the induced $G$-module structure of $V$ is non-trivial, then the third Hochschild homology group of $A$ is non-trivial.

A remark on the Hochschild dimension of liberated quantum groups

Abstract

Let be a Hopf algebra equipped with a projection onto the coordinate Hopf algebra of a semisimple algebraic group . It is shown that if admits a suitably non-degenerate comodule and the induced -module structure of is non-trivial, then the third Hochschild homology group of is non-trivial.
Paper Structure (12 sections, 6 theorems, 42 equations)

This paper contains 12 sections, 6 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The comodule $V$
  4. The pairing $\langle -,- \rangle$
  5. The Lie algebra $\mathfrak{g} _A$
  6. Hochschild (co)homology
  7. Proof of the theorem
  8. The Hochschild 3-cycle $c_V$
  9. The cap product $c_V \smallfrown \varphi$
  10. Evaluation in $\varepsilon$
  11. The Casimir operator
  12. The class $\pi _*([c_V])$

Key Result

Theorem 1

Let $G$ be a semisimple algebraic group over a field $\mathbb{F}$ of characteristic 0, $\pi \colon A \longrightarrow \mathcal{O}(G)$ be a Hopf algebra map, and $V$ be a right $A$-comodule with a non-degenerate symmetric or antisymmetric invariant bilinear form. If the representation of $G$ on $V$ in

Theorems & Definitions (11)

  • Theorem
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 1 more