Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Derivations and Hochschild cohomology of quantum nilpotent algebras

Stéphane Launois, Samuel A. Lopes, Isaac Oppong

Abstract

We compute the derivations of Quantum Nilpotent Algebras under a technical (but necessary) assumption on the center. As a consequence, we give an explicit description of the first Hochschild cohomology group of $U_q^+(\mathfrak{g})$, the positive part of the quantized enveloping algebra of a finite-dimensional complex simple Lie algebra $\mathfrak{g}$. Our results are obtained leveraging an initial cluster constructed by Goodearl and Yakimov.

Derivations and Hochschild cohomology of quantum nilpotent algebras

Abstract

We compute the derivations of Quantum Nilpotent Algebras under a technical (but necessary) assumption on the center. As a consequence, we give an explicit description of the first Hochschild cohomology group of , the positive part of the quantized enveloping algebra of a finite-dimensional complex simple Lie algebra . Our results are obtained leveraging an initial cluster constructed by Goodearl and Yakimov.
Paper Structure (15 sections, 21 theorems, 73 equations)

This paper contains 15 sections, 21 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Structure theorem for derivations of partially localized quantum affine spaces
  4. Prime elements and localizations of QNAs
  5. Homogeneous prime elements
  6. Partially localized quantum affine space associated to a QNA
  7. Normal elements cannot be multiples of a non-prime $y_i$
  8. Intersections of localizations
  9. Centers--the zeroth Hochschild cohomology group
  10. The first Hochschild cohomology group of a QNA
  11. The inner component of a derivation of $R$
  12. Homogeneous derivations
  13. Application to quantized enveloping algebras
  14. Proof of Theorem \ref{['b4']} without the symmetry assumption
  15. The normal elements in $U_q^+(\mathfrak{g})$

Key Result

Theorem 2.1

Let $A$ be a ${\mathbb K}$-algebra and set $R=A[X_1, \ldots, X_m]$, the polynomial algebra over $A$ on $m$ commuting variables. Then Additionally, assume that $A$ is finitely generated and that for some $\operatorname{\mathsf{Z}}(A)$-module $M$. Then where

Theorems & Definitions (43)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • ...and 33 more