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Universal enveloping of a graded Lie algebra

Felipe Yukihide Yasumura

Abstract

In this paper we construct a graded universal enveloping algebra of a $G$-graded Lie algebra, where $G$ is not necessarily an abelian group. If the grading group is abelian, then it coincides with the classical construction. We prove the existence and uniqueness of the graded enveloping algebra. As consequences, we prove a graded variant of Witt's Theorem on the universal enveloping algebra of the free Lie algebra, and the graded version of Ado's Theorem, which states that every finite-dimensional Lie algebra admits a faithful finite dimensional representation. Furthermore we investigate if a Lie grading is equivalent to an abelian grading.

Universal enveloping of a graded Lie algebra

Abstract

In this paper we construct a graded universal enveloping algebra of a -graded Lie algebra, where is not necessarily an abelian group. If the grading group is abelian, then it coincides with the classical construction. We prove the existence and uniqueness of the graded enveloping algebra. As consequences, we prove a graded variant of Witt's Theorem on the universal enveloping algebra of the free Lie algebra, and the graded version of Ado's Theorem, which states that every finite-dimensional Lie algebra admits a faithful finite dimensional representation. Furthermore we investigate if a Lie grading is equivalent to an abelian grading.
Paper Structure (14 sections, 37 theorems, 47 equations)

This paper contains 14 sections, 37 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Universal group of a grading
  4. Refinement and coarsening
  5. Free algebra
  6. Free pair
  7. Graded Universal Enveloping Algebra
  8. Strong pair
  9. A PBW basis for $\mathrm{SU}_G(\mathcal{L})$
  10. Free graded Lie algebra
  11. Free graded algebras and pairs
  12. Strong Witt's Theorem
  13. A graded version of Ado's Theorem
  14. Group gradings on Lie algebras

Key Result

Lemma 2

Let $\Gamma$ and $\Gamma'$ be gradings on $\mathcal{A}$, and assume that $\Gamma'$ is realized as a $H$-grading and is a coarsening of $\Gamma$. Then, there exists a group epimorphism $p:U(\Gamma)\to H$ such that $p(\mathrm{Supp}\,\Gamma)=\mathrm{Supp}\,\Gamma'$.

Theorems & Definitions (79)

  • Definition 1: EK13
  • Lemma 2: EK13
  • Definition 3
  • Remark
  • Lemma 4
  • proof
  • Corollary 5
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 69 more