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About enveloping algebras of direct sums

Gérard Henry Edmond Duchamp, Christophe Tollu, Jean-Gabriel Luque, Vu Nguyen Dinh

TL;DR

The paper addresses the PBW-like normal ordering problem for enveloping algebras of a direct sum $\mathfrak{g}=\mathfrak{g}_1\oplus\mathfrak{g}_2$ by developing a universal-construction framework and a normal-form calculator that transports a potential inverse of the natural map $\mu_{state}$ to a $\mathfrak{g}$-action on $\mathcal{U}(\mathfrak{g}_1)\otimes\mathcal{U}(\mathfrak{g}_2)$ via an explicit action $g\ast_U$ and the map $\mu_{state}^U$. The main result shows that $\mu_{state}^U$ is bijective, providing a constructive PBW-type decomposition of $\mathcal{U}(\mathfrak{g})$ in terms of $\mathcal{U}(\mathfrak{g}_1)$ and $\mathcal{U}(\mathfrak{g}_2)$. The approach relies on universal properties, free-object constructions, and transport-of-structure arguments, with potential extensions to quantized enveloping algebras and Lie superalgebras.

Abstract

We solve the PBW-like problem of normal ordering for enveloping algebras of direct sums.

About enveloping algebras of direct sums

TL;DR

The paper addresses the PBW-like normal ordering problem for enveloping algebras of a direct sum by developing a universal-construction framework and a normal-form calculator that transports a potential inverse of the natural map to a -action on via an explicit action and the map . The main result shows that is bijective, providing a constructive PBW-type decomposition of in terms of and . The approach relies on universal properties, free-object constructions, and transport-of-structure arguments, with potential extensions to quantized enveloping algebras and Lie superalgebras.

Abstract

We solve the PBW-like problem of normal ordering for enveloping algebras of direct sums.
Paper Structure (8 sections, 3 theorems, 27 equations)

This paper contains 8 sections, 3 theorems, 27 equations.

Table of Contents

  1. A question about the enveloping algebra of a direct sum.
  2. Universal Constructions
  3. General principle.
  4. Step-by-Step construction of a normal form calculator
  5. Construction of the actions $g\ast$
  6. $g\ast_U$ is a $\mathfrak{g}$-action on ${\mathcal{U}}(\mathfrak{g}_1)\otimes {\mathcal{U}}(\mathfrak{g}_2)$.
  7. The linear map $\mu_{state}^U$ is bijective.
  8. Conclusion and future

Key Result

Theorem 1

With the notations as above, i) there exists a unique linear map (in the sequel, $\Phi(g\otimes t\bigotimes m)$ will be alternatively noted $g\ast_T (t\bigotimes m)$) such that (Nota : For the sake of clarity, we have used the blue tensor product as explained in Remark Quot-Rem.States-bimod.) ii) This map is filtered in the following sense iii) It is compatible with a) The ${\mathcal{U}}(\mathf

Theorems & Definitions (7)

  • Example 1
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Theorem 2
  • proof