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Lie algebras in $\text{Ver}_4^+$

Serina Hu

TL;DR

This work develops Lie theory in the symmetric tensor category $ ext{Ver}_4^+$ over characteristic $2$, establishing a Koszul-based PBW criterion that identifies Lie algebras as operadic Lie algebras with $[x,x]=0$ on $ ext{ker } d$. It provides a systematic classification of low-dimensional Lie algebras in $ ext{Ver}_4^+$, including structures on $P$, $1+P$, $2\cdot1+P$, and $2P$, and introduces a $4$-center analogue for $U( rak{gl}(m\cdot\mathbf{1}+nP))$, along with quadratic Casimir elements. The paper then develops the representation theory of $ rak{gl}(P)$, identifying two families of irreducibles $L(0,a,b)$ and $L(1,a,b)$, computing their tensor products, and describing how $ ext{GL}(P)$-representations restrict to $ rak{gl}(P)$-representations with parameters in $ ext{GF}(2)$. Overall, the results extend super-Lie theory concepts to characteristic $2$ via Verlinde categories, connecting structural classifications with centers, Casimirs, and representation theory in a coherent framework. The methods leverage the Koszul deformation principle, explicit cohomological data, and detailed analyses of the indecomposable projective $P$ and its role in $ ext{Ver}_4^+$.

Abstract

We develop Lie theory in the category $\text{Ver}_4^+$ over a field of characteristic 2, the simplest tensor category which is not Frobenius exact, as a continuation of arXiv:2406.10201. We provide a conceptual proof that an operadic Lie algebra in $\text{Ver}_4^+$ is a Lie algebra, i.e. satisfies the PBW theorem, exactly when its invariants form a usual Lie algebra. We then classify low-dimensional Lie algebras in $\text{Ver}_4^+$, construct elements in the center of $U(\mathfrak{gl}(X))$ for $X \in \text{Ver}_4^+$, and study representations of $\mathfrak{gl}(P)$, where $P$ is the indecomposable projective of $\text{Ver}_4^+$.

Lie algebras in $\text{Ver}_4^+$

TL;DR

This work develops Lie theory in the symmetric tensor category over characteristic , establishing a Koszul-based PBW criterion that identifies Lie algebras as operadic Lie algebras with on . It provides a systematic classification of low-dimensional Lie algebras in , including structures on , , , and , and introduces a -center analogue for , along with quadratic Casimir elements. The paper then develops the representation theory of , identifying two families of irreducibles and , computing their tensor products, and describing how -representations restrict to -representations with parameters in . Overall, the results extend super-Lie theory concepts to characteristic via Verlinde categories, connecting structural classifications with centers, Casimirs, and representation theory in a coherent framework. The methods leverage the Koszul deformation principle, explicit cohomological data, and detailed analyses of the indecomposable projective and its role in .

Abstract

We develop Lie theory in the category over a field of characteristic 2, the simplest tensor category which is not Frobenius exact, as a continuation of arXiv:2406.10201. We provide a conceptual proof that an operadic Lie algebra in is a Lie algebra, i.e. satisfies the PBW theorem, exactly when its invariants form a usual Lie algebra. We then classify low-dimensional Lie algebras in , construct elements in the center of for , and study representations of , where is the indecomposable projective of .

Paper Structure

This paper contains 18 sections, 39 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Preliminaries
  4. Lie algebras in $\mathop{\mathrm{Ver}}\nolimits_4^+$ and the PBW theorem
  5. Determination of Lie algebras of low dimensionalities
  6. Classification of ordinary Lie algebra structures of dimension up to 4
  7. Classification of Lie algebra structures on $P$
  8. Classification of Lie algebra structures on $m \cdot \mathds{1} + nP$
  9. Classification of Lie algebra structures on $\mathds{1} + P$
  10. Classification of Lie algebra structures on $2 \cdot \mathds{1} + P$
  11. Classification of Lie algebra structures on $2P$
  12. A propsed analog of the $p$-center for Lie algebras in $\mathop{\mathrm{Ver}}\nolimits_4^+$
  13. Quadratic Casimir elements of $\mathfrak{gl}(m \cdot \mathds{1} + P)$
  14. Representations of $\mathfrak{gl}(P)$
  15. $\mathfrak{gl}(P)$ and $U(\mathfrak{gl}(P))$
  16. ...and 3 more sections

Key Result

Proposition 3.1

$P$ has exact Koszul complex, so it is Koszul.

Theorems & Definitions (98)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Example 2.9
  • Definition 2.10
  • ...and 88 more