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Higher structures for Lie $H$-pseudoalgebras

Apurba Das

Abstract

Let $H$ be a cocommutative Hopf algebra. The notion of Lie $H$-pseudoalgebra is a multivariable generalization of Lie conformal algebras. In this paper, we study some higher structures related to Lie $H$-pseudoalgebras where we increase the flexibility of the Jacobi identity. Namely, we first introduce $L_\infty$ $H$-pseudoalgebras (also called strongly homotopy Lie $H$-pseudoalgebras) as the homotopy analogue of Lie $H$-pseudoalgebras. We give several equivalent descriptions of such homotopy algebras and show that some particular classes of these homotopy algebras are closely related to the cohomology of Lie $H$-pseudoalgebras and crossed modules of Lie $H$-pseudoalgebras. Next, we introduce another higher structure, called Lie-$2$ $H$-pseudoalgebras which are the categorification of Lie $H$-pseudoalgebras. Finally, we show that the category of Lie-$2$ $H$-pseudoalgebras is equivalent to the category of certain $L_\infty$ $H$-pseudoalgebras.

Higher structures for Lie $H$-pseudoalgebras

Abstract

Let be a cocommutative Hopf algebra. The notion of Lie -pseudoalgebra is a multivariable generalization of Lie conformal algebras. In this paper, we study some higher structures related to Lie -pseudoalgebras where we increase the flexibility of the Jacobi identity. Namely, we first introduce -pseudoalgebras (also called strongly homotopy Lie -pseudoalgebras) as the homotopy analogue of Lie -pseudoalgebras. We give several equivalent descriptions of such homotopy algebras and show that some particular classes of these homotopy algebras are closely related to the cohomology of Lie -pseudoalgebras and crossed modules of Lie -pseudoalgebras. Next, we introduce another higher structure, called Lie- -pseudoalgebras which are the categorification of Lie -pseudoalgebras. Finally, we show that the category of Lie- -pseudoalgebras is equivalent to the category of certain -pseudoalgebras.
Paper Structure (17 sections, 17 theorems, 100 equations)

This paper contains 17 sections, 17 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Lie H-pseudoalgebras
  3. Strongly homotopy algebras and categorifications
  4. Graded Lie H-pseudoalgebras and strongly homotopy Lie H-pseudoalgebras
  5. Categorification of Lie H-pseudoalgebras
  6. Lie H-pseudoalgebras and their cohomology
  7. Strongly homotopy Lie H-pseudoalgebras
  8. Representations and cohomology of strongly homotopy Lie $H$-pseudoalgebras
  9. $\Gamma$-actions
  10. Annihilation algebras
  11. Examples of strongly homotopy Lie H-pseudoalgebras
  12. $L_\infty$ conformal algebras
  13. Current $L_\infty$ $H$-pseudoalgebras
  14. Graded Lie $H$-pseudoalgebras whose each arity is a free $H$-module of rank $1$
  15. Strongly homotopy associative $H$-pseudoalgebras
  16. ...and 2 more sections

Key Result

Proposition 3.7

Let $\mathcal{L} = \oplus_{n \in \mathbb{Z}} \mathcal{L}^n$ be a graded left $H$-module. An $L_\infty$$H$-pseudoalgebra structure on $\mathcal{L}$ is equivalent to a collection of degree $-1$ polylinear maps $\{ \eta_k \in \mathrm{Hom}^{-1}_{H^{\otimes k}} ( (\mathcal{L} [-1])^{\otimes k}, H^{\ot

Theorems & Definitions (50)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.6
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • ...and 40 more