The Controlling L_infty -Algebra, Cohomology and Homotopy of Embedding Tensors and Lie\u2013Leibniz Triples

Yunhe Sheng,Chenchang Zhu,Rong Tang

doi:10.1007/s00220-021-04032-y

Yunhe Sheng, Chenchang Zhu + Show 1 more

Open Access

https://doi.org/10.1007/s00220-021-04032-y

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Abstract

In this paper, we first construct the controlling algebras of embedding tensors and Lie–Leibniz triples, which turn out to be a graded Lie algebra and an L_infty -algebra respectively. Then we introduce representations and cohomologies of embedding tensors and Lie–Leibniz triples, and show that there is a long exact sequence connecting various cohomologies. As applications, we classify infinitesimal deformations and central extensions using the second cohomology groups. Finally, we introduce the notion of a homotopy embedding tensor which will induce a Leibniz_infty -algebra. We realize Kotov and Strobl’s construction of an L_infty -algebra from an embedding tensor, as a functor from the category of homotopy embedding tensors to that of Leibniz_infty -algebras, and a functor further to that of L_infty -algebras.

Highlights

An embedding tensor on a Lie algebra representation (g, V ) is a linear map T : V → g satisfying a quadratic equivariancy constraint
We develop the theory of controlling algebras, further the theory of cohomology and homotopy for embedding tensors and Lie–Leibniz triples
At the end of this section, we study the relation between the cohomology of an embedding tensor T : V −→ g and the cohomology of the underlying Leibniz algebra (V, [·, ·]T ) given in Proposition 2.4

Summary

Introduction

An embedding tensor on a Lie algebra representation (g, V ) is a linear map T : V → g satisfying a quadratic equivariancy constraint (see Definition 2.2). T is an embedding tensor on the general linear Lie algebra gl(V ) with respect to the natural representation on V if and only if the graph of T , denoted by GT , is an integrable subspace of the omni-Lie algebra (gl(V ) ⊕ V, [·, ·]ol, (·, ·)+). This embedding tensor comes from a strict 2-Lie algebra structure on the endomorphisms of a 2-term complex described in [46]. This can be generalized to any strict Lie 2-algebra as follows. (Lie objects in the infinitesimal tensor category of linear maps) Let (g, [·, ·]g) be a Lie algebra and (V ; ρ) a representation. These two cases exhaust all the possibilities of embedding tensors on H3(C) with respect to the adjoint representation

The Controlling Graded Lie Algebra and Cohomology of Embedding Tensors

Cohomology of Lie–Leibniz Triples and Applications

Homotopy Embedding Tensors and Higher Structures