Lie n-algebras and cohomologies of relative Rota-Baxter operators on n-Lie algebras

doi:10.1016/j.geomphys.2023.104785

Journal of Geometry and Physics

Volume 187, May 2023, 104785

https://doi.org/10.1016/j.geomphys.2023.104785 Get rights and content

Abstract

Based on the differential graded Lie algebra controlling deformations of an n-Lie algebra with a representation (called an n- $LieRep$ pair), we construct a Lie n-algebra, whose Maurer-Cartan elements characterize relative Rota-Baxter operators on n- $LieRep$ pairs. The notion of an n-pre-Lie algebra is introduced, which is the underlying algebraic structure of the relative Rota-Baxter operator. We give the cohomology of relative Rota-Baxter operators and study infinitesimal deformations and extensions of order m deformations to order $m + 1$ deformations of relative Rota-Baxter operators through the cohomology groups of relative Rota-Baxter operators. Moreover, we build the relation between the cohomology groups of relative Rota-Baxter operators on n- $LieRep$ pairs and those on $(n + 1)$ - $LieRep$ pairs by certain linear functions.

Introduction

In this paper, we use Maurer-Cartan elements of Lie n-algebras to characterize the relative Rota-Baxter operators on n- $LieRep$ pairs and study the cohomology and deformations of relative Rota-Baxter operators on n- $LieRep$ pairs.

The notion of an n-Lie algebra, or a Filippov algebra was introduced in [21]. It is the algebraic structure corresponding to Nambu mechanics ([38], [46]). n-Lie algebras, or more generally, n-Leibniz algebras, have attracted attention from several fields of mathematics and physics due to their close connection with dynamics, geometries as well as string theory ([3], [4], [10], [12], [16], [17], [19], [20], [30], [41]). For example, the structure of n-Lie algebras is applied to the study of the supersymmetry and gauge symmetry transformations of the word-volume theory of multiple M2-branes and the generalized identity for n-Lie algebras can be regarded as a generalized Plücker relation in the physics literature. See the review article [14] for more details.

The notion of a Lie n-algebra was introduced by Hanlon and Wachs in [29]. Lie n-algebra is a special $L_{\infty}$ -algebra, in which only the n-ary bracket is nonzero. See [15], [43], [45] for more details on Lie n-algebras and $L_{\infty}$ -algebras. One useful method for constructing $L_{\infty}$ -algebras is given by Voronov's higher derived bracket ([50]) and another one is by twisting with the Maurer-Cartan elements of a given $L_{\infty}$ -algebra ([26]). In this paper, we will use these methods to construct Lie n-algebras and $L_{\infty}$ -algebras that characterize relative Rota-Baxter operators on n-Lie algebras as Maurer-Cartan elements and deformations of relative Rota-Baxter operators.

The classical Yang-Baxter equation plays a significant role in many fields in mathematics and mathematical physics such as quantum groups ([11], [18]) and integrable systems ([44]). In order to gain better understanding of the relationship between the classical Yang-Baxter equation and the related integrable systems, the more general notion of a relative Rota-Baxter operator (also called $O$ -operator) on a $LieRep$ pair was introduced by Kupershmidt ([33]). To study solutions of 3-Lie classical Yang-Baxter equation, the notion of a relative Rota-Baxter operator on a 3- $LieRep$ pair was introduced in [5]. A relative Rota-Baxter operator on a 3- $LieRep$ pair $(g; ad)$ , where ad is the adjoint representation of the 3-Lie algebra $g$ , is exactly the Rota-Baxter operator on the 3-Lie algebra $g$ introduced in [6]. See the book [28] for more details and applications about Rota-Baxter operators.

In [48], the authors showed that a relative Rota-Baxter operator on a Lie algebra is a Maurer-Cartan element of a graded Lie algebra. Recently, it was shown in [49] that a relative Rota-Baxter operator on a 3-Lie algebra is a Maurer-Cartan element of a Lie 3-algebra. The first purpose in this paper is to realize the relative Rota-Baxter operators on n-Lie algebras as Maurer-Cartan elements of Lie n-algebras.

Pre-Lie algebras are a class of nonassociative algebras coming from the study of convex homogeneous cones, affine manifolds and affine structures on Lie groups, and the cohomologies of associative algebras. See the survey [9] and the references therein for more details. The beauty of a pre-Lie algebra is that the commutator gives rise to a Lie algebra and the left multiplication gives rise to a representation of the commutator Lie algebra. Conversely, a relative Rota-Baxter operator action on a Lie algebra gives rise to a pre-Lie algebra ([27], [33]) and thus the pre-Lie algebra can be seen as the underlying algebraic structure of a relative Rota-Baxter operator. In this paper, we introduce the notion of an n-pre-Lie algebra, which gives an n-Lie algebra naturally and its left multiplication operator gives rise to a representation of this n-Lie algebra. An n-pre-Lie algebra can also be obtained through the action of a relative Rota-Baxter operator on an n-Lie algebra.

The theory of deformation plays a prominent role in mathematics and physics. In physics, the ideal of deformation appears in the perturbative quantum field theory and quantizing classical mechanics. The idea of treating deformation as a tool to study the algebraic structures was introduced by Gerstenhaber in his work of associative algebras ([23], [24]) and then was extended to Lie algebras by Nijenhuis and Richardson ([39], [40]). Deformations of 3-Lie algebras and n-Lie algebras are studied in [1], [20], [34], [47]. See the review paper [14], [35] for more details. Recently, people pay more attention on the deformations of morphisms ([1], [22], [36], [51]), relative Rota-Baxter operators ([13], [48], [49]) and diagrams of algebras ([8], [25], [37]).

Usually the cohomology theory is an important tool in the study of deformations of a mathematical structure. Typically, infinitesimal deformations are classified by a suitable second cohomology group and the obstruction of an order m deformation extending to an order $m + 1$ deformation can be controlled by the third cohomology group. Cohomology and deformations of relative Rota-Baxter operators ( $O$ -operators) on associative algebras, Lie algebras and 3-Lie algebras were studied in [13], [48], [49].

In the present paper, we study the cohomology theory of relative Rota-Baxter operators on n- $LieRep$ pairs. By using the underlying n-Lie algebra of a relative Rota-Baxter operator, we introduce the cohomology of a relative Rota-Baxter operator on an n- $LieRep$ pair. Then we study infinitesimal deformations and order m deformations of a relative Rota-Baxter operator on an n- $LieRep$ pair. We show that infinitesimal deformations of relative Rota-Baxter operators are classified by the first cohomology group and that a higher order deformation of a relative Rota-Baxter operator is extendable if and only if its obstruction class in the second cohomology group of the relative Rota-Baxter operator is trivial.

In Section 2, we first recall representations and cohomologies of n-Lie algebras and then construct a graded Lie algebra whose Maurer-Cartan elements are precisely n- $LieRep$ pairs. In Section 3, we use Voronov's higher derived brackets to construct a Lie n-algebra from an n- $LieRep$ pair and show that relative Rota-Baxter operators on n- $LieRep$ pairs can be characterized by Maurer-Cartan elements of the constructed Lie n-algebra. We give the notion of an n-pre-Lie algebra and show that a relative Rota-Baxter operator on an n- $LieRep$ pair can give rise to an n-pre-Lie algebra naturally. In Section 4, we define the cohomology of a relative Rota-Baxter operator on an n- $LieRep$ pair using a certain n- $LieRep$ pair constructed by the relative Rota-Baxter operator. In Section 5, we use the cohomology theory of relative Rota-Baxter operators to study deformations of relative Rota-Baxter operators. In Section 6, we study the relation between the cohomology groups of relative Rota-Baxter operators on n- $LieRep$ pairs and those on $(n + 1)$ - $LieRep$ pairs by certain linear functions.

In this paper, all the vector spaces are over an algebraically closed field $K$ of characteristic 0, and finite dimensional.

Section snippets

Representations and cohomology of n-Lie algebras

Definition 2.1

([21]) An n-Lie algebra is a vector space $g$ together with a skew-symmetric linear map ${[-, \dots, -]}_{g} : \land^{n} g \to g$ such that for $x_{i}, y_{j} \in g, 1 \leq i \leq n - 1, 1 \leq j \leq n$ , the following identity holds: ${[x_{1}, \dots, x_{n - 1}, {[y_{1}, \dots, y_{n}]}_{g}]}_{g} = \sum_{i = 1}^{n} {[y_{1}, \dots, y_{i - 1}, {[x_{1}, \dots, x_{n - 1}, y_{i}]}_{g}, y_{i + 1}, \dots, y_{n}]}_{g} .$

For

x_{1}, x_{2}, \dots, x_{n - 1} \in g

, define

{ad}_{x_{1}, x_{2}, \dots, x_{n - 1}} \in gl (g)

{ad}_{x_{1}, x_{2}, \dots, x_{n - 1}} y : = {[x_{1}, x_{2}, \dots, x_{n - 1}, y]}_{g}, \forall y \in g

. Then

{ad}_{x_{1}, x_{2}, \dots, x_{n - 1}}

is a derivation, i.e.

{ad}_{x_{1}, x_{2}, \dots, x_{n - 1}} {[y_{1}, \dots, y_{n}]}_{g} = \sum_{i = 1}^{n} {[y_{1}, \dots, y_{i - 1}, {ad}_{x_{1}, x_{2}, \dots, x_{n - 1}} y_{i}, y_{i + 1}, \dots, y_{n}]}_{g} .

Definition 2.2

([32]) A representation of an n-Lie algebra $(g, {[-, \dots, -]}_{g})$ on

Maurer-Cartan characterization of relative Rota-Baxter operators on n- $LieRep$ pairs

In this section, we apply higher derived brackets introduced by Voronov in [50] to construct a Lie n-algebra that characterizes relative Rota-Baxter operators as Maurer-Cartan elements.

Cohomologies of relative Rota-Baxter operators on n- $LieRep$ pairs

Let T be a relative Rota-Baxter operator on an n- $LieRep$ pair $(g, {[-, \dots, -]}_{g}; ρ)$ . By Proposition 3.18, $(V, {[-, \dots, -]}_{T})$ is an n-Lie algebra, where the bracket ${[-, \dots, -]}_{T}$ is given by (27). Furthermore, we have

Lemma 4.1

Let T be a relative Rota-Baxter operator on an n- $LieRep$ pair $(g, {[-, \dots, -]}_{g}; ρ)$ . Define $ρ_{T} : \land^{n - 1} V \to gl (g)$ by $ρ_{T} (u_{1}, \dots, u_{n - 1}) x = {[T u_{1}, \dots, T u_{n - 1}, x]}_{g} - \sum_{i = 1}^{n - 1} {(- 1)}^{n - i} T ρ (T u_{1}, \dots, \hat{T u_{i}}, \dots, T u_{n - 1}, x) (u_{i}),$ where $u_{1}, \dots, u_{n - 1} \in V$ and $x \in g$ . Then $(g; ρ_{T})$ is a representation of the n-Lie algebra $(V, {[-, \dots, -]}_{T})$ .

Proof

Let T be a relative

Deformations of relative Rota-Baxter operators on n- $LieRep$ pairs

Let $V [[t]]$ denote the vector space of formal power series in t with coefficients in V. If in addition, $(g, {[-, \dots, -]}_{g})$ is an n-Lie algebra over $K$ , then there is an n-Lie algebra structure over the ring $K [[t]]$ on $g [[t]]$ given by ${[\sum_{j_{1} = 0}^{+ \infty} a_{1}^{j_{1}} t^{j_{1}}, \dots, \sum_{j_{n} = 0}^{+ \infty} a_{n}^{j_{n}} t^{j_{n}}]}_{g} = \sum_{s = 0}^{+ \infty} \sum_{j_{1} + \dots + j_{n} = s} {[a_{1}^{j_{1}}, \dots, a_{n}^{j_{n}}]}_{g} t^{s},$ where $a_{1}^{j_{1}}, \dots, a_{n}^{j_{n}} \in g$ .

For any representation $(V; ρ)$ of $(g, {[-, \dots, -]}_{g})$ , there is a natural representation of the n-Lie algebra $g [[t]]$ on the $K [[t]]$ -module $V [[t]]$ given by $ρ (\sum_{j_{1} = 0}^{+ \infty} a_{1}^{j_{1}} t^{j_{1}}, \dots, \sum_{j_{n - 1} = 0}^{+ \infty} a_{n - 1}^{j_{n - 1}} t^{j_{n - 1}}) ($

From cohomology groups of relative Rota-Baxter operators on n-Lie algebras to those on $(n + 1)$ -Lie algebras

Motivated by the construction of 3-Lie algebras from the general linear Lie algebras with trace forms in [31], the authors in [7] provide a construction of $(n + 1)$ -Lie algebras from n-Lie algebras and certain linear functions. See [2] for more details on constructions of n-Lie algebras.

Lemma 6.1

([7]) Let $(g, {[-, \dots, -]}_{g})$ be an n-Lie algebra and $g^{⁎}$ the dual space of $g$ . Suppose $f \in g^{⁎}$ satisfies $f ({[x_{1}, \dots, x_{n}]}_{g}) = 0$ for all $x_{i} \in g$ . Then $(g, {-, \dots, -})$ is an $(n + 1)$ -Lie algebra, where the bracket is given by ${x_{1}, \dots, x_{n + 1}} = \sum_{i = 1}^{n + 1}$

Acknowledgements

This research was supported by the National Key Research and Development Program of China (2021YFA1002000), the Fundamental Research Funds for the Central Universities (2412022QD033) and the China Postdoctoral Science Foundation (2022T150109). We give our warmest thanks to Yunhe Sheng and Rong Tang for very useful comments and discussions.

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Lie n-algebras and cohomologies of relative Rota-Baxter operators on n-Lie algebras

Abstract

Introduction

Section snippets

Representations and cohomology of n-Lie algebras

Maurer-Cartan characterization of relative Rota-Baxter operators on n-LieRep pairs

Cohomologies of relative Rota-Baxter operators on n-LieRep pairs

Deformations of relative Rota-Baxter operators on n-LieRep pairs

From cohomology groups of relative Rota-Baxter operators on n-Lie algebras to those on (n+1)-Lie algebras

Acknowledgements

J. Geom. Phys.

J. Algebra

Adv. Math.

J. Pure Appl. Algebra

J. Algebra

J. Pure Appl. Algebra

J. Algebra

Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras

J. Math. Phys.

Gauge symmetry and supersymmetry of multiple M2-branes gauge theories

Phys. Rev. D

Three-algebras and N=6 Chern-Simons gauge theories

Phys. Rev. D

Bialgebras, the classical Yang-Baxter equation and Manin triples for 3-Lie algebras

Adv. Theor. Math. Phys.

Rota-Baxter 3-Lie algebras

J. Math. Phys.

Constructing (n+1)-Lie algebras from n-Lie algebras

J. Phys. A

Deformation-obstruction theory for a Lies of algebras and applications to geometry

J. Noncommut. Geom.

Left-symmetric algebras and pre-Lie algebras in geometry and physics

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Leibniz n-algebras

Forum Math.

A Guide to Quantum Groups

Multiple M2-branes and generalized 3-Lie algebras

Phys. Rev. D

n-ary algebras: a review with applications

J. Phys. A, Math. Theor.

Higher-order simple Lie algebras

Commun. Math. Phys.

Metric Lie 3-algebras in Bagger-Lamber theory

J. High Energy Phys.

On the Lie-algebraic origin of metric 3-algebras

Commun. Math. Phys.

Quantum groups

Lorentzia Lie n-algebras

J. Math. Phys.

Deformations of 3-algebras

J. Math. Phys.

Maurer-Cartan characterization of relative Rota-Baxter operators on n- $LieRep$ pairs

Cohomologies of relative Rota-Baxter operators on n- $LieRep$ pairs

Deformations of relative Rota-Baxter operators on n- $LieRep$ pairs

From cohomology groups of relative Rota-Baxter operators on n-Lie algebras to those on $(n + 1)$ -Lie algebras

Three-algebras and $N = 6$ Chern-Simons gauge theories

Constructing $(n + 1)$ -Lie algebras from n-Lie algebras