Infinite-dimensional 3-Lie algebras and their connections to Harish-Chandra modules

doi:10.1007/s11464-017-0606-7

Front. Math. China

2017, Vol. 12

Issue (3) : 515-530 https://doi.org/10.1007/s11464-017-0606-7

RESEARCH ARTICLE

Infinite-dimensional 3-Lie algebras and their connections to Harish-Chandra modules

Ruipu BAI¹, Zhenheng LI^1,²(

), Weidong WANG¹

¹. College of Mathematics and Information Science, Hebei University, Baoding 071002, China
². Department of Mathematical Sciences, University of South Carolina Aiken, Aiken, SC 29801, USA

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Abstract

We construct two kinds of infinite-dimensional 3-Lie algebras from a given commutative associative algebra, and show that they are all canonical Nambu 3-Lie algebras. We relate their inner derivation algebras to Witt algebras, and then study the regular representations of these 3-Lie algebras and the natural representations of the inner derivation algebras. In particular, for the second kind of 3-Lie algebras, we find that their regular representations are Harish-Chandra modules, and the inner derivation algebras give rise to intermediate series modules of the Witt algebras and contain the smallest full toroidal Lie algebras without center.

Keywords 3-Lie algebra Harish-Chandra module Witt algebra intermediate series module toroidal Lie algebra inner derivation algebra

Corresponding Author(s): Zhenheng LI

Issue Date: 20 April 2017

Cite this article:

Ruipu BAI,Zhenheng LI,Weidong WANG. Infinite-dimensional 3-Lie algebras and their connections to Harish-Chandra modules[J]. Front. Math. China, 2017, 12(3): 515-530.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-017-0606-7
https://academic.hep.com.cn/fmc/EN/Y2017/V12/I3/515

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