Bracket width of current Lie algebras

Boris Kunyavskii; Ievgen Makedonskyi; Andriy Regeta

doi:10.1016/j.jalgebra.2025.03.023

Bracket width of current Lie algebras

Boris Kunyavskii
, Ievgen Makedonskyi
, Andriy Regeta

Department of Mathematics

Yanqi Lake Beijing Institute of Mathematical Sciences and Applications (BIMSA)
Friedrich Schiller University Jena

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

The length of an element z of a Lie algebra L is defined as the smallest number s needed to represent z as a sum of s brackets. The bracket width of L is defined as supremum of the lengths of its elements. Given a finite-dimensional simple Lie algebra g over an algebraically closed field k of characteristic zero, we study the bracket width of current Lie algebras L=g⊗A. We show that for an arbitrary A the bracket width is at most 2. For A=k[[t]] and A=k[t] we compute the bracket width for algebras isomorphic to sl_n and sp_2n.

Original language	English
Pages (from-to)	139-149
Number of pages	11
Journal	Journal of Algebra
Volume	676
DOIs	https://doi.org/10.1016/j.jalgebra.2025.03.023
State	Published - 15 Aug 2025

Bibliographical note

Publisher Copyright:
© 2025 The Author(s)

Keywords

Almost commuting variety
Bracket width
Lie algebra
Slice theorem

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10.1016/j.jalgebra.2025.03.023

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title = "Bracket width of current Lie algebras",

abstract = "The length of an element z of a Lie algebra L is defined as the smallest number s needed to represent z as a sum of s brackets. The bracket width of L is defined as supremum of the lengths of its elements. Given a finite-dimensional simple Lie algebra g over an algebraically closed field k of characteristic zero, we study the bracket width of current Lie algebras L=g⊗A. We show that for an arbitrary A the bracket width is at most 2. For A=k[[t]] and A=k[t] we compute the bracket width for algebras isomorphic to sln and sp2n.",

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AU - Makedonskyi, Ievgen

AU - Regeta, Andriy

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N2 - The length of an element z of a Lie algebra L is defined as the smallest number s needed to represent z as a sum of s brackets. The bracket width of L is defined as supremum of the lengths of its elements. Given a finite-dimensional simple Lie algebra g over an algebraically closed field k of characteristic zero, we study the bracket width of current Lie algebras L=g⊗A. We show that for an arbitrary A the bracket width is at most 2. For A=k[[t]] and A=k[t] we compute the bracket width for algebras isomorphic to sln and sp2n.

AB - The length of an element z of a Lie algebra L is defined as the smallest number s needed to represent z as a sum of s brackets. The bracket width of L is defined as supremum of the lengths of its elements. Given a finite-dimensional simple Lie algebra g over an algebraically closed field k of characteristic zero, we study the bracket width of current Lie algebras L=g⊗A. We show that for an arbitrary A the bracket width is at most 2. For A=k[[t]] and A=k[t] we compute the bracket width for algebras isomorphic to sln and sp2n.

KW - Almost commuting variety

KW - Bracket width

KW - Lie algebra

KW - Slice theorem

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JO - Journal of Algebra

JF - Journal of Algebra

ER -

Bracket width of current Lie algebras

Abstract

Bibliographical note

Keywords

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