THE LENGTH AND DEPTH OF ASSOCIATIVE ALGEBRAS

Damian Sercombe*, Aner Shalev

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Recently there has been considerable interest in studying the length and the depth of finite groups, algebraic groups and Lie groups. We introduce and study similar notions for algebras. Let k be a field and let A be an associative, not necessarily unital, algebra over k. An unrefinable chain of A is a chain of subalgebras A = A0 > A1 > … > At = 0 for some integer t, where each Ai is a maximal subalgebra of Ai−1. The maximal (respectively, minimal) length of such an unrefinable chain is called the length (respectively, depth) of A. It turns out that finite length, finite depth and finite dimension are equivalent properties for A. For A finite dimensional, we give a formula for the length of A, we bound the depth of A, and we study when the length of A equals its dimension and its depth respectively. Finally, we investigate under what circumstances the dimension of A is bounded above by a function of its length, or its depth, or its length minus its depth.

Original languageAmerican English
Pages (from-to)197-220
Number of pages24
JournalPacific Journal of Mathematics
Volume311
Issue number1
DOIs
StatePublished - 2021

Bibliographical note

Publisher Copyright:
© 2021. Mathematical Sciences Publishers.

Keywords

  • associative algebras
  • chain conditions
  • depth
  • length

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