Maximal subalgebras of finite-dimensional algebras

Miodrag Cristian Iovanov; Alexander Harris Sistko

doi:10.1515/forum-2019-0033

Article

Maximal subalgebras of finite-dimensional algebras

Miodrag Cristian Iovanov and Alexander Harris Sistko

Published/Copyright: June 14, 2019

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From the journal Forum Mathematicum Volume 31 Issue 5

Abstract

We study maximal associative subalgebras of an arbitrary finite-dimensional associative algebra B over a field 𝕂 and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case and then lifting to non-semisimple algebras. The results are sharpest in the case of algebraically closed fields and take special forms for algebras presented by quivers with relations. We also relate representation theoretic properties of the algebra and its maximal and other subalgebras and provide a series of embeddings between quivers, incidence algebras and other structures which relate indecomposable representations of algebras and some subalgebras via induction/restriction functors. Some results in literature are also re-derived as a particular case, and other applications are given.

Keywords: Maximal subalgebra; semisimple algebra; separable functor; separable; split; split-by-nilpotent

MSC 2010: 16S99; 16G60; 16G10; 16S50; 16W20

Communicated by Freydoon Shahidi

Acknowledgements

The authors would like to thank Ryan Kinser for a careful reading of a preliminary version of this paper and many useful suggestions which improved the paper; they would also like to thank Victor Camillo for encouraging discussions and suggesting a few additional references.

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Received: 2019-02-05

Published Online: 2019-06-14

Published in Print: 2019-09-01

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https://doi.org/10.1515/forum-2019-0033

Keywords for this article

Maximal subalgebra; semisimple algebra; separable functor; separable; split; split-by-nilpotent