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arXiv · 1311.6232

Classifying complements for associative algebras

A. L. Agore
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Abstract

For a given extension $A \subset E$ of associative algebras we describe and classify up to an isomorphism all $A$-complements of $E$, i.e. all subalgebras $X$ of $E$ such that $E = A + X$ and $A \cap X = \{0\}$. Let $X$ be a given complement and $(A, \, X, \, \triangleright, \triangleleft, \leftharpoonup, \rightharpoonup \bigl)$ the canonical matched pair associated with the factorization $E = A + X$. We introduce a new type of deformation of the algebra $X$ by means of the given matched pair and prove that all $A$-complements of $E$ are isomorphic to such a deformation of $X$. Several explicit examples involving the matrix algebra are provided.

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A. L. Agore. 2013-11-25. Classifying complements for associative algebras. https://doi.org/10.1016/j.laa.2014.01.012

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