Bilinear maps and graphs

Let V be a linear space of arbitrary dimension and over an arbitrary base field F, endowed with a bilinear map f:V×V→V. A basis B={vi}i∈I of V is an f-basis if for any i,j∈I we have that f(vi,vj)∈Fvk for some k∈I. We associate to any triplet (V,f,B) an adequate graph (V,E). By arguing on this graph...

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Published in:Discrete Applied Mathematics 2019-06, Vol.263, p.69-78
Main Authors: Martín, Antonio J. Calderón, Izquierdo, Francisco J. Navarro
Format: Article
Language:English
Subjects:
Algebra
Bilinear map
Decomposition theorem
Graph
Linear space
Subspaces
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description Let V be a linear space of arbitrary dimension and over an arbitrary base field F, endowed with a bilinear map f:V×V→V. A basis B={vi}i∈I of V is an f-basis if for any i,j∈I we have that f(vi,vj)∈Fvk for some k∈I. We associate to any triplet (V,f,B) an adequate graph (V,E). By arguing on this graph we show that V decomposes as a direct sum of strongly f-invariant linear subspaces, each one being associated to one connected component of (V,E). Also the B-semisimplicity and the B-simplicity of V are characterized in terms of the associated graph.
doi_str_mv 10.1016/j.dam.2018.03.020
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subjects Algebra
Bilinear map
Decomposition theorem
Graph
Linear space
Subspaces
title Bilinear maps and graphs
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