Birrepresentations in a locally nilpotent variety
It is known that commutative algebras satisfying the identity of degree four ((yx)x)x + γ((xx)x) = 0, with γ in the field and γ ≠ —1 are locally nilpotent. In this paper we study the birrepresentations of an algebra A that belongs to a variety ν...
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| Lenguaje: | English |
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Universidad Católica del Norte, Departamento de Matemáticas
2014
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oai:scielo:S0716-091720140001000092014-01-06Birrepresentations in a locally nilpotent varietyArenas,ManuelLabra,AliciaIt is known that commutative algebras satisfying the identity of degree four ((yx)x)x + γ((xx)x) = 0, with γ in the field and γ ≠ —1 are locally nilpotent. In this paper we study the birrepresentations of an algebra A that belongs to a variety ν of locally nilpotent algebras. We prove that if the split null extension of a birrepresentation of an algebra A ∈ ν by a vector space M is locally nilpotent, then it is trivial or reducible. As corollaries we get that if A is finitely generated, then every birrepresentation is trivial or reducible and that every finite-dimensional birrepresentation is equivalent to a birrepre-sentation consisting of strictly upper triangular matrices. We also prove that the multiplicative universal envelope of a finitely generated algebra in V is nilpotent, therefore it is finite-dimensional.info:eu-repo/semantics/openAccessUniversidad Católica del Norte, Departamento de MatemáticasProyecciones (Antofagasta) v.33 n.1 20142014-03-01text/htmlhttp://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172014000100009en10.4067/S0716-09172014000100009 |
| institution |
Scielo Chile |
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Scielo Chile |
| language |
English |
| description |
It is known that commutative algebras satisfying the identity of degree four ((yx)x)x + γ((xx)x) = 0, with γ in the field and γ ≠ —1 are locally nilpotent. In this paper we study the birrepresentations of an algebra A that belongs to a variety ν of locally nilpotent algebras. We prove that if the split null extension of a birrepresentation of an algebra A ∈ ν by a vector space M is locally nilpotent, then it is trivial or reducible. As corollaries we get that if A is finitely generated, then every birrepresentation is trivial or reducible and that every finite-dimensional birrepresentation is equivalent to a birrepre-sentation consisting of strictly upper triangular matrices. We also prove that the multiplicative universal envelope of a finitely generated algebra in V is nilpotent, therefore it is finite-dimensional. |
| author |
Arenas,Manuel Labra,Alicia |
| spellingShingle |
Arenas,Manuel Labra,Alicia Birrepresentations in a locally nilpotent variety |
| author_facet |
Arenas,Manuel Labra,Alicia |
| author_sort |
Arenas,Manuel |
| title |
Birrepresentations in a locally nilpotent variety |
| title_short |
Birrepresentations in a locally nilpotent variety |
| title_full |
Birrepresentations in a locally nilpotent variety |
| title_fullStr |
Birrepresentations in a locally nilpotent variety |
| title_full_unstemmed |
Birrepresentations in a locally nilpotent variety |
| title_sort |
birrepresentations in a locally nilpotent variety |
| publisher |
Universidad Católica del Norte, Departamento de Matemáticas |
| publishDate |
2014 |
| url |
http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172014000100009 |
| work_keys_str_mv |
AT arenasmanuel birrepresentationsinalocallynilpotentvariety AT labraalicia birrepresentationsinalocallynilpotentvariety |
| _version_ |
1718439796261519360 |