2 results
Degenerations of Leibniz and Anticommutative Algebras
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 62 / Issue 3 / September 2019
- Published online by Cambridge University Press:
- 29 January 2019, pp. 539-549
- Print publication:
- September 2019
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- Article
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We describe all degenerations of three-dimensional anticommutative algebras
$\mathfrak{A}\mathfrak{c}\mathfrak{o}\mathfrak{m}_{3}$ and of three-dimensional Leibniz algebras 
$\mathfrak{L}\mathfrak{e}\mathfrak{i}\mathfrak{b}_{3}$ over 
$\mathbb{C}$. In particular, we describe all irreducible components and rigid algebras in the corresponding varieties.
(ANTI)COMMUTATIVE ALGEBRAS WITH A MULTIPLICATIVE BASIS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 91 / Issue 2 / April 2015
- Published online by Cambridge University Press:
- 15 December 2014, pp. 211-218
- Print publication:
- April 2015
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- Article
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A basis
${\mathcal{B}}=\{u_{i}\}_{i\in I}$ of a commutative or anticommutative algebra 
$\mathfrak{C},$ over an arbitrary base field 
$\mathbb{F}$, is called multiplicative if for any 
$i,j\in I$ we have that 
$u_{i}u_{j}\in \mathbb{F}u_{k}$ for some 
$k\in I$. We show that if a commutative or anticommutative algebra 
$\mathfrak{C}$ admits a multiplicative basis then it decomposes as the direct sum 
$\mathfrak{C}=\bigoplus _{j}\mathfrak{i}_{j}$ of well-described ideals each one of which admits a multiplicative basis. Also the minimality of 
$\mathfrak{C}$ is characterised in terms of the multiplicative basis and it is shown that, under a mild condition, the above direct sum is indexed by the family of its minimal ideals admitting a multiplicative basis.
