Varieties of associative algebras with an identity of third degree
We give a complete description of the varieties of associative algebras over a field of characteristic zero which satisfy a polynomial identity of third degree.
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arXiv subjects
Vesselin S. Drensky
Publications and source records attributed to Vesselin S. Drensky.
On varieties of Lie algebras with infinite basis property
Over an arbitrary field of positive characteristic we construct an example of a locally finite variety of Lie algebras which does not have a finite basis of its polynomial identities. As a consequence we construct varieties of Lie algebras with prescribed properties.
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Weak identities in the algebra of symmetric matrices of order two
We describe the weak polynomial identities of the Jordan algebra of symmetric $2\times 2$ matrices over a field of characteristic zero. The corresponding weak verbal ideal is generated by the standard identity of degree four and the metabelian identity.
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Weak polynomial identities for a vector space with a symmetric bilinear form
Let $V_k$ be a $k$-dimensional vector space with a non-degenerate symmetric bilinear form over a field $K$ of characteristic 0 and let $C_k$ be the Clifford algebra on $V_k$. We study the weak polynomial identities of the pair $(C_k,V_k)$. We establish that all they follow from $[x_1^2,x_2]=0$ when $k=\infty$ and from $[x_1^2,x_2]=0$ and $S_{k+1}(x_1,\ldots,x_{k+1})=0$ when $k<\infty$. We also prove that the weak identity $[x_1^2,x_2]=0$ satisfies the Specht property. As a consequence we obtain a new proof of the theorem of Razmyslov that the weak Lie polynomial identities of the pair $(M_2(K),sl_2(K))$ follow from $[x_1^2,x_2]=0$.
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