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Register a new userOn simple Filippov superalgebras of type A(0,n)
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It is proved that there exist no simple finite-dimensional Filippov superalgebras of type A(0,n) over an algebraically closed field of characteristic 0.
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It is proved that there exist no simple finite-dimensional Filippov superalgebras of type A(m,n) over an algebraically closed field of characteristic 0.
In this paper we investigate the description of the complex Leibniz superalgebras with nilindex n+m, where n and m ($m eq 0$) are dimensions of even and odd parts, respectively. In fact, such superalgebras with characteristic sequence equal to $(n_1, ..., n_k | m_1, ..., m_s)$ (where $n_1+... +n_k=n, m_1+ ... + m_s=m$) for $n_1geq n-1$ and $(n_1, ..., n_k | m)$ were classified in works cite{FilSup}--cite{C-G-O-Kh1}. Here we prove that in the case of $(n_1, ..., n_k| m_1, ..., m_s)$, where $n_1leq n-2$ and $m_1 leq m-1$ the Leibniz superalgebras have nilindex less than n+m. Thus, we complete the classification of Leibniz superalgebras with nilindex n+m.
We present the description up to isomorphism of Leibniz superalgebras with characteristic sequence $(n|m_1,...,m_k)$ and nilindex $n+m,$ where $m=m_1+ >...+m_k,$ $n$ and $m$ ($m eq 0$) are dimensions of even and odd parts, respectively.
In this work we investigate the complex Leibniz superalgebras with characteristic sequence $(n_1,...,n_k|m)$ and nilindex n+m, where $n=n_1+...+n_k,$ n and m (m is not equal to zero) are dimensions of even and odd parts, respectively. Such superalgebras with condition n_1 > n-2 were classified in cite{FilSup}--cite{C-G-O-Kh}. Here we prove that in the case $n_1 < n-1$ the Leibniz superalgebras have nilindex less than $n+m.$ Thus, we get the classification of Leibniz superalgebras with characteristic sequence $(n_1, ...,n_k|m)$ and nilindex n+m.
We consider the variety of Filippov ($n$-Lie) algebra structures on an $(n+1)$-dimensional vector space. The group $GL_n(K)$ acts on it, and we study the orbit closures with respect to the Zariski topology. This leads to the definition of Filippov algebra degenerations. We present some fundamental results on such degenerations, including trace invariants and necessary degeneration criteria. Finally, we classify all orbit closures in the variety of complex $(n+1)$-dimensional Filippov $n$-ary algebras.
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