论文标题
特殊线性谎言超级对称的超级对象
Left-symmetric Superalgebras on Special Linear Lie Superalgebras
论文作者
论文摘要
在本文中，我们研究了特殊线性lie superalgebras $ {\ mathfrak {s sl}}}（m | n）$的左对称超级级别的存在和分类问题，并使用$ m \ neq n $。本文的主要三个结果是：（i）$ {\ mathfrak {sl}}}（2 | 1）$，（ii）$ {\ mathfrak {s sl}}（m | 1）$完全分类的左2级超级gera $ {\ mathfrak {sl}}（m+1 | m）$允许每$ m \ geq 1 $允许左2级超级algebra。为了证明这些结果，我们将有关左对称代数的存在和分类的现有结果结合在一起，在lie代数$ {\ mathfrak {\ mathfrak {gl}} _ m $和对Lie Superalgebras $ {\ Mathfrak {sl}}}（M | 1）$的较小表示的详细分析中。我们还推测$ {\ mathfrak {sl}}}（m | n）$在且仅当$ m = n+1 $时才允许左2级超级ger骨。
In this paper, we study the existence and classification problems of left-symmetric superalgebras on special linear Lie superalgebras ${\mathfrak{sl}}(m|n)$ with $m\neq n$. The main three results of this paper are: (i) a complete classification of the left-symmetric superalgebras on ${\mathfrak{sl}}(2|1)$, (ii) ${\mathfrak{sl}}(m|1)$ does not admit left-symmetric superalgebras for $m \geq 3$, and (iii) ${\mathfrak{sl}}(m+1|m)$ admits a left-symmetric superalgebra for every $m \geq 1$. To prove these results we combine existing results on the existence and classification of left-symmetric algebras on the Lie algebras ${\mathfrak{gl}}_m$ with a detailed analysis of small representations of the Lie superalgebras ${\mathfrak{sl}}(m|1)$. We also conjecture that ${\mathfrak{sl}}(m|n)$ admits left-symmetric superalgebras if and only if $m = n+1$.
