$D(2,1;a)$型李超群的奇异退化
Singular Degenerations of Lie Supergroups of Type $D(2,1;a)$
摘要：
$D(2,1;a)$型的复李超代数$\mathfrak{g}$-也记作$\mathfrak{osp}(4,2;a) $ -通常被认为是参数$a$的“非奇异”值，因此它们是单的。在本文中，我们介绍了五个合适的积分形式的$\mathfrak{g}$，这是良好的定义在奇异值，引起“奇异专业化”，不再简单：这扩展了家庭的简单对象类型$D(2,1;a)$在五个不同的方式。由此产生的五个家庭符合一般值的$a$，但不同的“奇异”的：在这里，他们提供了非单李超代数，其结构，我们明确描述。我们还进行了平行建设复杂的李超群和描述其奇异的专业化（或“退化”）在奇异值$a$。虽然人们可以用单个复参数$a$来工作，但为了强调整个情形的总体$\mathfrak{S}_3$对称性，我们将（遵循卡普兰斯基）用复仿射平面$σ_1 + σ_2 + σ_3 = 0$中的二维参数$\boldsymbolσ = (σ_1,σ_2,σ_3)$来工作。
英文摘要：
The complex Lie superalgebras $\mathfrak{g}$ of type $D(2,1;a)$ - also denoted by $\mathfrak{osp}(4,2;a) $ - are usually considered for "non-singular" values of the parameter $a$, for which they are simple. In this paper we introduce five suitable integral forms of $\mathfrak{g}$, that are well-defined at singular values too, giving rise to "singular specializations" that are no longer simple: this extends the family of simple objects of type $D(2,1;a)$ in five different ways. The resulting five families coincide for general values of $a$, but are different at "singular" ones: here they provide non-simple Lie superalgebras, whose structure we describe explicitly. We also perform the parallel construction for complex Lie supergroups and describe their singular specializations (or "degenerations") at singular values of $a$. Although one may work with a single complex parameter $a$, in order to stress the overall $\mathfrak{S}_3$-symmetry of the whole situation, we shall work (following Kaplansky) with a two-dimensional parameter $\boldsymbolσ = (σ_1,σ_2,σ_3)$ ranging in the complex affine plane $σ_1 + σ_2 + σ_3 = 0$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
