AI 中文总结
该研究围绕格拉斯曼代数相关的涅夫 - 施瓦茨李超代数展开，通过定义其格拉斯曼化及复化，明确描述相关超群和半群，在参数不等式条件下，将超群的酉表示扩展到半群，且不要求读者有相关先验知识。
AI 中文摘要
用\({\mathcal A}\)表示具有可数个生成元的格拉斯曼代数，\({\mathcal A}_{\overline 0}\)、\({\mathcal A}_{\overline 1}\)为其偶部和奇部。将超群视为\(\mathcal A\)上的群。考虑涅夫 - 施瓦茨李超代数\(\mathfrak{ns}= \mathfrak{ns}_{\overline 0}\oplus \mathfrak{ns}_{\overline 1}\)及其格拉斯曼化\(\mathfrak{ns}(\mathcal{A})\)和相应超群\(\mathrm{NS}(\mathcal {A})\)，并以不同方式明确描述该群。接着定义\(\mathrm{NS}({\mathcal A})\)的复化\(\Gamma({\mathcal A})\)，它是一个半群，其元素是配备接触结构的\(1|1\)维超环面，乘法是此类超环面的胶合。在参数的一些不等式下，证明\(\mathrm{NS}({\mathcal A})\)的酉表示可扩展为\(\Gamma(\mathcal {A})\)的表示。不假定读者有李超代数和超群的先验知识。
英文摘要
Denote by ${\mathcal A}$ the Grassmann algebra with a countable number of generators, by ${\mathcal A}_{\overline 0}$, ${\mathcal A}_{\overline 1}$ its even and odd parts. We consider supergroups as groups over $\mathcal A$. Consider the Neveu--Schwarz Lie superalgebra $\mathfrak{ns}= \mathfrak{ns}_{\overline 0}\oplus \mathfrak{ns}_{\overline 1}$. Consider its Grassmannization $\mathfrak{ns}(\mathcal{A}):= (\mathfrak{ns}_{\overline 0}\otimes {\mathcal A}_{\overline 0}) \oplus (\mathfrak{ns}_{\overline 1}\otimes {\mathcal A}_{\overline 1})$ and the corresponding supergroup $\mathrm{NS}(\mathcal {A})$. We describe this group explicitly in different ways. Next, we define a complexification $Γ({\mathcal A})$ of $\mathrm{NS}({\mathcal A})$. It is a semigroup, whose elements are superannuli of dimension $1|1$ equipped with contact structures; the multiplication is gluing of such superannuli. Under some inequalities for parameters, we show that unitary representations of $\mathrm{NS}({\mathcal A})$ admit extensions to representations of $Γ(\mathcal {A})$. We do not assume that the reader has prior knowledge of Lie superalgebras and supergroups.
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