标题： $\mathbb{Z}_2^2$分级的$osp(1|2)$的仿射扩张和维拉索罗代数 显示英文标题

标题： Affine extensions of $\mathbb{Z}_2^2$-graded $osp(1|2)$ and Virasoro algebra

摘要： 已知存在两个非等价的$\mathbb{Z}_2^2$-graded$osp(1|2)$李超代数。它们的仿射扩张被研究，并表明其中一种允许两个中心元素，一个是非分次的，另一个是$(1,1)$-graded 的。仿射$\mathbb{Z}_2^2$-$osp(1|2)$代数被用于 Sugawara 构造来研究 Virasoro 代数可能的$\mathbb{Z}_2^2$-graded 扩张。我们得到一个$\mathbb{Z}_2^2$-graded 的 Virasoro 代数，其中包含一个非平凡分次的中心元素。 在整个研究过程中，$\mathbb{Z}_2^2$-分次超代数上的不变双线性形式起着关键作用，因此也发展了不变双线性形式的理论。 显示英文摘要

摘要： It is known that there are two inequivalent $\mathbb{Z}_2^2$-graded $osp(1|2)$ Lie superalgebras. Their affine extensions are investigated and it is shown that one of them admits two central elements, one is non-graded and the other is $(1,1)$-graded. The affine $\mathbb{Z}_2^2$-$osp(1|2)$ algebras are used by the Sugawara construction to study possible $\mathbb{Z}_2^2$-graded extensions of the Virasoro algebra. We obtain a $\mathbb{Z}_2^2$-graded Virasoro algebra with a non-trivially graded central element. Throughout the investigation, invariant bilinear forms on $\mathbb{Z}_2^2$-graded superalgebras play a crucial role, so a theory of invariant bilinear forms is also developed.