Quantum queer supergroups via υ-differential operators
Quantum queer supergroups via υ-differential operators
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通过 Ï-微分算子的量子酷儿超群
DOI：
10.1016/j.jalgebra.2022.01.017
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发表时间：
2022
影响因子：
0.9
通讯作者：
zhongguo Zhou
中科院分区：
文献类型：
--
作者：
Jie Du;Yanan Lin;zhongguo Zhou
利用量子微分算子，构造了量子酷儿超群Uυ(qn)的超表示。该表示的下空间是一个2n2变量的变形多项式超代数，其齐次分量可以作为酷儿q-舒尔超代数的下空间。然后，我们将其扩展到包含一个（超）的形式幂级数代数。子模块与Uυ(qn)的正则表示同构。Uυ(qn)的单项式基M在证明其同构中起着关键作用。这样，我们可以用另一种新的基L来表示量子酷儿超群Uυ(qn)，并由生成器给出一些显式的乘法公式。作为应用，通过上述齐次分量得到了酷儿q-舒尔超代数的类似表示。碱基M和L的存在以及新的表述表明，量子gln的开创性结构已经建立。三十年前由Beilinson-Lusztig-MacPherson发起的研究，通过一种完全不同的方法扩展到了这个“奇怪的”量子超群。
By using certain quantum differential operators, we construct a super representation for the quantum queer supergroup Uυ(qn). The underlying space of this representation is a deformed polynomial superalgebra in 2n2 variables whose homogeneous components can be used as the underlying spaces of queer q-Schur superalgebras. We then extend the representation to its formal power series algebra which contains a (super) .submodule isomorphic to the regular representation of Uυ(qn). A monomial basis M for Uυ(qn) plays a key role in proving the isomorphism. In this way, we may present the quantum queer supergroup Uυ(qn) by another new basis L together with some explicit multiplication formulas by the generators. As an application, similar presentations are obtained for queer q-Schur superalgebras via the above mentioned homogeneous .components. The existence of the bases M and L and the new presentation show that the seminal construction of quantum gln established .by Beilinson–Lusztig–MacPherson thirty years ago extends to this “queer” quantum supergroup via a completely different approach.