Reflection equation algebras, coideal subalgebras, and their centres
Reflection equation algebras, coideal subalgebras, and their centres
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DOI：
10.1007/s00029-009-0007-1
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发表时间：
2008-12
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影响因子：
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通讯作者：
S. Kolb;J. Stokman
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文献类型：
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作者：
S. Kolb;J. Stokman
反射方程代数和相关余模代数出现在量子齐次空间的各种构造中，并且可以通过嬗变或等效地通过余循环扭转来获得。在本文中，我们研究了这种所谓的“协变”代数的代数和表示理论性质，特别是它们的中心、不变量和特征。相对于左伴随作用的局部有限部分是协变代数的一个特殊例子。概括 Noumi 的量子对称对的构造，我们为协变代数的每个特征定义了一个共理想子代数 B。我们证明，对于中心 Z(Bf) 的任何特征，规范地包含半单李代数的表示环。此外，我们还表明，对于此类特征，可以从反射方程的任何可逆解构造出来，因此我们获得了许多新的内部显式实现。作为一个例子，我们讨论对应于 m 维子空间的格拉斯曼流形 Gr(m,2m) 的反射方程的解。
Reflection equation algebras and related-comodule algebras appear in various constructions of quantum homogeneous spaces and can be obtained via transmutation or equivalently via twisting by a cocycle. In this paper we investigate algebraic and representation theoretic properties of such so called ‘covariantized’ algebras, in particular concerning their centres, invariants, and characters. The locally finite partofwith respect to the left adjoint action is a special example of a covariantized algebra. Generalising Noumi’s construction of quantum symmetric pairs we define a coideal subalgebraBfoffor each characterfof a covariantized algebra. We show that for any characterfofthe centreZ(Bf) canonically contains the representation ringof the semisimple Lie algebra. We show moreover that forsuch characters can be constructed from any invertible solution of the reflection equation and hence we obtain many new explicit realisations ofinside. As an example we discuss the solutions of the reflection equation corresponding to the Grassmannian manifoldGr(m,2m) ofm-dimensional subspaces in.