Generalization and exact deformations of quantum groups
Generalization and exact deformations of quantum groups
复制标题
量子群的推广和精确变形
DOI：
10.2977/prims/1195145535
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发表时间：
1996
影响因子：
1.2
通讯作者：
C. Frønsdal
中科院分区：
文献类型：
--
作者：
C. Frønsdal
研究了一类“标准”上边界Hopf代数.当参数处于一般位置时，证明了泛R-矩阵的存在性。参数空间中的代数曲面的特征在于某些理想的出现;在这种情况下，泛R-矩阵存在于相关的代数商上。在特殊情况下，商是一个“标准”量子群;所有熟悉的量子群，包括扭曲的量子群，都是这样得到的。在其他特殊情况下，人们发现新的类型的共边界双代数。“标准”的通用R-矩阵是一个非常简单的线性递归关系的唯一解决方案。在有限仿射型量子化Kac-Moody代数的情况下，得到了经典极限。回到一般情况下，我们研究的标准R-矩阵和相关的Hopf代数的变形。初步调查的一阶变形揭示了一类变形incompass量子化的所有Kac-Moody代数的有限和仿射型。相应的精确变形被描述为广义扭曲，$ R_\n =（F^t）^{-1}RF$，其中$R$是标准R-矩阵，上循环$F$（变形参数$\n $中的幂级数）是与确定$R$的线性递归关系相同类型的线性递归关系的解。这里包含了与$sl（n）$相关的椭圆量子群的普适R-矩阵，这是一个很大的惊喜！再次专门化，量子化卡茨-穆迪代数的情况下，并采取这些深奥的量子群的经典极限，一个重新发现所有的三角和椭圆r-矩阵的贝拉文和德林费尔德。所得公式比原来的公式更易于使用，而且经典r-矩阵空间的结构也更透明。这里得到的r-矩阵是更一般的，因为它们被定义在完整的Kac-Moody代数上，循环群的中心扩张。
A large family of "standard" coboundary Hopf algebras is investigated. The existence of a universal R-matrix is demonstrated for the case when the parameters are in general position. Algebraic surfaces in parameter space are characterized by the appearance of certain ideals; in this case the universal R-matrix exists on the associated algebraic quotient. In special cases the quotient is a "standard" quantum group; all familiar quantum groups including twisted ones are obtained in this way. In other special cases one finds new types of coboundary bi-algebras. The "standard" universal R-matrix is shown to be the unique solution of a very simple, linear recursion relation. The classical limit is obtained in the case of quantized Kac-Moody algebras of finite and affine type. Returning to the general case, we study deformations of the standard R-matrix and the associated Hopf algebras. A preliminary investigation of the first order deformations uncovers a class of deformations that incompasses the quantization of all Kac-Moody algebras of finite and affine type. The corresponding exact deformations are described as generalized twists, $ R_\epsilon = (F^t)^{-1}RF$, where $R$ is the standard R-matrix and the cocycle $F$ (a power series in the deformation parameter $\epsilon$) is the solution of a linear recursion relation of the same type as that which determines $R$. Included here is the universal R-matrix for the elliptic quantum groups associated with $sl(n)$, a big surprise! Specializing again, to the case of quantized Kac-Moody algebras, and taking the classical limit of these esoteric quantum groups, one re-discovers all the trigonometric and elliptic r-matrices of Belavin and Drinfeld. The formulas obtained here are easier to use than the original ones, and the structure of the space of classical r-matrices is more transparent. The r-matrices obtained here are more general in that they are defined on the full Kac-Moody algebras, the central extensions of the loop groups.