论文标题
R-Matrix形式主义，用于量化代数
The R-matrix formalism for quantized enveloping algebras
论文作者
论文摘要
令$ u_ \ hbar \ mathfrak {g} $表示与复杂的semisimple lie lie algebra $ \ mathfrak {g} $相关的Drinfeld-Jimbo量子组。我们将$ r $ -Matrix构造的修改用于量子组的$ r $ r $ -matrix $ u_ \ hbar \ hbar \ mathfrak {g} $的通用$ r $ -matrix在其任何有限量表表示的张量广场上。这会产生一个量化的包络代数$ \ mathrm {u_r}（\ mathfrak {g}）$，其定义是根据两个生成矩阵满足众所周知的$ rll $ resptions的变体的定义。我们证明$ \ mathrm {u_r}（\ Mathfrak {g}）$是borel subalgebra $ u_ \ u_ \ hbar \ hbar \ hbar \ hbar \ hbar \ mathfrak {b} \ subset U_ \ subset u_ \ hbar \ hbar \ hbar \ mathfra和一个量子的量子的量子的量子量的张量同构的。 $ \ mathfrak {g} $ - 与$ u_ \ hbar \ hbar \ mathfrak {g} $的基础有限维表示的半经典限制$ v $相关的不变。使用此描述，我们将$ u_ \ hbar \ Mathfrak {g} $和$ u_ \ hbar \ hbar \ Mathfrak {b} $的量子双重量为$ \ mathrm {u_r {u_r}（\ mathfrak {g}）$的hopf商，并作为固定点subalgebras inseflass insely witly witlys to sysermos of sysermors of autos autos autos saip of saip of sausiss。作为另一种推论，我们推断出$ \ mathrm {u_r}（\ mathfrak {g}）$是quasitriangular，而当$ v $的不可还原的总和是不同的。
Let $U_\hbar\mathfrak{g}$ denote the Drinfeld-Jimbo quantum group associated to a complex semisimple Lie algebra $\mathfrak{g}$. We apply a modification of the $R$-matrix construction for quantum groups to the evaluation of the universal $R$-matrix of $U_\hbar\mathfrak{g}$ on the tensor square of any of its finite-dimensional representations. This produces a quantized enveloping algebra $\mathrm{U_R}(\mathfrak{g})$ whose definition is given in terms of two generating matrices satisfying variants of the well-known $RLL$ relations. We prove that $\mathrm{U_R}(\mathfrak{g})$ is isomorphic to the tensor product of the quantum double of the Borel subalgebra $U_\hbar\mathfrak{b}\subset U_\hbar\mathfrak{g}$ and a quantized polynomial algebra encoded by the space of $\mathfrak{g}$-invariants associated to the semiclassical limit $V$ of the underlying finite-dimensional representation of $U_\hbar\mathfrak{g}$. Using this description, we characterize $U_\hbar\mathfrak{g}$ and the quantum double of $U_\hbar\mathfrak{b}$ as Hopf quotients of $\mathrm{U_R}(\mathfrak{g})$ and as fixed-point subalgebras with respect to certain natural automorphisms. As an additional corollary, we deduce that $\mathrm{U_R}(\mathfrak{g})$ is quasitriangular precisely when the irreducible summands of $V$ are distinct.
