$GL_q^+(N,R)$的Gauss-Lusztig分解及q-环面表示
Gauss-Lusztig Decomposition for $GL_q^+(N,R)$ and Representation by q-Tori
摘要：
利用正的本质自伴算子构造了正量子群$GL_q^+(N,\R)$及其模偶$GL_{q\til[q]}^+(N,\R)$的表示.推广Lusztig的参数化，我们发现了一个高斯型分解的全正量子群$GL_q^+(N,\R)$的$|q|=1$，参数化的最长元素$w_0\in W=S_{N-1}$的标准分解。在这种参数化下，我们明确地找到了标准量子变量之间的关系，量子簇变量之间的关系，并利用正的本质自伴算子，用$q$ -环面$uv=q^2 vu$的非紧生成元实现了它们.模二重化自然地产生于超越关系，并且在冯·诺依曼设置中的$L^2(GL_{q\til[q]}^+(N,\R))$空间也可以被定义。
英文摘要：
We found an explicit construction of a representation of the positive quantum group $GL_q^+(N,\R)$ and its modular double $GL_{q\til[q]}^+(N,\R)$ by positive essentially self-adjoint operators. Generalizing Lusztig's parametrization, we found a Gauss type decomposition for the totally positive quantum group $GL_q^+(N,\R)$ for $|q|=1$, parametrized by the standard decomposition of the longest element $w_0\in W=S_{N-1}$. Under this parametrization, we found explicitly the relations between the standard quantum variables, the relations between the quantum cluster variables, and realizing them using non-compact generators of the $q$-tori $uv=q^2 vu$ by positive essentially self-adjoint operators. The modular double arises naturally from the transcendental relations, and an $L^2(GL_{q\til[q]}^+(N,\R))$ space in the von Neumann setting can also be defined.
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