标题： 与$U_q(gl_n)$和$U_q(sl_n)$相关的量子李代数 显示英文标题

标题： Quantum Lie algebras associated to $U_q(gl_n)$ and $U_q(sl_n)$

摘要： 量子李代数$\qlie{g}$是非结合代数，它们被嵌入到德林费尔德和吉姆布的量子包络代数$U_q(g)$中的方式与普通李代数被嵌入到它们的包络代数中的方式相同。 $\qlie{g}$上的量子李括号由$U_q(g)$的量子伴随作用诱导。 我们构造了与$U_q(gl_n)$和$U_q(sl_n)$相关的量子李代数。 我们确定了结构常数和量子根系，这些现在都是量子参数$q$的函数。 它们在$q\leftrightarrow 1/q$下表现出有趣的对偶对称性。 显示英文摘要

摘要： Quantum Lie algebras $\qlie{g}$ are non-associative algebras which are embedded into the quantized enveloping algebras $U_q(g)$ of Drinfeld and Jimbo in the same way as ordinary Lie algebras are embedded into their enveloping algebras. The quantum Lie product on $\qlie{g}$ is induced by the quantum adjoint action of $U_q(g)$. We construct the quantum Lie algebras associated to $U_q(gl_n)$ and $U_q(sl_n)$. We determine the structure constants and the quantum root systems, which are now functions of the quantum parameter $q$. They exhibit an interesting duality symmetry under $q\leftrightarrow 1/q$.