论文标题
Drinfeld-Jimbo代数及其复杂分析形式的Banach空间表示
Banach space representations of Drinfeld-Jimbo algebras and their complex-analytic forms
论文作者
论文摘要
我们证明，在$ | q | | q | q | q | q | q | q | \ ne 1 $时，$ | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | \ ne 1 $时，$ | mathfrak $ | \ ne ne ne emente dimensions of drinfeld-jimbo代数$ u_q（\ mathfrak {g}）$的每个非脱位Banach空间表示。作为推论，我们找到了$ u_q（\ mathfrak {g}）$的Arens-Michael信封的明确形式，该形式与Joseph Taylor在70年代获得的$ U（\ Mathfrak {G}）$相似。在$ \ mathfrak {g} = \ mathfrak {s} \ mathfrak {l} _2 $的情况下，我们还考虑了相应的分析形式的表示理论$ \ wideTilde u（\ m mathfrak {\ s} \ mathfrak {s} \ mathfrak {l} _2 _2 _2 _2 _2 _2）_ \ hbar $（yis $ e^y $ e^with $ e^）对于$ U_Q（\ Mathfrak {s} \ Mathfrak {l} _2）$。例如，$ \ widetilde u的所有不可约连续表示（\ Mathfrak {s} \ Mathfrak {l} _2 _2）_ \ hbar $对于复杂参数$ \ hbar $的每个可允许的值是有限的维度，而$ \ hbar $，而$ u_q（\ mathfrak $ usfrak $ s t op toperagionally”当$ | q | = 1 $和$ q $的不可约无限维表示不是统一的根源。
We prove that every non-degenerate Banach space representation of the Drinfeld-Jimbo algebra $U_q(\mathfrak{g})$ of a semisimple complex Lie algebra $\mathfrak{g}$ is finite dimensional when $|q|\ne 1$. As a corollary, we find an explicit form of the Arens-Michael envelope of $U_q(\mathfrak{g})$, which is similar to that of $U(\mathfrak{g})$ obtained by Joseph Taylor in 70s. In the case when $\mathfrak{g}=\mathfrak{s}\mathfrak{l}_2$, we also consider the representation theory of the corresponding analytic form $\widetilde U(\mathfrak{s}\mathfrak{l}_2)_\hbar$ (with $e^\hbar=q$) and show that it is simpler than for $U_q(\mathfrak{s}\mathfrak{l}_2)$. For example, all irreducible continuous representations of $\widetilde U(\mathfrak{s}\mathfrak{l}_2)_\hbar$ are finite dimensional for every admissible value of the complex parameter $\hbar$, while $U_q(\mathfrak{s}\mathfrak{l}_2)$ has a topologically irreducible infinite-dimensional representation when $|q|= 1$ and $q$ is not a root of unity.
