Abstract
We prove that every nondegenerate Banach space representation of the Drinfeld–Jimbo algebra of a semisimple complex Lie algebra is finite dimensional when . As a corollary, we find an explicit form of the Arens–Michael envelope of , which is similar to that of obtained by Joseph Taylor in 1970s. In the case when , we also consider the representation theory of the corresponding analytic form, the Arens–Michael algebra (with ) and show that it is simpler than for . For example, all irreducible continuous representations of are finite dimensional for every admissible value of the complex parameter ℏ, while has a topologically irreducible infinite-dimensional representation when and q is not a root of unity.
Dedication
To the memory of Majya Zhegalova
Citation
O. Yu. Aristov. "Banach space representations of Drinfeld–Jimbo algebras and their complex-analytic forms." Illinois J. Math. 67 (2) 363 - 382, June 2023. https://doi.org/10.1215/00192082-10592466
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