The geometric reductivity of the quantum group $SL_q(2)$

Michał Kępa ¹   ; Andrzej Tyc ¹

¹ Faculty of Mathematics and Computer Sciences N. Copernicus University Chopina 12/18 87-100 Toruń, Poland

Colloquium Mathematicum, Tome 124 (2011) no. 2, pp. 169-190

Résumé

We introduce the concept of geometrically reductive quantum group which is a generalization of the Mumford definition of geometrically reductive algebraic group. We prove that if $G$ is a geometrically reductive quantum group and acts rationally on a commutative and finitely generated algebra $A$, then the algebra of invariants $A^G$ is finitely generated. We also prove that in characteristic $0$ a quantum group $G$ is geometrically reductive if and only if every rational $G$-module is semisimple, and that in positive characteristic every finite-dimensional quantum group is geometrically reductive. Both the concept of geometrically reductive quantum group and the above mentioned theorems are formulated in the language of Hopf algebras and generalize the results of Borsai and Ferrer Santos. The main theorem of the paper says that in positive characteristic the quantum group $SL_q(2)$ is geometrically reductive for any parameter $q$.

Détail

DOI : 10.4064/cm124-2-3

Keywords: introduce concept geometrically reductive quantum group which generalization mumford definition geometrically reductive algebraic group prove geometrically reductive quantum group acts rationally commutative finitely generated algebra algebra invariants finitely generated prove characteristic quantum group geometrically reductive only every rational g module semisimple positive characteristic every finite dimensional quantum group geometrically reductive concept geometrically reductive quantum group above mentioned theorems formulated language hopf algebras generalize results borsai ferrer santos main theorem paper says positive characteristic quantum group geometrically reductive parameter nbsp