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Extensions of finite quantum groups by finite groups. (English) Zbl 1187.17006

In a previous work, the authors determined all Hopf algebra quotients of \(O_q(G)\), the complex form of the quantized coordinate algebra of \(G\), \(q\) a primitive \(l\)-th root of 1, where \(G\) is a connected simply connected complex simple Lie group, and \(l\) is an odd integer at least 3 which is relatively prime to 3 if \(G\) contains a \(G_2\)-component [“Quantum subgroups of a simple quantum group at roots of 1”, Compos. Math. (to appear), http://arxiv.org/abs/0707.0070]. A byproduct was the discovery of many new examples of Hopf algebras of finite dimension or of finite Gelfand-Kirillov dimension.
The paper under review studies the question of isomorphisms between the Hopf algebra quotients of the previous work, and also whether they are “new”, i.e., neither semisimple, nor pointed, nor dual to pointed. The Hopf quotients \(H\) are presented as central extensions \(1\to K\to A\to H\to 1\), where \(K\) is the Hopf center of \(A\) and \(H\) has trivial Hopf center. Determining when two such \(H\) are isomorphic is generally difficult. Here it is answered assuming that \(A\) is Noetherian and \(H\)-Galois over \(K\), and that any Hopf algebra isomorphism of \(H\) lifts to \(A\).
Whether all the quotients of the \(O_q(G)\) satisfy the lifting condition is not known, but the authors verify their conditions for many of the quotients of the \(O_q(G)\), and classify those up to isomorphism. They use this to show that there are infinitely many nonisomorphic Hopf algebras of the same dimension, and that they form a family of nonsemisimple, nonpointed Hopf algebras with nonpointed duals. For \(G=\text{SL}_2\), such an infinite family was obtained by E. Müller [Proc. Lond. Math. Soc. (3) 81, No. 1, 190–210 (2000; Zbl 1030.20030)].

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
20G42 Quantum groups (quantized function algebras) and their representations
16T05 Hopf algebras and their applications

Citations:

Zbl 1030.20030
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References:

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