Since I am not an expert in algebraic groups; Is there a description of algebraic groups with semisimple category of finite dimensional representations such that they have only finitely many irreducible representations? Maybe there is a classification?
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$\begingroup$ See the answer: mathoverflow.net/questions/155617/ $\endgroup$– abzCommented Feb 7, 2014 at 4:54
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2 Answers
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This type of question has probably come up previously on MO, but in any case I'd suggest one especially natural example arising in connection with semisimple algebraic groups or semisimple Lie algebras. This involves so-called "fusion rules", and one of several approaches gets a nice overview in a paper by H.H. Andersen and his student J. Paradowski Fusion categories arising from semisimple Lie algebras,
Comm. Math. Phys. 169 (1995), no. 3, 563–588. This comes up in influential work in mathematical physics associated with the name Verlinde, but has purely mathematical developments in connection with quantum groups at a root of unity.
Roughly speaking, the idea is to start with ordinary tensor products of finite dimensional highest weight modules, but then truncate in a certain way that leaves a semisimple category with only finitely many simple objects. Following Ringel, Donkin, Mathieu, one gets a theory of "tilting modules" for semisimple algebraic groups in prime characteristic.
answered Feb 6, 2014 at 13:36
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$\begingroup$ Out of curiosity: Are the categories arising this way actually the categories of representations for some algebraic group (my guess would be usually not)? $\endgroup$ Commented Feb 6, 2014 at 13:42
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$\begingroup$ @Tobias: In prime characteristic, you get a fusion category for a semisimple algebraic group by looking only at simple modules with highest weights in the lowest $p$-alcove or its closure; but the product is a truncated version of the standard tensor product. Naturally these are not all possible simple modules for the group, but the formalism agrees well with the case of a quantum group at a root of unity. $\endgroup$ Commented Feb 6, 2014 at 14:28
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These are the finite linearly reductive groups. In characteristic zero, they are just the finite algebraic groups.
In positive characteristic, Nagata's theorem gives the following characterization of linearly reductive groups over a field of characteristic $p$: $G$ is linearly reductive if and only if there is a finite index subgroup $H$ that is a subgroup of a torus, and the quotient $G/H$ has order prime to $p$. For our purposes, it is therefore necessary and sufficient that the diagonalizable normal subgroup $H$ be finite, and the quotient $G/H$ be finite étale and tame. By Lemma 2.11 of Tame stacks in positive characteristic (this paper also generalizes to arbitrary base schemes), there is a finite purely inseparable extension $k'$ of the base field such that $G_{k'}$ is isomorphic to the semidirect product $H_{k'} \rtimes (G/H)_{k'}$.
You can also find a discussion in Demazure-Gabriel Chapter IV section 3, Theorem 3.6.
answered Feb 6, 2014 at 14:05
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$\begingroup$ But not all of them in characterisic $p$ have finitely many irreducible representations... $\endgroup$– AleksaCommented Feb 6, 2014 at 15:15
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$\begingroup$ This characterization over fields with positive characteristic is due to Nagata from long ago (Olsson-Vistoli extend it to work over rings/schemes); this is proved in somewhat "elementary" terms somewhere in the huge book of Demazure-Gabriel, but I am unable to relocated the exact location in there at the moment. $\endgroup$ Commented Feb 6, 2014 at 16:45
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$\begingroup$ @Aleksa Okay, I've added the finiteness conditions. $\endgroup$– S. Carnahan ♦Commented Feb 7, 2014 at 0:55
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$\begingroup$ @user76758 Thanks, I've added the references. (It's near the end of DG, and I was starting to think you had sent me on a wild goose chase...) $\endgroup$– S. Carnahan ♦Commented Feb 7, 2014 at 3:07
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$\begingroup$ @S.Carnahan: Very good. Every time I try to dig up that reference I have a devil of a time tracking it down again inside DG because it is inside a section called "Commutative groups" which I skim past, thinking it cannot be in there...though the subsection name in the table of contents gives it away if one looks closely enough. $\endgroup$ Commented Feb 7, 2014 at 5:26