Support varieties and representation type of small quantum groups

Feldvoss, Joerg; Witherspoon, Sarah

doi:10.48550/arxiv.0910.1383

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“…By definition, H ev (u(g), κ) = i≥0 Ext 2i u(g)#κZ 2 (κ, κ) and now u(g)#κZ 2 is an ordinary finite dimensional Hopf algebra. So this lemma is just a corollary of Proposition 2.3 in [15].…”

Section: Support Varieties and Representation Type Of Lie Superalgebrasmentioning

confidence: 82%

“…The proof is base on the estimation of the number C u(g)#κZ 2 (κ). By Proposition 2.1 in [15], we have…”

Section: Support Varieties and Representation Type Of Lie Superalgebrasmentioning

confidence: 87%

“…Let H be an arbitrary finite dimensional Hopf algebra such that C H (N ) ≥ 3 for some H-module N . Then Theorem 3.1 in [15] implies that H is wild provided H * (H, κ) is finitely generated and H * (H, N ′ ) is a Noetherian module over H * (H, κ) for any finite dimensional H-module N ′ . So the lemma is proved due to our Theorem 3.1.…”

Section: Support Varieties and Representation Type Of Lie Superalgebrasmentioning

confidence: 99%

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Support varieties and representation types for basic classical Lie superalgebras

Liu

2012

Journal of Algebra

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Let κ be an algebraically closed field of characteristic p > 3 and g a restricted Lie superalgebra over κ. We introduce the definition of restricted cohomology for g and show its cohomology ring is finitely generated provided g is a basic classical Lie superalgebra. As a consequence, we show that the restricted enveloping algebra of a basic classical Lie superalgebra g is always wild except g = sl2 or g = osp(1|2) or g = C(2). All finite dimensional indecomposable restricted representations of u(osp(1|2)), the restricted enveloping algebra of Lie superalgebra osp(1|2), are determined.where n → n −1 ⊗ n 0 , N → J ⊗ N denotes the comodule structure, as usual. Let A be a braided Hopf algebra in J J Y D. By definition, it is an algebra as well as coalgebra in J J Y D such that its comultiplication and counit are algebra morphism, and such that the identity morphism has a convolution inverse in J J Y D. When we say that the comultiplication ∆ : A → A ⊗ A should be an algebra morphism, the braiding defined as above arises in the definition of the algebra structure of A ⊗ A and so A is not an ordinary Hopf algebra in general. Through the Radford-Majid bosonization [21,27], it gives rise to an ordinary Hopf algebra A ⋊ J. As an algebra, this is the smash product A#J, and it is the smash coproduct as a coalgebra. Lemma 2.1. Let J be a Hopf algebra with bijective antipode and A a braided Hopf algebra in J J Y D. Then the cohomology ring H * (A, κ) := i≥0 Ext i A (κ, κ) is a braided graded commutative algebra in J J Y D. 4 GONGXIANG LIU Proof. By Theorem 3.12 in [23], the Hochschild cohomology ring HH * (A, κ) := i≥0 Ext i A⊗A op (A, κ)is a braided graded commutative algebra in J J Y D. By the standard bar resolution for computing these extension groups, one can see that Ext i A (κ, κ) ∼ = Ext i A⊗A op (A, κ) for i ≥ 0 (see also subsection 2.4 in [23]). The proof is complete.2.2. Cohomology of restricted Lie superalgebras. We fix some notions at first. By definition, a superalgebra is nothing but a Z 2 -graded algebra. By forgetting the grading we may consider any superalgebra A as a usual algebra and this algebra will be denoted by |A|. For any two Z 2 -graded vector spaces V, W , we use Hom κ (V, W ) to represent the set of all linear maps from V to W and Hom κ (V, W ) to denote that of all even linear maps. Now let A = A 0 ⊕ A 1 be a superalgebra. Then there is a natural action of Z 2 = g|g 2 = 1 on A given byNote that this definition makes sense as stated only for homogeneous elements, it should be interpreted via linearity in the general case. Thus A is a κZ 2 -module algebra (for definition, see Section 4.1 in [25]) and the smash product A#κZ 2 is a usual algebra. We use A-smod to denote the category of all finitely generated left A-supermodules with even homomorphisms and A#κZ 2 -mod the usual finitely generated left A#κZ 2 -modules category.Lemma 2.2. Let A be a superalgebra. Then A-smod is equivalent to A#κZ 2mod.Proof. Let M = M 0 ⊕ M 1 be an A-supermodule and g ∈ Z 2 the generator of Z 2 . Through assigning g · m...

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Section: Support Varieties and Representation Type Of Lie Superalgebrasmentioning

confidence: 82%

“…The proof is base on the estimation of the number C u(g)#κZ 2 (κ). By Proposition 2.1 in [15], we have…”

Section: Support Varieties and Representation Type Of Lie Superalgebrasmentioning

confidence: 87%

Section: Support Varieties and Representation Type Of Lie Superalgebrasmentioning

confidence: 99%

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Support varieties and representation types for basic classical Lie superalgebras

Liu

2012

Journal of Algebra

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Support varieties and representation type of small quantum groups

Cited by 1 publication

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Support varieties and representation types for basic classical Lie superalgebras

Support varieties and representation types for basic classical Lie superalgebras

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