Abelian and non-abelian second cohomologies of quantized enveloping algebras

Abstract. We complete the classification of Hopf algebras whose infinitesimal braiding is a principal Yetter-Drinfeld realization of a braided vector space of Cartan type G2 over a cosemisimple Hopf algebra.We develop a general formula for a class of liftings in which the quantum Serre relations hold. We give a detailed explanation of the procedure for finding the relations, based on the recent work of Andruskiewitsch, Angiono and Rossi Bertone. IntroductionThis article belongs to the series initiated in [AAG], and followed by [AG2], with the purpose of computing all liftings of braided vector spaces of diagonal type (V, c) whose Nichols algebra B(V ) is finite-dimensional.In this part we focus on:That is, we assume there are N > 3, q a primitive N -th root of 1 and a basis {x 1 , x 2 } of V such that the braiding is determined by a matrix q = q q 12 q 21 q 3 ∈ k 2×2 with q 12 q 21 = q −3 as:• H a cosemisimple Hopf algebra with a principal realizationWe recall that a Hopf algebra L is called a lifting of V ∈ H H YD when gr L ≃ B(V )#H. We refer the reader to the article [AAG], the first of the series, based on [A+], for a description of the program for computing all liftings of braided vector spaces of diagonal type with a realization V ∈ H H YD. Set B = B(V )#H. In a sentence, the program consists in constructing a subset Cleft ′ (B) ⊆ Cleft(B) of right cleft objects for B in such a way that for every X ∈ Cleft ′ (B) the left Schauenburg Hopf algebra A = L(X, B), see [S], satisfies gr A ≃ B and checking that indeed, every lifting can be obtained in this way.2000 Mathematics Subject Classification. 16W30.

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“…If H = kΓ, then this is [AS2, Classification Theorem 0.1], using [A2,Theorem 2]. This also extends [M,Theorem 7.8].…”

Section: Liftings Of Generic Cartan Typementioning

confidence: 60%

Liftings of Nichols algebras of diagonal type III. Cartan type G2

Iglesias

Giraldi

2017

Journal of Algebra

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“…Conjectures. If g = sl 2 (C) and ε ∈ C g , then it follows from Poincaré duality and [Mas,Remark 7.17] that H…”

Section: Roots Of Unitymentioning

confidence: 99%

Cohomology rings for quantized enveloping algebras

Drupieski

2013

Proc. Amer. Math. Soc.

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Abstract. We compute the structure of the cohomology ring for the quantized enveloping algebra (quantum group) Uq associated to a finite-dimensional simple complex Lie algebra g. We show that the cohomology ring is generated as an exterior algebra by homogeneous elements in the same odd degrees as generate the cohomology ring for the Lie algebra g. Partial results are also obtained for the cohomology rings of the non-restricted quantum groups obtained from Uq by specializing the parameter q to a non-zero value ε ∈ C.

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“…It is well-known that the category of representations of H = B(V )#kC n is tensor equivalent to Rep(H q ), see for example [19]. Therefore we shall describe module categories over Rep(H ).…”

Section: Module Categories Over Rep(u Q (Sl 2 ))mentioning

confidence: 99%