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Register a new userOn the computation of Hopf 2-cocycles, with an example of diagonal type
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We present a framework for the computation of the Hopf 2-cocycles involved in the deformations of Nichols algebras over semisimple Hopf algebras. We write down a recurrence formula and investigate the extent of the connection with invariant Hochschild cohomology in terms of exponentials. As an example, we present detailed computations leading to the explicit description of the Hopf 2-cocycles involved in the deformations of a Nichols algebra of Cartan type $A_2$ with $q=-1$, a.k.a. the positive part of the small quantum group $mathfrak{u}^+_{sqrt{text{-1}}}(mathfrak{sl}_3)$. We show that these cocycles are generically pure, that is they are not cohomologous to exponentials of Hochschild 2-cocycles.
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We generalize the theory of the second invariant cohomology group $H^2_{rm inv}(G)$ for finite groups $G$, developed in [Da2,Da3,GK], to the case of affine algebraic groups $G$, using the methods of [EG1,EG2,G]. In particular, we show that for connected affine algebraic groups $G$ over an algebraically closed field of characteristic $0$, the map $Theta$ from [GK] is bijective (unlike for some finite groups, as shown in [GK]). This allows us to compute $H^2_{rm inv}(G)$ in this case, and in particular show that this group is commutative (while for finite groups it can be noncommutative, as shown in [GK]).
The quiver Hopf algebras are classified by means of ramification systems with irreducible representations. This leads to the classification of Nichols algebras over group algebras and pointed Hopf algebras of type one.
Let H be a non-semisimple Hopf algebra of dimension 2p^2 over an algebraically closed field of characteristic zero, where p is an odd prime. We prove that H or H^* is pointed, which completes the classification for Hopf algebras of these dimensions.
Let $V$ be a braided vector space of diagonal type. Let $mathfrak B(V)$, $mathfrak L^-(V)$ and $mathfrak L(V)$ be the Nichols algebra, Nichols Lie algebra and Nichols braided Lie algebra over $V$, respectively. We show that a monomial belongs to $mathfrak L(V)$ if and only if that this monomial is connected. We obtain the basis for $mathfrak L(V)$ of arithmetic root systems and the dimension for $mathfrak L(V)$ of finite Cartan type. We give the sufficient and necessary conditions for $mathfrak B(V) = Foplus mathfrak L^-(V)$ and $mathfrak L^-(V)= mathfrak L(V)$. We obtain an explicit basis of $mathfrak L^ - (V)$ over quantum linear space $V$ with $dim V=2$.
We introduce a new filtration on Hopf algebras, the standard filtration, generalizing the coradical filtration. Its zeroth term, called the Hopf coradical, is the subalgebra generated by the coradical. We give a structure theorem: any Hopf algebra with injective antipode is a deformation of the bosonization of the Hopf coradical by its diagram, a connected graded Hopf algebra in the category of Yetter-Drinfeld modules over the latter. We discuss the steps needed to classify Hopf algebras in suitable classes accordingly. For the class of co-Frobenius Hopf algebras, we prove that a Hopf algebra is co-Frobenius if and only if its Hopf coradical is so and the diagram is finite dimensional. We also prove that the standard filtration of such Hopf algebras is finite. Finally, we show that extensions of co-Frobenius (resp. cosemisimple) Hopf algebras are co-Frobenius (resp. cosemisimple).
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