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Register a new userTwisting of affine algebraic groups, I
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Authors
Shlomo Gelaki
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We continue the study of twisting of affine algebraic groups G (i.e., of Hopf 2-cocycles J for the function algebra O(G)), which was started in [EG1,EG2], and initiate the study of the associated one-sided twisted function algebras O(G)_J. We first show that J is supported on a closed subgroup H of G (defined up to conjugation), and that O(G)_J is finitely generated with center O(G/H). We then use it to study the structure of O(G)_J for connected nilpotent G. We show that in this case O(G)_J is a Noetherian domain, which is a simple algebra if and only if J is supported on G, and describe the simple algebras that arise in this way. We also use [EG2] to obtain a classification of Hopf 2-cocycles for connected nilpotent G, hence of fiber functors Rep(G)to Vect. Along the way we provide many examples, and at the end formulate several ring-theoretical questions about the structure of the algebras O(G)_J for arbitrary G.
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We use cite{G} to study the algebra structure of twisted cotriangular Hopf algebras ${}_Jmathcal{O}(G)_{J}$, where $J$ is a Hopf $2$-cocycle for a connected nilpotent algebraic group $G$ over $mathbb{C}$. In particular, we show that ${}_Jmathcal{O}(G)_{J}$ is an affine Noetherian domain with Gelfand-Kirillov dimension $dim(G)$, and that if $G$ is unipotent and $J$ is supported on $G$, then ${}_Jmathcal{O}(G)_{J}cong U(g)$ as algebras, where $g={rm Lie}(G)$. We also determine the finite dimensional irreducible representations of ${}_Jmathcal{O}(G)_{J}$, by analyzing twisted function algebras on $(H,H)$-double cosets of the support $Hsubset G$ of $J$. Finally, we work out several examples to illustrate our results.
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]).
We show that except in several cases conjugacy classes of classical Weyl groups $W(B_n)$ and $W(D_n)$ are of type {rm D}. We prove that except in three cases Nichols algebras of irreducible Yetter-Drinfeld ({rm YD} in short )modules over the classical Weyl groups are infinite dimensional.
We construct a model of the affine nil-Hecke algebra as a subalgebra of the Nichols-Woronowicz algebra associated to a Yetter-Drinfeld module over the affine Weyl group. We also discuss the Peterson isomorphism between the homology of the affine Grassmannian and the small quantum cohomology ring of the flag variety in terms of the braided differential calculus.
The Donald--Flanigan problem for a finite group H and coefficient ring k asks for a deformation of the group algebra kH to a separable algebra. It is solved here for dihedral groups and for the classical Weyl groups (whose rational group algebras are also computed), leaving but six finite reflection groups with solutions unknown. We determine the structure of a wreath product of a group with a sum of central separable algebras and show that if there is a solution for H over k which is a sum of central separable algebras then there is also a solution for the wreath product of H with any symmetric group, abelian group, or dihedral group. The theorems suggested by the Donald-Flanigan conjecture and subsequently proven follow, we also show, from a geometric conjecture which although weaker for groups applies to a broader class of algebras than group algebras.
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