Proc. Nat. Acad. Sei

This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion category is always positive, and that the S-matrix of any (not necessarily hermitian) modular category is unitary. We also show that the category of module functors between two module categories over a fusion category is semisimple, and that fusion categories and tensor functors between them are undeformable (generalized Ocneanu rigidity). In particular the number of such categories (functors) realizing a given fusion datum is finite. Finally, we develop the theory of Frobenius-Perron dimensions in an arbitrary fusion category. At the end of the paper we generalize some of these results to positive characteristic.

http://annals.math.princeton.edu/wp-content/uploads/annals-v162-n2-p01.pdf

On fusion categories

Operators on Hilbert Space.- C*-Algebras.- Von Neumann Algebras.- Further Structure.- K-Theory and Finiteness.

Operator Algebras: Theory of C*-Algebras and von Neumann Algebras

To any finite group Γ⊂Sp(V) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, Hκ of the algebra ℂ[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of ℙ
 r
 , where r=number of conjugacy classes of symplectic reflections in $Γ$. The algebra Hκ, called a symplectic reflection algebra, is related to the coordinate ring of a Poisson deformation of the quotient singularity V/Γ. This leads to a symplectic analogue of McKay correspondence, which is most complete in case of wreath-products. If Γ is the Weyl group of a root system in a vector space ? and V=?⊕?*, then the algebras Hκ are certain ‘rational’ degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let $Γ=S

 n
 , the Weyl group of ?=??
 n
 . We construct a 1-parameter deformation of the Harish-Chandra homomorphism from ?(?)?, the algebra of invariant polynomial differential operators on ??
 n
 , to the algebra of S

 n
 -invariant differential operators with rational coefficients on the space ℂ
 n
 of diagonal matrices. The second order Laplacian on ? goes, under the deformed homomorphism, to the Calogero-Moser differential operator on ℂ
 n
 , with rational potential. Our crucial idea is to reinterpret the deformed Harish-Chandra homomorphism as a homomorphism: ?(?)? ↠ spherical subalgebra in Hκ, where Hκ is the symplectic reflection algebra associated to the group Γ=S

 n
 . This way, the deformed Harish-Chandra homomorphism becomes nothing but a description of the spherical subalgebra in terms of ‘quantum’ Hamiltonian reduction. In the ‘classical’ limit κ→∞, our construction gives an isomorphism between the spherical subalgebra in H∞ and the coordinate ring of the Calogero-Moser space. We prove that all simple H∞-modules have dimension n!, and are parametrised by points of the Calogero-Moser space. The family of these modules forms a distinguished vector bundle on the Calogero-Moser space, whose fibers carry the regular representation of S

 n
 . Moreover, we prove that the algebra $H∞ is isomorphic to the endomorphism algebra of that vector bundle.

Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism

Definitions and examples Integrals and semisimplicity Freeness over subalgebras Action of finite-dimensional Hopf algebras and smash products Coradicals and filtrations Inner actions Crossed products Galois extensions Duality New constructions from quantum groups Some quantum groups.

Hopf algebras and their actions on rings

/pdf/group-graded-rings-smash-products-and-group-actions-g9iiqijxow.pdf

Group-graded rings, smash products, and group actions

This paper develops a theory of crossed products and inner (weak) actions of arbitrary Hopf algebras on noncommutative algebras. The theory covers the usual examples of inner automorphisms and derivations, and in addition is general enough to include \"inner\" group gradings of algebras. We prove that if 7r : H —► H is a Hopf algebra epimorphism which is split as a coalgebra map, then H is algebra isomorphic to A #„ H, a crossed product of H with the left Hopf kernel A of it. Given any crossed product A #CT H with H (weakly) inner on A, then A #CT if is isomorphic to a twisted product AT[H] with trivial action. Finally, if H is a finite dimensional semisimple Hopf algebra, we consider when semisimplicity or semiprimeness of A implies that of A #„ H; in particular this is true if the (weak) action of H is inner. Introduction. The purpose of this paper is to begin to lay the foundations of a general theory of actions of Hopf algebras on noncommutative algebras. The importance of such a theory derives from three special cases in which the Hopf algebra is a group algebra, an enveloping algebra of a Lie algebra, or the dual of a group algebra. These cases show that the theory encompasses the study of actions of groups as automorphisms of algebras, the study of actions of Lie algebras as derivations of algebras, and the study of group graded rings, respectively. Each of these areas of study have been quite active lately (see [16, 9, 5]). A fundamental concept in the first two cases has been the notion of inner action. One of the major purposes of this paper is to study inner actions of Hopf algebras; as a new example, we will investigate what is meant by an inner grading. Another important concept in the first two cases is that of a semidirect product (smash product): for group actions we have skew group rings and for Lie algebra actions we have differential polynomial rings. This notion is also defined for Hopf algebra actions. More generally one can consider crossed products A ffa H of an algebra A with a Hopf algebra H, where the multiplication of the copy of H in A ffa H is twisted by a cocycle o, and their study is the other major purpose of this paper. It turns out that these two concepts (inner actions and crossed products) are closely interrelated. Now inner actions and crossed products of Hopf algebras were both studied by Sweedler [24] in the context of the cohomology theory of Hopf algebras. The present paper owes a great debt to Sweedler's work, especially in §§1 and 4. However, his set-up was restricted at crucial places to the situation where H is a cocommutative Received by the editors December 12, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A24; Secondary 16A72, 16A03, 46L40. The third author was partially supported by NSF Grant DMS 8500959 and by a Guggenheim Memorial Foundation Fellowship. ©1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page

/pdf/crossed-products-and-inner-actions-of-hopf-algebras-3dhpvb864q.pdf

Crossed products and inner actions of Hopf algebras

Preliminaries.- Rings with no |G|-torsion: Existence of fixed elements, the prime radical and Jacobson radical of RG.- Simple rings: Inner and outer automorphisms, simple fixed rings, R as an RG-module.- Inner and outer automorphisms of semiprime rings.- Rings with no nilpotent elements: Existence of fixed elements and traces.- Chain conditions.- Polynomial identities.- Skew group rings and applications to modules.

Fixed Rings of Finite Automorphism Groups of Associative Rings

https://projecteuclid.org/download/pdf_1/euclid.pjm/1102650535

Susan Montgomery

Papers

Hopf algebras and their actions on rings

Group-graded rings, smash products, and group actions

Crossed products and inner actions of Hopf algebras

Fixed Rings of Finite Automorphism Groups of Associative Rings

Crossed products and Galois extensions of Hopf algebras.