We apply so-called "fine" gradings of certain classical finite-dimensional Lie algebras and superalgebras to the determination of the center of the universal enveloping algebra. We then extend the same approach to much wider class of Lie algebras and superalgebras possessing linear bases labeled by the elements of a finite abelian group, with the commutator determined in terms of a fixed bicharacter on this group. These results are joint with Alexander Molev.
For H a finite dimensional bisemisimple Hopf algebra over an algebraically closed field, the induced representations from H and H* to the Drinfeld double D(H) are studied. The product of two such representations is a sum of copies of the regular representation of D(H). The action of certain irreducible central characters of H* on the simple modules of H is considered. The modules that receive trivial action from each such irreducible central character are precisely the constituents of the tensor powers of the adjoint representation of H.
A coalgebra is said to be hereditary if every homomorphic image of every injective comodule is injective. Path coalgebras of arbitrary quivers are hereditary, and it turns out that these are the only pointed examples. Thus, over an algebraically closed field, every hereditary coalgebra is Morita-Takeuchi equivalent to the path coalgebra of its Ext-quiver.
This report is about Hopf algebras with Jacobson radical a Hopf ideal, joint work with Mitja Mastnak. We consider in particular the case when the quotient by the radical is cosemisimple, and we explore how these Hopf algebras dualize. Such Hopf algebras occur naturally, for example as duals of certain pointed Hopf algebras.
Let k be a field, and H a Hopf algebra with bijective antipode. If H is commutative, noetherian, semisimple and cosemisimple, then the category of left-right Yetter-Drinfeld modules is semisimple. We also prove a similar statement for the category of Long dimodules, without the assumption that H is commutative.
More or less recently two important generalizations appeared of the Hopf algebra of symmetric functions. They are the Hopf algebra of noncommutative symmetric functions, NSymm, and its graded dual, the Hopf algebra of quasisymmetric functions. The Hopf algebra of permutations, MPR, is a selfdual generalization of both. It can be interpreted, in a certain way, as a Hopf algebra of (module) endomorphisms of the Shuffle Hopf algebra. This way of looking at MPR permits generalizations such as the word Hopf algebra WHA and the double word Hopf algebra dWHA (and many others). These may have room enough for such extra structures as Frobenius and Verschiebung endomorphisms and lambda-ring structures (which is not the case for MPR).
We consider algebras over a field K with a presentation K áx1, ..., xn : R ñ, where R consists of \binomn2 square-free relations of the form xi xj = xk xl with every monomial xi xj, i ¹ j, appearing in one of the relations (or briefly, quadratic algebras of skew type). Such algebras first appeared in the work of Gateva-Ivanova and Van den Bergh, inspired by earlier work of Tate and Van den Bergh. In this case the monoids and groups with the same presentation satisfy some natural non-degenerate conditions and they yield a set theoretical solution to the quantum Yang-Baxter equation.
In this talk we present some recent results on the structure of such algebras; this is joint work with F. Cedo and J. Okninski. First we describe the algebraic structure of the monoids and groups determined by such and related presentations. Several interesting group theoretic results and open problems will be discussed. Second we discuss the algebraic structure of the algebra. Special attention is given to the Gelfand-Kirillov dimension and prime ideals. In particular, it follows that there exist examples on 4n generators so that the algebra has Gelfand-Kirillov dimension one while the algebra is noetherian PI and semiprime in case the field K has characteristic zero.
Frobenius-Schur indicators appear as important invariants of finite groups. Using generalized power map we can extend this notion to semisimple Hopf algebras. However it turns out that unlike in the group theory case, higher Frobenius-Schur indicators may not be integers. In this talk we are going to construct an example of a Hopf algebra, obtained as a certain abelian extension, for which the higher Frobenius-Schur indicators are not real.
This talk is a part of joint work with Yorck Sommerhäuser and Yongchang Zhu.
This talk will discuss the structure of quantum invariants of virtual knots and links. Formally, virtual knots are obtained from classical knots by allowing a crossing that is geometrically a detour from one point to another. One associates a Yang-Baxter operator or order two to the virtual crossing (so that it satisfies appropriate conditions with respect to the usual Yang-Baxter operator assigned to the classical crossings). This talk will review work of the presenter with David Radford on bioriented quantum algebras, and will discuss new invariants of virtual knots recently discovered by the presenter.
A. I. Shirshov  and E. Witt  proved that every subalgebra of a free Lie algebra is free. This result has been generalized to colored Lie superalgebras (A. A. Mikhalev , A. S. Shtern ).
The Shirshov-Witt theorem admits a formulation it terms a free associative algebra: Every subHopfalgebra of a free algebra káxi ñ with the diagonal coproduct, d(xi) = xiÄ1+1 Äxi, is free and it is freely generated by some Freidrichs-primitive elements. If we consider the free algebra as a braided Hopf algebra with a very special braiding, we get a reformulation of the Mikhalev-Shtern generalization as well. Our aim is to extend these results to free algebras with arbitrary braidings.
Let R be a ring. A subset P Ì R is called an ordering if P+P Ì P, P·P Ì P, PÈ-P=R, and supp P:=PÇ-P is a prime ideal of R. The set of all orderings of R is called the real spectrum of R. The study of real spectra of noncommutative rings is known as noncommutative real algebraic geometry. An ordering with zero support gives rise to a total order relation on R.
Many important examples of Hopf algebras, such as group algebras, (quantized) enveloping algebras, and (quantized) function algebras, often admit a zero support ordering. This observation motivates the problem of finding criteria of orderability for an arbitrary Hopf algebra (viewed as a ring). We will investigate this problem for cocommutative Hopf algebras.
In the context of rings with involution, there is a notion of the so called *-orderings. Many Hopf algebras have a natural involution. We will consider the problem of existence of a zero support *-ordering for cocommutative Hopf algebras.
Joint work with J. Cimpric and M. Marshall.
Let k be a field of characteristic 0 containing a primitive root of unity z of an odd order. We denote by uz the Frobenius-Lusztig kernel for the simple Lie algebra of rank 1. We let D(uz) stand for the quantum (= Drinfel'd) double of uz.
We present a complete description of the simple D( uz)-modules. This is based on the notion of a primitive weight vector in a D(uz)-module. Our results show that the simple modules are classified by the weights of their primitive vectors, an analog of the theorem of Curtis-Lusztig for the simple modules for the Frobenius-Lusztig kernels of any type and rank.
The talk is part of joint work with D. Radford.
For each finite group G, an integral domain R of characteristic 0 with the property that no prime divisor of the order of G is invertible is called a G-adapted ring. In this talk, we consider for which finite groups G and G-adapted rings R, RG has only trivial units. Since G-adapted rings contain Z, this can only occur if ZG contains only trivial units, and such groups were classified by Higman. These groups are abelian groups of exponent dividing 4 or 6, and Hamiltonian 2-groups. For such groups, we establish ring-theoretic conditions under which the group ring RG has nontrivial units. Several examples of rings satisfying the conditions and rings not satisfying the conditions are given. In addition, we extend a well-known result for fields by showing that if R is a commutative ring of finite characteristic and RG has only trivial units, then G has order at most 3.
This is joint work with L. Grunenfelder. We study the structure of Yetter Drinfeld modules over those Hopf algebras, that are crossed products of a group algebra and the dual of a group algebra. In particular, if a group T is acting on a group N and c:N×N® k· is an anti-symmetric bicharacter, then we describe explicitly some interesting examples of Yetter-Drinfeld modules over (kN)* \rtimesc kT and their liftings.
We will make a (more) Hopf-algebraic approach to the basics of affine (super)group theory, discussing Hopf modules, crossed products, equivariant smoothness, and duality (including Takeuchi's hyperalgebras). Non-commutative Hopf algebras as well as Hopf algebras in braided monoidal categories are within our scope.
Let U(RG) denote the group ring of a group G over a commutative ring with unity R. In the case when the coefficient ring is a field F or a ring of algebraic 1ntegers, the existence of free subgroups of rank 2 in U(RG) has been studied and explicit constructions of such groups were given by several authors.
Recently, Gonçalvez and Passman investigated the existence of free groups in the subgroup of unitary units with respect to the natural involution of FG induced by g® g-1, for all g Î G.
We shall discuss the existence of free groups in another significant subgroup of U (FG): the subgroup U2(FG) generated by all units of order 2 of FG. These results were obtained in joint work with Prof. A. Giambruno.
Let H be a finite-dimensional Hopf algebra over a field k and R an H-module algebra. We consider when the Jacobson radical J(R) is H-stable. This is true trivially for group actions, and is true for gradings by a finite group G if |G| is invertible in k, by a old result of Cohen and the speaker.
Counterexamples exist when H is not semisimple. We prove it is true if H is semisimple, k has characteristic 0, and all irreducible R-modules are finite-dimensional. As a consequence it is true if R is an affine PI algebra.
This is joint work with Vitaly Linchenko and Lance Small.
In this talk, we will discuss the Frobenius-Schur (FS) indicator cV for an irreducible representation V of a semi-simple quasi-Hopf algebra H. There exists a canonical central element nH in H which is invariant under gauge transformations. The FS indicator cV is defined to be c(nH) where c is the character afforded by V. The scalar c(nH) is non-zero if, and only if, V is self-dual. Moreover, the set of all FS indicators for H is an invariant of the tensor category H-mod. The tensor category H-mod also admits a canonical pivotal structure which can be described via the character of the regular representation and the normalized integral of H. The pivotal structure implies that the FS indicators for H can only be 0, 1 or -1.
A polynomial form f is a not necessarily linear map, from an infinite module to a finite abelian group of exponent n. We show that for a form of degree d then nd-1 Wf is a submodule of A, where Wf is the set of zeros of f. Among all Z-submodules of finite index, there is a submodule B such that |f(B)| (the order of the subset f(B)) is as small as possible. f(B) is called the final value of f and Passman asks if f(B) is necessarily a subgroup of S. This paper shows that if the degree of f £ 2 then the final value is a subgroup and if the form f has arbitrary degree from a non-torsion finitely generated abelian group, then the final value is 0. We will also discuss the zeros of f.
Assume that an R-algebra A (where R is a commutative ring with 1) is graded by a finite abelian group H and let b: H×H ® R* be a skew symmetric bicharacter on H with values in R. If x Î Ag, y Î Ah are homogeneous elements of A (g,h Î H) define a generalized Lie bracket by [x,y]b = xy- b(g,h) yx and extend this bracket to all of A by linearity. A is said to be b-commutative if [a1, a2]b = 0 for all a1, a2 in A. Similarly the usual notions of Lie nilpotence and Lie solvability can be extended to b-nilpotence and b-solvability on A.
In this talk we discuss the above concepts when A is the integral group ring ZG of a finite group G. The results described are part of ongoing joint work with Yuri Bahturin.
There is an extensive class of pointed Hopf algebras which consists of quotients of 2-cocyle twists of tensor products of pointed Hopf algebras with relatively simple structures. Examples of Hopf algebras obtained by 2-cocycle twists are the finite-dimensional quantum doubles.
We will describe the general outline of the theory of the irreducible representations of Hopf algebras which belong to the class. Our results apply to many of the Hopf algebras described by Andruskiewitsch and Schneider in their classification program for pointed Hopf algebras. Most of the work described in this talk is joint with Schneider.
Kurosh was the first to pose certain ring-theoretic analogues of the famous Burnside problem for groups. Specifically, he asked whether or not every nil algebra is locally nilpotent. While Golod eventually constructed a counterexample to the general problem, Kaplansky had already given a positive solution for all the class of all algebras satisfying a nontrivial polynomial identity (PI). In my talk, I shall first discuss why Kaplansky's PI condition can be weaken to "infinitesimally PI". The proof uses strong Lie-theoretic results of Zelmanov. Applications will then be made to the Kurosh problem for group algebras: if the augmentation ideal of a group algebra is nil, is it locally nilpotent? More generally, I shall address the following problem raised by Kaplansky: if a group algebra has the property that its augmentation ideal coincides with its Jacobson radical, is the augmentation ideal locally nilpotent?
Let H be a Hopf algebra, semisimple over the complex numbers. Kashina initiated the study of the Hopf power maps on H, defined by h[n] = h(1) · ¼ ·h(2). She conjectured-and proved in interesting special cases-that the exponent exp(H), defined as the smallest number n such that all n-th powers are trivial, always divides the dimension of H. Such a conjecture is easily motivated by comparison to the group case. Etingof and Gelaki proved several general results on the exponent, in particular that exp(H) | dim(H)3, and that the exponent is invariant under twists.
We report on computer experiments (using Maple) with the Hopf power maps in specific examples-bismash products and Drinfeld doubles of groups. In particular, we ask for which n there exist nontrivial elements whose n-th Hopf power is trivial, or, more specifically, elements of Hopf order n; we also investigate whether the answers to these questions are twisting-invariant. Our results show that the behavior of the Hopf powers of individual elements is much less predictable than that of the exponent, and deviates further from expectations one might have from comparing to the group case.
Let H be a Hopf algebra and A an H-simple right H-comodule algebra. It is shown that under certain hypotheses every (H,A)-Hopf module is either projective or free as an A-module and A is either a quasi-Frobenius or a semisimple ring. As an application it is proved that every weakly finite (in particular, every finite dimensional) Hopf algebra is free both as a left and a right module over its finite dimensional right coideal subalgebras, and the latter are Frobenius algebras. Similar results are obtained for H-simple H-module algebras.
A module algebra for a bialgebra may be deformed by a twisting element or a universal deformation formula over the bialgebra. Classically, a polynomial ring carries the action of its Lie algebra of derivations, and resulting deformations include quantum space and the Weyl algebra. More recently crossed products of polynomial rings (and their deformations) with finite groups have been of interest in relation to the corresponding geometry. For these crossed product algebras, the infinitesimal versions of deformations may be described explicitly. Some of the bialgebras involved are Taft algebras, their Drinfel'd doubles, and other related finite quantum groups. An example of a universal deformation formula arising in this context will be given.