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Shlomo Gelaki

Publications and source records attributed to Shlomo Gelaki.

At least 55 records · Page 3Linked to original sources
arXiv2003

Finite dimensional quasi-Hopf algebras with radical of codimension 2

It is shown in math.QA/0301027 that a finite dimensional quasi-Hopf algebra with radical of codimension 1 is semisimple and 1-dimensional. On the other hand, there exist quasi-Hopf (in fact, Hopf) algebras, whose radical has codimension 2. Namely, it is known that these are exactly the Nichols Hopf algebras H_{2^n} of dimension 2^n, n\ge 1 (one for each value of n). The main result of this paper is that if H is a finite dimensional quasi-Hopf algebra over C with radical of codimension 2, then H is twist equivalent to a Nichols Hopf algebra H_{2^n}, n\ge 1, or to a lifting of one of the four special quasi-Hopf algebras H(2), H_+(8), H_-(8), H(32) of dimensions 2, 8, 8, and 32, defined in Section 3. As a corollary we obtain that any finite tensor category which has two invertible objects and no other simple object is equivalent to \Rep(H_{2^n}) for a unique n\ge 1, or to a deformation of the representation category of H(2), H_+(8), H_-(8), or H(32). As another corollary we prove that any nonsemisimple quasi-Hopf algebra of dimension 4 is twist equivalent to H_4.

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arXiv2003

On Hopf algebras of dimension pq

In this paper we contribute to the classification of Hopf algebras of dimension pq, where p,q are distinct prime numbers. More precisely, we prove that if p and q are odd primes with p<q<2p+3, then any complex Hopf algebra of dimension pq is semisimple and hence isomorphic to either a group algebra or to the dual of a group algebra by the previous works math.QA/9801129 and math.QA/9801128.

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arXiv2002

Triangular Hopf algebras with the Chevalley property

We say that a Hopf algebra has the Chevalley property if the tensor product of any two simple modules over this Hopf algebra is semisimple. In this paper we classify finite dimensional triangular Hopf algebras with the Chevalley property, over the field of complex numbers. Namely, we show that all of them are twists of triangular Hopf algebras with R-matrix having rank <=2, and explain that the latter ones are obtained from group algebras of finite supergroups by a simple modification procedure. We note that all examples of finite dimensional triangular Hopf algebras which are known to the authors, do have the Chevalley property, so one might expect that our classification potentially covers all finite dimensional triangular Hopf algebras.

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arXiv2002

On twisting of finite-dimensional Hopf algebras

In this paper we study the properties of Drinfeld's twisting for finite-dimensional Hopf algebras. We determine how the integral of the dual to a unimodular Hopf algebra $H$ changes under twisting of $H$. We show that the classes of cosemisimple unimodular, cosemisimple involutive, cosemisimple quasitriangular finite-dimensional Hopf algebras are stable under twisting. We also prove the cosemisimplicity of a coalgebra obtained by twisting of a cosemisimple unimodular Hopf algebra by two different twists on two sides (such twists are closely related to biGalois extensions), and describe the representation theory of its dual. Next, we define the notion of a non-degenerate twist for a Hopf algebra $H$, and set up a bijection between such twists for $H$ and $H^*$. This bijection is based on Miyashita-Ulbrich actions of Hopf algebras on simple algebras. It generalizes to the non-commutative case the procedure of inverting a non-degenerate skew-symmetric bilinear form on a vector space. Finally, we apply these results to classification of twists in group algebras and of cosemisimple triangular finite-dimensional Hopf algebras in positive characteristic, generalizing the previously known classification in characteristic zero.

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arXiv2001

An affine PI Hopf algebra not finite over a normal commutative Hopf subalgebra

In formulating a generalized framework to study certain noncommutative algebras naturally arising in representation theory, K. A. Brown asked if every finitely generated Hopf algebra satisfying a polynomial identity was finite over a normal commutative Hopf subalgebra. In this note we show that Radford's biproduct, applied to the enveloping algebra of the Lie superalgebra pl(1,1), provides a noetherian prime counterexample.

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arXiv2001

On the quasi-exponent of finite-dimensional Hopf algebras

Recall (math.QA/9812151) that the exponent of a finite-dimensional complex Hopf algebra H is the order of the Drinfeld element u of the Drinfeld double D(H) of H. Recall also that while this order may be infinite, the eigenvalues of u are always roots of unity (math.QA/9812151, Theorem 4.8); i.e., some power of u is always unipotent. We are thus naturally led to define the quasi-exponent of a finite-dimensional Hopf algebra H to be the order of unipotency of u. The goal of the paper is to create a theory of quasi-exponent, which would be parallel to the theory of the exponent developed in math.QA/9812151. In particular, we give two other equivalent definitions of the quasi-exponent, and prove that it is invariant under twisting. Furthermore, we prove that the quasi-exponent of a finite-dimensional pointed Hopf algebra H is equal to the exponent of the group G(H) of grouplike elements of H. (In particular, the order of the squared antipode of H divides exp(G(H)).) As an application, we find that if H is obtained by twisting the quantum group at root of unity U_q(g) then the order of any grouplike element in H divides the order of q.

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arXiv2001

On families of triangular Hopf algebras

Following the ideas of our previous works math.QA/0008232 (joint with Andruskiewitsch) and math.QA/0101049, we study families of triangular Hopf algebras obtained by twisting finite supergroups by a twist lying entirely in the odd part. These families are parametrized by data (G,V,u,B), where G is a finite group, V its finite dimensional representation, u a central element of G of order 2 acting by -1 on V, and B an element of S^2V. We fix the discrete data G,V,u, and find the set of isomorphism classes of the members of the family as Hopf algebras, in terms of the continuous parameter B. This set is often infinite, which provides examples of nontrivial continuous families of triangular Hopf algebras. The lowest dimension in which such a family occurs is 32, in which case we get 3 families which are dual to the 3 families of pointed Hopf algebras of dimension 32 constructed recently by Grana. Furthermore, we show that if (S^2V)^G=0 then such continuous families are nontrivial not only up to a Hopf algebra isomorphism, but also up to twisting of the multiplication. Thus, they provide counterexamples to Masuoka's weakened Kaplansky's 10th conjecture, which claims that up to twisting, there are finitely many types of Hopf algebras in each dimension. Finally, we study the algebra structure of the duals of our families, and show they are direct sums of Clifford algebras. Since there are finitely many types of Clifford algebras in each dimension, this allows us to construct nontrivial families of rigid tensor structures on the abelian category of modules over a finite dimensional algebra, with a fixed Grothendieck ring.

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arXiv2001

Semisimple Triangular Hopf Algebras and Tannakian Categories

One of the most fundamental problems in the theory of finite- dimensional Hopf algebras is their classification over an algebraically closed field k of characteristic 0. This problem is extremely difficult, hence people restrict it to certain classes of Hopf algebras, e.g. to semisimple ones. Semisimple Hopf algebras deserve to be considered as "quantum" analogue of finite groups, but even so, the problem remains extremely hard (even in low dimensions) and very little is known. A great boost to the theory of Hopf algebras was given by Drinfeld who invented the fundamental class of (quasi)triangular Hopf algebras (A,R). Thus, it is natural to first try to classify semisimple triangular Hopf algebras (A,R) over k (i.e. R is unitary). The theory of such Hopf algebras is now essentially closed by a sequence of works by P. Etingof and the author. The key theorem in this theory states that A is obtained from a group algebra of a unique (up to isomorphism) finite group by twisting its usual comultiplication. The proof of this theorem relies on Deligne's theorem on Tannakian categories in an essentail way. The purpose of this paper is to explain the proof in full details. In particular, to discuss all the necessary background from category theory (e.g. symmetric, rigid, and Tannakian categories), and the properties of the category of representations of triangular (semisimple) Hopf algebras.

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arXiv2001

Classification of Finite-Dimensional Triangular Hopf Algebras With The Chevalley Property

Recall that a triangular Hopf algebra A is said to have the Chevalley property if the tensor product of any two simple A-modules is semisimple, or, equivalently, if the radical of A is a Hopf ideal. There are two reasons to study this class of triangular Hopf algebras: First, it contains all known examples of finite-dimensional triangular Hopf algebras; second, it can be, in a sense, "completely understood". Namely, it was shown in our previous work with Andruskiewitsch that any finite-dimensional triangular Hopf algebra with the Chevalley property is obtained by twisting of a finite-dimensional triangular Hopf algebra with R-matrix of rank at most 2 which, in turn, is obtained by "modifying" the group algebra of a finite supergroup. This provides a classification of such Hopf algebras. The goal of this paper is to make this classification more effective and explicit, i.e. to parameterize isomorphism classes of finite-dimensional triangular Hopf algebras with the Chevalley property by group-theoretical objects, similarly to how it was done in our previous work in the semisimple case. This is achieved in Theorem 2.2, where these classes are put in bijection with certain septuples of data. In the semisimple case, the septuples reduce to the quadruples, and we recover the result of our previous paper. In the minimal triangular pointed case, we recover a classification theorem from a previous paper of the second author.

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arXiv2000

On the Classification of Finite-Dimensional Triangular Hopf Algebras

A fundamental problem in the theory of Hopf algebras is the classification and explicit construction of finite-dimensional quasitriangular Hopf algebras over C. These Hopf algebras constitute a very important class of Hopf algebras, introduced by Drinfeld (e.g. the quantum double of a Hopf algebra). However, this intriguing problem turns out to be extremely hard and it is still widely open. One can hope that resolving the problem in the triangular case would contribute to the understanding of the general case. There is no complte classification of finite-dimensional triangular Hopf algebras yet. However, the problem was solved in the semisimple case by Etingof and the author, in the minimal triangular pointed case by the author, and more generally for triangular Hopf algebras with the Chevalley property by Andruskiewitsch, Etingof and the author. The purpose of this paper is to report on all of this progress, and to explain in full details the mathematics and ideas involved in this theory. The classification in the semisimple case relies on Deligne's theorem on Tannakian categories and on Movshev's theory in an essential way. We explian Movshev's theory in details, and refer the reader to math.QA/0007147 for a detailed discussion of the first aspect. We also discuss the existence of grouplike elements in quasitriangular semisimple Hopf algebras, and the representation theory of cotriangular semisimple Hopf algebras. We conclude the paper with a list of related open problems.

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arXiv2000

Some Properties and Examples of Triangular Pointed Hopf Algebras

A fundamental problem in the theory of Hopf algebras is the classification and construction of finite-dimensional (minimal) triangular Hopf algebras (A,R) introduced by Drinfeld. Only recently Etingof and the author completely solved this problem for semisimple A over algebraically closed fields of characteristics 0 and p>>dim(A) (any p if one assumes that A is also cosemisimple). In this paper we take the first step towards solving this problem for finite-dimensional pointed Hopf algebras over an algebraically closed field k of characteristic 0. We first prove that the fourth power of the antipode of any triangular pointed Hopf algebra A is the identity. We do that by focusing on minimal triangular pointed Hopf algebras (A,R) (every triangular Hopf algebra contains a minimal triangular sub Hopf algebra) and proving that the group algebra of the group of grouplike elements of A (which must be abelian) admits a minimal triangular structure and consequently that A has the structure of a biproduct. We also generalize our result on the order of the antipode to any finite-dimensional quasitriangular Hopf algebra A whose Drinfeld element u acts as a scalar in any irreducible representation of A (e.g. when A^* is pointed). Second, we describe a method of construction of finite-dimensional pointed Hopf algebras which admit a minimal triangular structure, and classify all their minimal triangular structures. We conclude the paper by proving that any minimal triangular Hopf algebra which is generated as an algebra by grouplike elements and skew primitive elements is isomorphic to a minimal triangular Hopf algebra constructed using our method.

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arXiv2000

Isocategorical groups

It is well known that if two finite groups have the same symmetric tensor categories of representations over C, then they are isomorphic. We study the following question: when do two finite groups G1,G2 have the same tensor categories of representations over C (without regard for the commutativity constraint). We call two groups with such property isocategorical. We give an example of two groups which are isocategorical but not isomorphic: the affine symplectic group of a vector space over the field of two elements, and an appropriate "affine pseudosymplectic group" introduced by R.Griess (containing the "pseudosymplectic group" of A.Weil). On the other hand, we give a classification of groups isocategorical to a given group. In particular, we show that if G has no nontrivial normal subgroups of order 2^{2m} then any group isocategorical to G must actually be isomorphic to G. The proofs use the theory of triangular Hopf algebras. We also apply the notion of isocategorical groups to studying the question: when are two triangular semisimple Hopf algebras isomorphic as Hopf algebras?

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arXiv2000

On cotriangular Hopf algebras

In 1997 we proved that any triangular semisimple Hopf algebra over an algebraically closed field k of characteristic 0 is obtained from the group algebra k[G] of a finite group G, by twisting its comultiplication by a twist in the sense of Drinfeld. In this paper, we generalize this result to not necessarily finite-dimensional cotriangular Hopf algebras. Namely, our main result says that a cotriangular Hopf algebra A over k is obtained from a function algebra of a pro-algebraic group by twisting by a Hopf 2-cocycle, and possibly changing its R-form by a central grouplike element of A^* of order <=2, IF AND ONLY IF the trace of squared antipode on any finite-dimensional subcoalgebra of A is the dimension of this subcoalgebra. This generalization, like the original theorem, is proved using Deligne's theorem on Tannakian categories. In the second half of the paper, we give examples of twisted function algebras. In particular, we show that in the infinite- dimensional case, the squared antipode may not equal the identity. On the other hand, we show that in all of our examples of twisted function algebras, the squared antipode is unipotent, and conjecture it to be the case for any twisted function algebra. We prove this conjecture in a large number of special cases, using the quantization theory of the first author and D.Kazhdan.

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arXiv1999

Biperfect Hopf Algebras

Recall that a finite group is called perfect if it does not have non-trivial 1-dimensional representations (over the field of complex numbers C). By analogy, let us say that a finite dimensional Hopf algebra H over C is perfect if any 1-dimensional H-module is trivial. Let us say that H is biperfect if both H and H^* are perfect. Note that, H is biperfect if and only if its quantum double D(H) is biperfect. It is not easy to construct a biperfect Hopf algebra of dimension >1. The goal of this note is to describe the simplest such example we know. The biperfect Hopf algebra H we construct is based on the Mathiew group of degree 24, and it is semisimple. Therefore, it yields a negative answer to Question 7.5 from a previous paper of the first two authors (math.QA/9905168). Namely, it shows that Corollary 7.4 from this paper stating that a triangular semisimple Hopf algebra over C has a non-trivial group-like element, fails in the quasitriangular case. The counterexample is the quantum double D(H).

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arXiv1999

On the exponent of finite-dimensional Hopf algebras

One of the classical notions of group theory is the notion of the exponent of a group. The exponent of a group is the least common multiple of orders of its elements. In this paper we generalize the notion of exponent to Hopf algebras. We give five equivalent definitions of the exponent. Two of them are: 1) the exponent of H equals the order of the Drinfeld element u of the quantum double D(H); 2) the exponent of H is the order of the squared brading acting in the tensor square of the regular representation of D(H). We show that the exponent is invariant under twisting. We prove that for semisimple and cosemisimple Hopf algebras H, the exponent is finite and divides dim(H)^3. For triangular Hopf algebras in characteristic zero, we show that the exponent divides dim(H). We conjecture that if H is semisimple and cosemisimple then the exponent always divides dim(H). At the end we formulate some open questions, in particular suggest a formulation for a possible Hopf algebraic analogue of Sylow's theorem.

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arXiv1998

The Representation Theory of Co-triangular Semisimple Hopf Algebras

In a previous paper we prove that any semisimple triangular Hopf algebra A over an algebraically closed field of characteristic 0 (say the field of complex numbers C) is obtained from a finite group after twisting the ordinary comultiplication of its group algebra in the sense of Drinfeld; that is A=C[G]^J for some finite group G and a twist J\in C[G]\ot C[G]. In this paper we explicitly describe the representation theory of co-triangular semisimple Hopf algebras A^*=(C[G]^J)^* in terms of representations of some associated groups. As a corollary we prove that Kaplansky's 6th conjecture from 1975 holds for A^*; that is that the dimension of any irreducible representation of A^* divides the dimension of A.

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arXiv1998

Semisimple Hopf algebras of dimension pq are trivial

Masuoka proved that for a prime p, semisimple Hopf algebras of dimension 2p over an algebraically closed field k of characteristic 0, are trivial (i.e. are either group algebras or the dual of group algebras). Westreich and the second author obtained the same result for dimension 3p, and then pushed the analysis further and among the rest obtained the same result for semisimple Hopf algebras H of dimension pq so that H and H^* are of Frobenius type (i.e. the dimensions of their irreducible representations divide the dimension of H). They concluded with the conjecture that any semisimple Hopf algebra H of dimension pq over k, is trivial. In this paper we use Theorem 1.4 in our previous paper q-alg/9712033 to prove that both H and H^* are of Frobenius type, and hence prove this conjecture.

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