Richard G. Larson

Finite dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple

Finite dimensional cosemisimple Hopf algebras are a natural generalization of group algebras. Much of the representation theory of finite groups can be extended to such Hopf algebras. However, there are still many unanswered basic questions for such Hopf algebras. One such question is a conjecture mentioned by Kaplansky [I]: the square of the antipode of a finite dimensional cosemisimple Hopf algebra is the identity. In this paper we prove that the fourth power of the antipode is the identity in a finite dimensional cosemisimple Hopf algebra over a field of characteristic 0. In the course of proving this, we investigate two elements of the Hopf algebra which are roughly analogous to the sum of the group elements in a group algebra. We show that Kaplansky’s conjecture is equivalent to these two elements being equal. For a finite dimensional Hopf algebra A with antipode s over a field, let L(p)(q) =pq for p, q E A* denote the left module action of A* on itself. In this paper we study two elements of A defined by the trace function: the element ;i defined by p(A) = Tr(L(p) (s’)*) for all PE A*, and the element x defined by p(x) = Tr(L(p)) for all p E A*. We show that 2 is a left integral and use it to prove Theorem 3.3, the main result of this paper: a finite dimensional cosemisimple Hopf algebra over a field of characteristic 0 is also semisimple (hence its antipode has order 1, 2, or 4). We extend the first corollary to [3, Proposition 91 in Theorems 4.3 and 4.4 by finding necessary and sufficient conditions for s to be a non-zero left integral (in which case x = A), and also study how x relates to the structure of A. In particular, when (dim A) 1 # 0 the right ideal xA may be thought of as a measure of the extent to which s* # I. The elements ji and .K are two examples of elements of A associated with an endomorphism of A. In general, for f~ End(A) we define A, E A by p(A&=Tr(L(p)cf*) for all PEA *. Thus I? = AS2 and x = A,. In Section 2 267 0021-8693/88 33.00

Hopf-algebraic structure of families of trees

In this paper we describe Hopf algebras which are associated with certain families of trees. These Hopf algebras originally arose in a natural fashion: one of the authors [5] was investigating data structures based on trees, which could be used to efficiently compute certain differential operators. Given data structures such as trees which can be multiplied, and which act as higherorder derivations on an algebra, one expects to find a Hopf algebra of some sort. We were pleased to find that not only was there a Hopf algebra associated with these data structures, but that it could be used to give new proofs of enumerations of such objects as rooted trees and ordered rooted trees. Previous work applying Hopf algebras to combinatorial objects (such as [10], [13] or [14]) has concerned itself with algebraic structures on polynomial algebras and on partially ordered sets, rather than on trees themselves.

Characters of Hopf algebras

In the development of the character theory of finite groups, one of the key facts used in proving the orthogonality relations is the invariance property of cg E kG. (See, e.g., the proofs of Lemma 5.1.3 in [2] and of Eq. (31.8) in [l].) Since finite-dimensional Hopf algebras [5] and the dual algebras of certain infinite-dimensional Hopf algebras [7] contain elements with properties analogous to those of Cg, it is reasonable to expect that an orthogonality relation will hold for characters of such Hopf algebras. In Section 2 of this paper we prove such an orthogonality relation. In order to include a character theory for infinite dimensional Hopf algebras, we must consider characters as elements of the Hopf algebra which are associated with comodules over the Hopf algebra, rather than as functionals on the Hopf algebra which are associated with modules over the Hopf algebra. This point of view allows a simultaneous treatment of the characters of compact Lie groups and completely reducible affine algebraic groups (taking as the Hopf algebra the algebra of representative functions), of the characters of semisimple Lie algebras over an algebraically closed field of characteristic 0 (taking U(L)” as the Hopf algebra; the characters studied here differ from those of Expose 18 of [6] by a scalar multiple), and of the characters of finite groups (taking as the Hopf algebra the dual Hopf algebra to the group algebra). We then use our results to prove that the dimension of a simple comodule of an involutory cosemisimple Hopf algebra over an algebraically closed field is not divisible by the characteristic of the field. In Section 3 we prove that the antipode y of a cosemisimple Hopf algebra is bijective, and that y2 maps each simple subcoalgebra onto itself. In Section 4 we give a generalization of Maschke’s Theorem: If the characteristic of the field does not divide the dimension of a finite dimensional involutory Hopf algebra, then the Hopf algebra and its dual are semisimple; if the characteristic does divide the

Hopf-algebraic structure of combinatorial objects and differential operators

A Hopf-algebraic structure on a vector space which has as basis a family of trees is described, and we survey some applications of this structure to combinatorics and to differential operators. Some possible future directions for this work are indicated.

Hopf algebra orders determined by group valuations

Hopf algebra orders in the group algebra of a finite group can be used to get information on the representation theory of the group. In this paper, we describe a class of such orders that arises from group valuations on the group and use properties of these orders to get a new bound on the degrees of the absolutely irreducible representations of the group. In Section 1, we discuss the basic properties of group valuations. A group valuation is a real-valued function on the group that satisfies conditions that reflect the product and commutator relations on the group. We also introduce weighted filtrations. These correspond to group valuations and are sometimes more convenient for computation. Next, Hopf algebra orders are introduced and in Section 3, their relation to group valuations is discussed. There is a one-one correspondence between group valuations satisfying certain conditions reflecting the orders of the group elements and the pth power map (the p-adic order-bounded group valuations) and certain Hopf algebra orders in the group algebra. An important invariant associated with a Hopf algebra order A is e(LA), where LA is the ideal of left integrals in A. In Section 4, we compute E(L~) for those Hopf algebra orders associated with p-adic orderbounded group valuations. In Section 5, we apply the results of Section 4 to get a new bound on the degrees of the absolutely irreducible representations of a finite group. In Section 6, we compare the bound given in Section 5 with the bound given in Ito’s theorem (the degree of an absolutely irreducible representation must divide the index of any normal abelian subgroup) for the groups of order 2”, 0 < n < 6. The work in the first six sections is all local; in Section 7, we briefly describe the global situation, and discuss some of the open questions involving Hopf algebra orders. In this paper, we assume that the reader is familiar with the results and techniques of [S]. Throughout this paper, K is an algebraic number field and

An algebraic approach to hybrid systems

We propose an algebraic model for hybrid systems and illustrate its usefulness by proving theorems in realization theory using this viewpoint. By a hybrid system, we mean a collection of continuous nonlinear control systems associated with a discrete finite state automaton. The automaton switches between the continuous control systems, and this switching is a function of the discrete input symbols that it receives.

The realization of input-output maps using bialgebras

We use the theory of bialgebras to provide the algebraic background for state space realization theorems for input-output maps of control systems. This allows us to consider from a common viewpoint classical results about formal state space realizations of nonlinear systems and more recent results involving analysis related to families of trees. If H is a bialgebra, we say that p 2 H is dierentially produced by the algebra R with the augmentation if there is right H-module algebra structure on R and there exists f 2 R satisfying p(h) = (f ·h). We characterize those p 2 H which are dierentially produced.

The algebraic structure of linearly recursive sequences under hadamard product

We describe the algebraic structure of linearly recursive sequences under the Hadamard (point-wise) product. We characterize the invertible elements and the zero divisors. Our methods use the Hopf-algebraic structure of this algebra and classical results on Hopf algebras. We show that our criterion for invertibility is effective if one knows a linearly recursive relation for a sequence and certain information about finitely-generated subgroups of the multiplicitive group of the field.

Symbolic computation of derivations using labelled trees

We discuss the effective symbolic computation of operators under composition. We analyse data structures consisting of formal linear combinations of rooted labelled trees. We define a multiplication on rooted labelled trees, thereby making the set of these data structures into an associative algebra. We then define an algebra homomorphism from the original algebra of operators into this algebra of trees. The cancellation which occurs when non-commuting operators are expressed in terms of commuting ones occurs naturally when the operators are represented using this data structure. This leads to an algorithm which, for operators which are derivations, speeds up the computation exponentially in the degree of the operator.

Orders in Hopf algebras

Number-theoretic techniques are very useful in studying the representation theory of finite groups. In this paper we attempt to introduce such techniques into the representation theory of finite-dimensional semisimple involutory Hopf algebras over a number field. We consider Hopf algebras which contain an integral Hopf algebra order, and study the relation between the structure of the Hopf algebra and the number-theoretic properties of its orders. Let H be a finite-dimensional semisimple involutory Hopf algebra. An element /l E H is called a left integral if hfl = .~(h)/l for all h E H. It is known [5, 61 that there exist left integrals in H for which E(A) # 0. Let A be an order in H, and let IT, be the ideal of all left integrals in A. The ideal E(L) gives much information on the structure of H and A. It plays a role similar to that played by the order of the group in discussing the representation theory of a finite group. In fact it always is a divisor of dim H. More specifically, let A* be the dual order to A in H*, and let L* be the ideal of left integrals in A*. We prove that c(L) c(L*) is the ideal generated by dim H. A is a separable algebra if and only if c(L) = R. If B is an order containing A, and M is the ideal of left integrals contained in B, we show that (c(L) c(&Z)l) (B/A) = 0. In particular, if E(L) = c(M), it follows that B = A. We then prove the following generalization of a theorem of Frobenius: the degree of any absolutely irreducible representation of H divides E(L). If H is the group algebra of a group G, and A is the integral group ring, then E(L) = (1 G I), and this gives Frobenius’ theorem. If G has a normal abelian subgroup N, then it is possible to construct an order for which c(L) = ([G : N]), in which case our result gives us a theorem of Ito. This