Warren D. Nichols

The Grothendieck group of a Hopf algebra

Abstract Let H be a cosemisimple Hopf algebra over an algebraically closed field k . We show that if H contains a simple subcoalgebra of dimension 4, then H contains either a Hopf subalgebra of dimension 2, 12, or 60, or a simple subcoalgebra of dimension n 2 for each positive integer n . In particular, if H is finite dimensional, then it has even dimension.

The structure of the dual Lie coalgebra of the Witt algebra

Abstract Over a field K of characteristic different from 2, the Lie coalgebra dual to the Lie algebra of derivations of the polynomials in one variable over K consists of the linear recursive sequences. When K is algebraically closed, the structure of that Lie coalgebra is determined explicitly.

The grothendieck algebra of a hopf algebra, i

0 Introduction We began in 4] a study of the Grothendieck group G(H) of the category of nite dimensional right comodules of a Hopf algebra H. We begin here the study of the representation theory of the algebras G(H) K obtained from G(H) by extending the scalars. Our primary motivation has been to try to establish Kaplanskys conjecture that if a nite dimensional cosemisimple Hopf algebra H over an algebraically closed eld contains a simple subcoal-gebra of dimension n 2 , then n divides the dimension of H. As a more direct attack on the problem has not been successful, we have been studying the problem from a general

Seminormality in power series rings

Let R be a commutative unitary ring. Recently, several authors have been interested in the preservation of seminormality in passing from R to the polynomial ring R [X, ,..., X,,J. In this paper we give a proof in the case of power series rings and the proof is both short and applicable in the polynomial setting. The chief difficulty lies in proving that “relative” stability is preserved in passage to the power series ring. We briefly recall the pertinent terminology. If R is a ring, then following Swan [3] we say that R is seminormal if whenever b, c E R satisfy b3 = c2 there is an element a E R such that a2 = b and a3 = c. If R c T are rings, then R is said to be seminormal in T if and only if whenever u E T with u*, a3 E R, it follows that a E R. It can be seen that this is equivalent to saying that whenever (x E T with an, cP+ l,..., E R for some positive integer n, it follows that (r E R. Indeed, if R is seminormal in T and if n is taken to be minimal so that on, on+‘,..., E R, then (u”~‘)“, (c?‘)~ E R unless n = 1, in which case u E R, anyway. We refer the reader to [2, 31 for nice discussions of seminormality. We can now state the theorem.

On Lie and associative duals

The dual of the Lie algebra structure defined on a commutative algebra A by a derivation D of A is often the same as the Lie coalgebra structure defined by D on the coalgebra A°. This duality allows the Lie coalgebra structure map on the linearly recursive sequences to be calculated explicitly in terms of the known formula for the coalgebra structure map on those sequences.

The prime spectra of subalgebras of affine algebras and their localizations

Abstract Prime spectra of affine domains A over a field F are known to have especially nice properties. Here we investigate Spec(R) in the cases where either (a) R is a subalgebra of AM for some maximal ideal M of A, or (b) R is a subalgebra of A, requiring in neither case that A should be an integral domain. In case (a) we show that dim R=tr.deg.FR. In case (b) it is known that R[1/ƒ] is affine over F for some ƒ ϵ R, ƒ ≠ 0, if A is an integral domain; this yields nice properties on appropriate Zariski open subsets of Spec(R), but we show that globally, Spec(R) shares few of the attractive properties of Spec(A).

Finite-dimensional hopf algebras are free over grouplike subalgebras

Abstract Let H be a finite-dimensional Hopf algebra over a field k , and let G be a subgroup of the group of grouplikes of H . Then every left ( H , kG )-Hopf module is free as a left kG -module. If k is algebraically closed and C is a simple subcoalgebra of H with gC = C for all g ∈ G , then the exponent of G and (in characteristic p > 0) the p -part of | G | divide n , where n 2 is the dimension of C .

On series of ordinals and combinatorics

This paper deals mainly with generalizations of results in finitary combinatorics to infinite ordinals. It is well-known that for finite ordinals ∑bT<αβ is the number of 2-element subsets of an α-element set. It is shown here that for any well-ordered set of arbitrary infinite order type α, ∑bT<αβ is the ordinal of the set M of 2-element subsets, where M is ordered in some natural way. The result is then extended to evaluating the ordinal of the set of all n-element subsets for each natural number n ≥ 2. Moreover, series ∑β<αf(β) are investigated and evaluated, where α is a limit ordinal and the function f belongs to a certain class of functions containing polynomials with natural number coefficients. The tools developed for this result can be extended to cover all infinite α, but the case of finite α appears to be quite problematic.

On extending endomorphisms to automorphisms

A monic endomorphism of a structure A can be extended to an automorphism uf a larger structure A We investigate which properties are preserved by this process.