Robert G. Underwood

Duality for Hopf orders

In this paper we use duality to construct new classes of Hopf orders in the group algebra KC p3 , where K is a finite extension of Q p and C p3 denotes the cyclic group of order p 3 . Included in this collection is a subcollection of Hopf orders which are realizable as Galois groups.

Quasitriangular Structure of Myhill–Nerode Bialgebras

In computer science the Myhill–Nerode Theorem states that a set L of words in a finite alphabet is accepted by a finite automaton if and only if the equivalence relation ∼L, defined as x ∼L y if and only if xz ∈ L exactly when yz ∈ L, ∀z, has finite index. The Myhill–Nerode Theorem can be generalized to an algebraic setting giving rise to a collection of bialgebras which we call Myhill–Nerode bialgebras. In this paper we investigate the quasitriangular structure of Myhill–Nerode bialgebras.

The Group of Galois Extensions Over Orders in _

In this paper we characterize all Galois extensions over H where H is an arbitrary R-Hopf order in KCp2 . We conclude that the abelian group of H-Galois extensions is isomorphic to a certain quotient of units groups in R×R. This result generalizes the classification of H-Galois extensions, where H ⊂ KCp, due to Roberts, and also to Hurley and Greither.

Applications of Hopf Algebras

In this chapter we present three diverse applications of Hopf algebras. Our first application involves almost cocommutative bialgebras and quasitriangular bialgebras. We show that a quastitriangular bialgebra determines a solution to the Quantum Yang–Baxter Equation, and we give details on how to compute quastitriangular structures for certain two-dimensional bialgebras and Hopf algebras. We show that almost cocommutative Hopf algebras generalize Hopf algebras in which the coinverse has order 2. We then define the braid group on three strands (or more simply, the braid group) and show that a quasitriangular structure determines a representation of the braid group.

Algebras and Coalgebras

In this chapter we introduce algebras and coalgebras. We begin by generalizing the construction of the tensor product to define the tensor product of a finite collection of R-modules, where R is a commutative ring with unity.

On the Content Bound for Real Quadratic Field Extensions

Let K be a finite extension of ℚ and let S = {ν} denote the collection of K normalized absolute values on K. Let V K + denote the additive group of adeles over K and let c : V K + → ℝ ≥0 denote the content map defined as c( { a v } ) = ∏ v∈s v( a v ) for { a v }∈ V K + . A classical result of J. W. S. Cassels states that there is a constant c > 0 depending only on the field K with the following property: if { a v }∈ V K + with c( { a v } ) > c , then there exists a non-zero element b ∈ K for which v(b)≤v( a v ), ∀v∈ S . Let cK be the greatest lower bound of the set of all c that satisfy this property. In the case that K is a real quadratic extension there is a known upper bound for cK due to S. Lang. The purpose of this paper is to construct a new upper bound for cK in the case that K has class number one. We compare our new bound with Lang’s bound for various real quadratic extensions and find that our new bound is better than Lang’s in many instances.