Dissertation/Thesis Abstract

A class of topologically free locally convex spaces and related group Hopf algebras
by Brauner, Kalman George, Jr., Ph.D., University of California, Berkeley, 1972, 348; 3270245
Abstract (Summary)

A locally convex space E will be called p-reflexive provided the evaluation map from E to (E')' is a topological linear isomorphism, where the dual space in each case is given the topology of precompact convergence. Let [special characters omitted] denote the category of p-reflexive spaces and continuous linear maps. [special characters omitted] is shown to be complete, cocomplete, and self-dual.

Let Sets denote the category of sets and functions and ν:[special characters omitted] → Sets, the forgetful functor.

Theorem. There exist functors Hom: [special characters omitted] × [special characters omitted] → [special characters omitted] and ⊗: [special characters omitted] × [special characters omitted] → [special characters omitted] such that (1) ν∘ Hom = hom ; and (2) for all A, B, Cob [special characters omitted], Hom(AB, C) is naturally isomorphic to Hom(A, Hom(B, C)).

Thus [special characters omitted] can be extended to be a closed, symmetric, monoidal category in the sense of Eilenberg and Kelly.

Let [special characters omitted] denote the category of k-spaces and continuous maps, and let c: [special characters omitted] × [special characters omitted] → Sets denote the functor such that if X ∈ ob[special characters omitted] and E ∈ ob[special characters omitted], c(X,E) equals the set of continuous functions from X to E.

Theorem. There exist functors C: [special characters omitted] × [special characters omitted] → [special characters omitted] and M: [special characters omitted] → [special characters omitted] such that (1) ν ∘ C = c , and (2) for all Xob[special characters omitted] and Eob[special characters omitted], C(X, E) is naturally isomorphic to Hom(M(X), E).

It is known that [special characters omitted] can be extended to be a closed, symmetric, monoidal category. Let ☆: [special characters omitted] × [special characters omitted] → [special characters omitted] denote the product functor.

Theorem. If X, Yob[special characters omitted], then M(X☆Y) is naturally isomorphic to M(X) M(Y).

M can be extended to be a "strong," symmetric, monoidal functor. M will transform any algebraic structure that occurs at the level of [special characters omitted] to an analogous algebraic structure at the level of [special characters omitted].

Theorem. Let [special characters omitted] and [special characters omitted] be categories such that either (a) [special characters omitted] = k-spaces and [special characters omitted] = [special characters omitted]-coalgebras; (b) [special characters omitted] = k-monoids and [special characters omitted] = [special characters omitted]-bialgebras; or (c) [special characters omitted] = k-groups and [special characters omitted] = [special characters omitted]-Hopf algebras. Then M can be regarded as a functor from [special characters omitted] to [special characters omitted]. Furthermore, there exists a functor Ω: [special characters omitted] → [special characters omitted] such that if Gob[special characters omitted] and Hob[special characters omitted], then [special characters omitted](G, Ω(H)) is naturally isomorphic to [special characters omitted] (M(G),H). M is an adjoint of Ω.

Loosely speaking, a [special characters omitted]-algebra is a p-reflexive space with an associative multiplication and a unit; a [special characters omitted]-coalgebra is a p-reflexive space with a coassociative comultiplication and a counit; a [special characters omitted]-bialgebra is a p-reflexive space which simultaneously has the structure of a [special characters omitted]-algebra and a [special characters omitted]-coalgebra with the property that the comultiplication and counit maps are [special characters omitted]-algebra morphisms; and a [special characters omitted]-Hopf algebra is a [special characters omitted]-bialgebra which admits an antipode. k-monoids and k-groups are k-spaces which satisfy the axioms of topological monoids and topological groups respectively, except that the k-space product is used in place of the usual product.

Indexing (document details)
Advisor: Rieffel, Marc A.
Commitee:
School: University of California, Berkeley
School Location: United States -- California
Source: DAI-B 68/07, Dissertation Abstracts International
Source Type: DISSERTATION
Subjects: Mathematics
Keywords: Adjoint functor, Banach-Dieudonne theorem, Compactly supported measures, Dual of Frechet space, Group-like elements, Hopf algebras, Locally convex spaces, Topologically free, k-spaces
Publication Number: 3270245
ISBN: 978-0-549-10900-6
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