The shuffle Hopf algebra and noncommutative full completeness

1998 ◽  
Vol 63 (4) ◽  
pp. 1413-1436 ◽  
Author(s):  
R. F. Blute ◽  
P. J. Scott
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Linear Logic ◽  
Natural Extension ◽  
Additive Group ◽  
Completeness Theorem ◽  
Vector Spaces ◽  
Invariance Condition ◽  
Shuffle Algebra ◽  
Topological Vector Spaces

AbstractWe present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformations being equivariant under certain actions of a noncocommutative Hopf algebra called the shuffle algebra Multiplicative sequents are assigned a vector space of such dinaturals, and we show that this space has as a basis the denotations of cut-free proofs in CyLL + MIX. This can be viewed as a fully faithful representation of a free *-autonomous category, canonically enriched over vector spaces.This paper is a natural extension of the authors' previous work, “Linear Läuchli Semantics”, where a similar theorem is obtained for the commutative logic MLL + MIX. In that paper, we interpret proofs as dinaturals which are invariant under certain actions of the additive group of integers. Here we also present a simplification of that work by showing that the invariance criterion is actually a consequence of dinaturality. The passage from groups to Hopf algebras in this paper corresponds to the passage from commutative to noncommutative logic. However, in our noncommutative setting, one must still keep the invariance condition on dinaturals.

Hopf algebras and linear logic

1996 ◽  
Vol 6 (2) ◽  
pp. 189-212 ◽  
Author(s):  
Richard F. Blute
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Linear Logic ◽  
Vector Spaces ◽  
Monoidal Categories ◽  
Simple Manner ◽  
Linear Topology ◽  
Topological Vector Spaces ◽  
Expository Paper ◽  
Involutive Negation

It has recently become evident that categories of representations of Hopf algebras provide fundamental examples of monoidal categories. In this expository paper, we examine such categories as models of (multiplicative) linear logic. By varying the Hopf algebra, it is possible to model several variants of linear logic. We present models of the original commutative logic, the noncommutative logic of Lambek and Abrusci, the braided variant due to the author, and the cyclic logic of Yetter. Hopf algebras provide a unifying framework for the analysis of these variants. While these categories are monoidal closed, they lack sufficient structure to model the involutive negation of classical linear logic. We recall work of Lefschetz and Barr in which vector spaces are endowed with an additional topological structure, called linear topology. The resulting category has a large class of reflexive objects, which form a *-autonomous category, and so model the involutive negation. We show that the monoidal closed structure of the category of representations of a Hopf algebra can be extended to this topological category in a natural and simple manner. The models we obtain have the advantage of being nondegenerate in the sense that the two multiplicative connectives, tensor and par, are not equated. It has been recently shown by Barr that this category of topological vector spaces can be viewed as a subcategory of a certain Chu category. In an Appendix, Barr uses this equivalence to analyze the structure of its tensor product.

scholarly journals Multiplier Hopf algebras: Globalization for partial actions

2019 ◽  
Vol 30 (03) ◽  
pp. 539-565
Author(s):  
Graziela Fonseca ◽  
Eneilson Fontes ◽  
Grasiela Martini
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Action Theory ◽  
Natural Extension ◽  
Partial Action ◽  
Pertinent Question ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra

In partial action theory, a pertinent question is whenever given a partial action of a Hopf algebra [Formula: see text] on an algebra [Formula: see text], it is possible to construct an enveloping action. The authors Alves and Batista, in [M. Alves and E. Batista, Globalization theorems for partial Hopf (co)actions and some of their applications, groups, algebra and applications, Contemp. Math. 537 (2011) 13–30], have shown that this is always possible if [Formula: see text] is unital. We are interested in investigating the situation, where both algebras [Formula: see text] and [Formula: see text] are not necessarily unitary. A nonunitary natural extension for the concept of Hopf algebras was proposed by Van Daele, in [A. Van Daele, Multiplier Hopf algebras, Trans. Am. Math. Soc. 342 (1994) 917–932], which is called multiplier Hopf algebra. Therefore, we will consider partial actions of multipliers Hopf algebras on algebras with a nondegenerate product and we will present a globalization theorem for this structure. Moreover, Dockuchaev et al. in [Globalizations of partial actions on nonunital rings, Proc. Am. Math. Soc. 135 (2007) 343–352], have shown when group partial actions on nonunitary algebras are globalizable. Based on this paper, we will establish a bijection between globalizable group partial actions and partial actions of a multiplier Hopf algebra.

scholarly journals A quantum double construction in Rel

2012 ◽  
Vol 22 (4) ◽  
pp. 618-650 ◽  
Author(s):  
MASAHITO HASEGAWA
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Linear Logic ◽  
Binary Relations ◽  
Monoidal Categories ◽  
Quantum Double ◽  
Closed Category ◽  
Category Of Modules ◽  
Extra Structure

We study bialgebras and Hopf algebras in the compact closed categoryRelof sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of bialgebras and Hopf algebras inRel. In particular, for any groupG, we derive a ribbon category of crossedG-sets as the category of modules of a Hopf algebra inRelthat is obtained by the quantum double construction. This category of crossedG-sets serves as a model of the braided variant of propositional linear logic.

On Köthe sequence spaces and linear logic

2002 ◽  
Vol 12 (5) ◽  
pp. 579-623 ◽  
Author(s):  
THOMAS EHRHARD
Keyword(s):  
Simple Structure ◽  
Linear Logic ◽  
Sequence Spaces ◽  
Vector Spaces ◽  
Topological Vector Spaces ◽  
Cartesian Closed ◽  
Linear Category ◽  
Köthe Sequence Spaces ◽  
Locally Convex ◽  
Standard Concept

We present a category of locally convex topological vector spaces that is a model of propositional classical linear logic and is based on the standard concept of Köthe sequence spaces. In this setting, the ‘of course’ connective of linear logic has a quite simple structure of a commutative Hopf algebra. The co-Kleisli category of this linear category is a cartesian closed category of entire mappings. This work provides a simple setting in which typed λ-calculus and differential calculus can be combined; we give a few examples of computations.

scholarly journals Bimonads and Hopf monads on categories

2010 ◽  
Vol 7 (2) ◽  
pp. 349-388 ◽  
Author(s):  
Bachuki Mesablishvili ◽  
Robert Wisbauer
Keyword(s):  
Hopf Algebra ◽  
Vector Space ◽  
Hopf Algebras ◽  
Natural Transformation ◽  
Vector Spaces ◽  
Baxter Equation ◽  
Basic Tool ◽  
Twist Map ◽  
Finite Dimensional ◽  
Base Category

AbstractThe purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal structure of the category of endofunctors on any category and by this we retain some of the combinatorial complexity which makes the theory so interesting. As a basic tool we use distributive laws between monads and comonads (entwinings) on : we define a bimonad on as an endofunctor B which is a monad and a comonad with an entwining λ : BB → BB satisfying certain conditions. This λ is also employed to define the category of (mixed) B-bimodules. In the classical situation, an entwining λ is derived from the twist map for vector spaces. Here this need not be the case but there may exist special distributive laws τ : BB → BB satisfying the Yang-Baxter equation (local prebraidings) which induce an entwining λ and lead to an extension of the theory of braided Hopf algebras.An antipode is defined as a natural transformation S : B → B with special properties. For categories with limits or colimits and bimonads B preserving them, the existence of an antipode is equivalent to B inducing an equivalence between and the category of B-bimodules. This is a general form of the Fundamental Theorem of Hopf algebras.Finally we observe a nice symmetry: If B is an endofunctor with a right adjoint R, then B is a (Hopf) bimonad if and only if R is a (Hopf) bimonad. Thus a k-vector space H is a Hopf algebra if and only if Homk(H,−) is a Hopf bimonad. This provides a rich source for Hopf monads not defined by tensor products and generalises the well-known fact that a finite dimensional k-vector space H is a Hopf algebra if and only if its dual H* = Homk(H,k) is a Hopf algebra. Moreover, we obtain that any set G is a group if and only if the functor Map(G,−) is a Hopf monad on the category of sets.

Finite dimensional locally convex cones

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5111-5116
Author(s):  
Davood Ayaseha
Keyword(s):  
Finite Dimension ◽  
Convex Cones ◽  
Vector Spaces ◽  
Topological Vector Spaces ◽  
Euclidean Topology ◽  
Finite Dimensional ◽  
Dimensional Cone ◽  
Locally Convex Cones ◽  
Locally Convex ◽  
Special Case

We study the locally convex cones which have finite dimension. We introduce the Euclidean convex quasiuniform structure on a finite dimensional cone. In special case of finite dimensional locally convex topological vector spaces, the symmetric topology induced by the Euclidean convex quasiuniform structure reduces to the known concept of Euclidean topology. We prove that the dual of a finite dimensional cone endowed with the Euclidean convex quasiuniform structure is identical with it?s algebraic dual.

scholarly journals EXAMPLES OF FINITE-DIMENSIONAL POINTED HOPF ALGEBRAS IN CHARACTERISTIC 2

2020 ◽  
pp. 1-14
Author(s):  
NICOLÁS ANDRUSKIEWITSCH ◽  
DIRCEU BAGIO ◽  
SARADIA DELLA FLORA ◽  
DAIANA FLÔRES
Keyword(s):  
Hopf Algebras ◽  
Abelian Groups ◽  
Vector Spaces ◽  
Group Algebras ◽  
Finite Abelian Groups ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Characteristic 2 ◽  
Nichols Algebras ◽  
Kirillov Dimension

Abstract We present new examples of finite-dimensional Nichols algebras over fields of characteristic 2 from braided vector spaces that are not of diagonal type, admit realizations as Yetter–Drinfeld modules over finite abelian groups, and are analogous to Nichols algebras of finite Gelfand–Kirillov dimension in characteristic 0. New finite-dimensional pointed Hopf algebras over fields of characteristic 2 are obtained by bosonization with group algebras of suitable finite abelian groups.

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