scholarly journals The Tensor Category of Linear Maps and Leibniz Algebras

1998 ◽  
Vol 5 (3) ◽  
pp. 263-276
Author(s):  
J. L. Loday ◽  
T. Pirashvili
Keyword(s):  
Tensor Product ◽  
Hopf Algebras ◽  
Type Theorem ◽  
Tensor Category ◽  
Leibniz Algebra ◽  
Vector Spaces ◽  
Associative Algebras ◽  
Enveloping Algebra ◽  
Leibniz Algebras ◽  
Linear Maps

Abstract We equip the category of linear maps of vector spaces with a tensor product which makes it suitable for various constructions related to Leibniz algebras. In particular, a Leibniz algebra becomes a Lie object in and the universal enveloping algebra functor UL from Leibniz algebras to associative algebras factors through the category of cocommutative Hopf algebras in . This enables us to prove a Milnor–Moore type theorem for Leibniz algebras.

scholarly journals Varieties of Null-Filiform Leibniz Algebras Under the Action of Hopf Algebras

2021 ◽  
Author(s):  
Lucio Centrone ◽  
Chia Zargeh
Keyword(s):  
Hopf Algebra ◽  
Free Algebra ◽  
Hopf Algebras ◽  
Leibniz Algebra ◽  
Module Structure ◽  
Leibniz Algebras ◽  
Finite Dimensional ◽  
Filiform Leibniz Algebra ◽  
Algebra Over A Field ◽  
Cocommutative Hopf Algebra

AbstractLet L be an n-dimensional null-filiform Leibniz algebra over a field K. We consider a finite dimensional cocommutative Hopf algebra or a Taft algebra H and we describe the H-actions on L. Moreover we provide the set of H-identities and the description of the Sn-module structure of the relatively free algebra of L.

scholarly journals Racks, Leibniz algebras and Yetter–Drinfel'd modules

2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Ulrich Krähmer ◽  
Friedrich Wagemann
Keyword(s):  
Hopf Algebra ◽  
Leibniz Algebra ◽  
Unified Framework ◽  
Leibniz Algebras ◽  
Linear Maps

AbstractA Hopf algebra object in Loday and Pirashvili's category of linear maps entails an ordinary Hopf algebra and a Yetter–Drinfel'd module. We equip the latter with a structure of a braided Leibniz algebra. This provides a unified framework for examples of racks in the category of coalgebras discussed recently by Carter, Crans, Elhamdadi and Saito.

scholarly journals The Chromatic Brauer Category and Its Linear Representations

2020 ◽  
Author(s):  
L. Felipe Müller ◽  
Dominik J. Wrazidlo
Keyword(s):  
Representation Theory ◽  
Tensor Product ◽  
Topological Field Theories ◽  
Monoidal Category ◽  
Field Theories ◽  
Vector Spaces ◽  
Real Vector ◽  
Linear Representations ◽  
Linear Maps ◽  
Topological Field

AbstractThe Brauer category is a symmetric strict monoidal category that arises as a (horizontal) categorification of the Brauer algebras in the context of Banagl’s framework of positive topological field theories (TFTs). We introduce the chromatic Brauer category as an enrichment of the Brauer category in which the morphisms are component-wise labeled. Linear representations of the (chromatic) Brauer category are symmetric strict monoidal functors into the category of real vector spaces and linear maps equipped with the Schauenburg tensor product. We study representation theory of the (chromatic) Brauer category, and classify all its faithful linear representations. As an application, we use indices of fold lines to construct a refinement of Banagl’s concrete positive TFT based on fold maps into the plane.

scholarly journals Comodules over weak multiplier bialgebras

2014 ◽  
Vol 25 (05) ◽  
pp. 1450037 ◽  
Author(s):  
Gabriella Böhm
Keyword(s):  
Tensor Product ◽  
Hopf Algebras ◽  
Monoidal Category ◽  
Base Algebra ◽  
Finite Dimensional ◽  
Linear Maps ◽  
Multiplier Hopf Algebras ◽  
Hopf Modules ◽  
Total Algebra ◽  
Module Structures

This is a sequel paper of [Weak multiplier bialgebras, Trans. Amer. Math. Soc., in press] in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full comultiplication. Replacing the usual notion of coassociative coaction over a (weak) bialgebra, a comodule is defined via a pair of compatible linear maps. Both the total algebra and the base (co)algebra of a regular weak multiplier bialgebra with a full comultiplication are shown to carry comodule structures. Kahng and Van Daele's integrals [The Larson–Sweedler theorem for weak multiplier Hopf algebras, in preparation] are interpreted as comodule maps from the total to the base algebra. Generalizing the counitality of a comodule to the multiplier setting, we consider the particular class of so-called full comodules. They are shown to carry bi(co)module structures over the base (co)algebra and constitute a monoidal category via the (co)module tensor product over the base (co)algebra. If a regular weak multiplier bialgebra with a full comultiplication possesses an antipode, then finite-dimensional full comodules are shown to possess duals in the monoidal category of full comodules. Hopf modules are introduced over regular weak multiplier bialgebras with a full comultiplication. Whenever there is an antipode, the Fundamental Theorem of Hopf Modules is proven. It asserts that the category of Hopf modules is equivalent to the category of firm modules over the base algebra.

scholarly journals A universal enveloping algebra for cocommutative rack bialgebras

2019 ◽  
Vol 31 (5) ◽  
pp. 1305-1315
Author(s):  
Ulrich Krähmer ◽  
Friedrich Wagemann
Keyword(s):  
Universal Enveloping Algebra ◽  
Leibniz Algebra ◽  
Enveloping Algebra ◽  
Linear Maps ◽  
Deformation Complex

AbstractWe construct a bialgebra object in the category of linear maps {\mathcal{LM}} from a cocommutative rack bialgebra. The construction does extend to some non-cocommutative rack bialgberas, as is illustrated by a concrete example. As a separate result, we show that the Loday complex with adjoint coefficients embeds into the rack bialgebra deformation complex for the rack bialgebra defined by a Leibniz algebra.

scholarly journals EXTENDED QUANTUM ENVELOPING ALGEBRAS OF (2)

2009 ◽  
Vol 51 (3) ◽  
pp. 441-465 ◽  
Author(s):  
WU ZHIXIANG
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Cyclic Group ◽  
Hopf Algebras ◽  
Infinite Cyclic Group ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Quantized Enveloping Algebra ◽  
Usual Quantum ◽  
Cocommutative Hopf Algebra

AbstractIn present paper we define a new kind of quantized enveloping algebra of (2). We denote this algebra by Ur,t, where r, t are two non-negative integers. It is a non-commutative and non-cocommutative Hopf algebra. If r = 0, then the algebra Ur,t is isomorphic to a tensor product of the algebra of infinite cyclic group and the usual quantum enveloping algebra of (2) as Hopf algebras. The representation of this algebra is studied.

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.

Positive Linear Maps on C*-Algebras

1972 ◽  
Vol 24 (3) ◽  
pp. 520-529 ◽  
Author(s):  
Man-Duen Choi
Keyword(s):  
Vector Spaces ◽  
Complex Numbers ◽  
Complex Matrices ◽  
Completely Positive ◽  
C Algebras ◽  
Linear Maps ◽  
Positive Linear Maps

The objective of this paper is to give some concrete distinctions between positive linear maps and completely positive linear maps on C*-algebras of operators.Herein, C*-algebras possess an identity and are written in German type . Capital letters A, B, C stand for operators, script letters for vector spaces, small letters x, y, z for vectors. Capital Greek letters Φ, Ψ stand for linear maps on C*-algebras, small Greek letters α, β, γ for complex numbers.We denote by the collection of all n × n complex matrices. () = ⊗ is the C*-algebra of n × n matrices over .

ON THE RIESZ REPRESENTATION THEOREM IN TOPOLOGICAL VECTOR SPACES

2000 ◽  
Vol 36 (3-4) ◽  
pp. 347-352
Author(s):  
M. A. Alghamdi ◽  
L. A. Khan ◽  
H. A. S. Abujabal
Keyword(s):  
Representation Theorem ◽  
Type Theorem ◽  
Representation Type ◽  
Vector Spaces ◽  
Riesz Representation Theorem ◽  
Topological Vector Spaces ◽  
Riesz Representation

I this paper we establish a Riesz representation type theorem which characterizes the dual of the space C rc (X,E)endowed with the countable-ope topologyi the case of E ot ecessarilya locallyconvex TVS.

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