scholarly journals A TRIANGULAR DEFORMATION OF THE TWO-DIMENSIONAL POINCARÉ ALGEBRA

1995 ◽  
Vol 10 (11) ◽  
pp. 873-883 ◽  
Author(s):  
M. KHORRAMI ◽  
A. SHARIATI ◽  
M.R. ABOLHASSANI ◽  
A. AGHAMOHAMMADI
Keyword(s):  
Hopf Algebra ◽  
Lorentz Transformation ◽  
Hopf Algebras ◽  
Universal Enveloping Algebra ◽  
Two Dimensional ◽  
R Matrix ◽  
Enveloping Algebra ◽  
Differential Structure ◽  
Algebra Of Functions ◽  
Poincaré Algebra

Contracting the h-deformation of SL(2, ℝ), we construct a new deformation of two-dimensional Poincaré's algebra, the algebra of functions on its group and its differential structure. It is seen that these dual Hopf algebras are isomorphic to each other. It is also shown that the Hopf algebra is triangular, and its universal R-matrix is also constructed explicitly. We then find a deformation map for the universal enveloping algebra, and at the end, give the deformed mass shells and Lorentz transformation.

On the Hopf algebra dual of an enveloping algebra

1982 ◽  
Vol 91 (2) ◽  
pp. 215-224 ◽  
Author(s):  
Stephen Donkin
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Algebraic Groups ◽  
Field Extension ◽  
Smash Product ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Polycyclic Groups ◽  
Algebra Over A Field

In (1) it is claimed that the main results of that paper have applications to the representation theory of algebraic groups, of polycyclic groups and of Lie algebras. An application to algebraic groups is given in Corollary 6·4 of (1), the applications to polycyclic groups are given in (2), the purpose of this work is to deal with the outstanding case of enveloping algebras. To make use of the results of (1), in this context, we show that the Hopf algebra dual of the enveloping algebra of a finite dimensional Lie algebra over a field of characteristic zero is quasi-affine (see § 1·5). This is done by an easy field extension argument and a generalization, to the Hopf algebra dual of the smash product of Hopf algebras, of Proposition 1·6·3 of (2) on the dual of the group algebra of a semidirect product of groups. Since this paper is aimed at those readers interested in enveloping algebras, the Hopf theoretic aspects are dealt with at a fairly leisurely pace.

scholarly journals JORDANIAN QUANTUM SUPERALGEBRA Uh(osp(2/1))

2003 ◽  
Vol 18 (13) ◽  
pp. 885-903 ◽  
Author(s):  
N. AIZAWA ◽  
R. CHAKRABARTI ◽  
J. SEGAR
Keyword(s):  
Algebraic Structure ◽  
Lie Superalgebra ◽  
Universal Enveloping Algebra ◽  
Borel Subalgebra ◽  
Quantum Deformation ◽  
R Matrix ◽  
Enveloping Algebra ◽  
Quantum Superalgebra ◽  
Tensor Operators

A quantum deformation of the Lie superalgebra osp(2/1) from the classical r-matrix including an odd generator is presented with its full Hopf algebraic structure. A class of deformation maps and the corresponding twisting elements, the interrelation between these twists, and the tensor operators are considered for the deformed osp(2/1) algebra. It is also shown that the Borel subalgebra of the universal enveloping algebra of the quantized osp(2/1) is self-dual.

scholarly journals QUASI THREE-PARAMETRIC R-MATRIX AND QUANTUM SUPERGROUPS GLp,q(1/1) AND Up,q[gl(1/1)]

2019 ◽  
Vol 29 (4) ◽  
pp. 511
Author(s):  
Nguyen Thi Hong Van ◽  
Nguyen Anh Ky
Keyword(s):  
Universal Enveloping Algebra ◽  
Present Approach ◽  
Baxter Equation ◽  
Quantum Deformation ◽  
R Matrix ◽  
Enveloping Algebra ◽  
Quantum Deformations ◽  
First Time

An overparametrized (three-parametric) R-matrix satisfying a graded Yang-Baxter equation is introduced. It turns out that such an overparametrization is very helpful. Indeed, this R-matrix with one of the parameters being auxiliary, thus, reducible to a two-parametric R-matrix, allows the construction of quantum supergroups GLp,q(1/1) and Up,q[gl(1/1)] which, respectively, are two-parametric deformations of the supergroup GL(1/1) and the universal enveloping algebra U[gl(1/1)]. These two-parametric quantum deformations GLpq(1/1) and Upq[gl(1/1)], to our knowledge, are constructed for the first time via the present approach. The quantum deformation Up,q[gl(1/1)] obtained here is a true two-parametric deformation of Drinfel’d-Jimbo’s type, unlike some other one obtained previously elsewhere.

The depth and LS category of a topological space

2018 ◽  
Vol 123 (2) ◽  
pp. 220-238
Author(s):  
Yves Félix ◽  
Steve Halperin
Keyword(s):  
Topological Space ◽  
Hopf Algebra ◽  
Lie Algebra ◽  
Finite Type ◽  
Universal Enveloping Algebra ◽  
Primitive Elements ◽  
Enveloping Algebra ◽  
Cw Complex ◽  
Lusternik Schnirelmann ◽  
Simply Connected

The depth of an augmented ring $\varepsilon \colon A\to k $ is the least $p$, or ∞, such that \begin {equation*} \Ext _A^p(k , A)\neq 0. \end {equation*} When $X$ is a simply connected finite type CW complex, $H_*(\Omega X;\mathbb {Q})$ is a Hopf algebra and the universal enveloping algebra of the Lie algebra $L_X$ of primitive elements. It is known that $\depth H_*(\Omega X;\mathbb {Q}) \leq \cat X$, the Lusternik-Schnirelmann category of $X$. For any connected CW complex we construct a completion $\widehat {H}(\Omega X)$ of $H_*(\Omega X;\mathbb {Q})$ as a complete Hopf algebra with primitive sub Lie algebra $L_X$, and define $\depth X$ to be the least $p$ or ∞ such that \[ \Ext ^p_{UL_X}(\mathbb {Q}, \widehat {H}(\Omega X))\neq 0. \] Theorem: for any connected CW complex, $\depth X\leq \cat X$.

THE PRIME IDEALS AND SIMPLE MODULES OF THE UNIVERSAL ENVELOPING ALGEBRA U(𝔟⋉V2)

2019 ◽  
Vol 62 (S1) ◽  
pp. S77-S98 ◽  
Author(s):  
VOLODYMYR V. BAVULA ◽  
TAO LU
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Direct Product ◽  
Prime Factor ◽  
Universal Enveloping Algebra ◽  
Explicit Description ◽  
Prime Ideals ◽  
Two Dimensional ◽  
Enveloping Algebra ◽  
Torsion Modules

AbstractLet 𝔟 be the Borel subalgebra of the Lie algebra 𝔰𝔩2 and V2 be the simple two-dimensional 𝔰𝔩2-module. For the universal enveloping algebra $\[{\cal A}: = U(\gb \ltimes {V_2})\]$ of the semi-direct product 𝔟⋉V2 of Lie algebras, the prime, primitive and maximal spectra are classified. Please approve edit to the sentence “The sets of completely prime…”.The sets of completely prime ideals of $\[{\cal A}\]$ are described. The simple unfaithful $\[{\cal A}\]$-modules are classified and an explicit description of all prime factor algebras of $\[{\cal A}\]$ is given. The following classes of simple U(𝔟⋉V2)-modules are classified: the Whittaker modules, the 𝕂[X]-torsion modules and the 𝕂[E]-torsion modules.

scholarly journals On representation theory of total (co)integrals

2016 ◽  
Vol 15 (10) ◽  
pp. 1650195
Author(s):  
Mohammad Hassanzadeh
Keyword(s):  
Representation Theory ◽  
Hopf Algebra ◽  
Polynomial Algebra ◽  
Universal Enveloping Algebra ◽  
Drinfeld Modules ◽  
Enveloping Algebra

In this paper, we show that total integrals and cointegrals are new sources of stable anti-Yetter–Drinfeld modules. We explicitly show that how special types of total (co)integrals can be used to provide both (stable) anti Yetter–Drinfeld and Yetter–Drinfeld modules. We use these modules to classify total (co)integrals and (cleft) Hopf Galois (co)extensions of the Connes–Moscovici Hopf algebra, and some examples of universal enveloping algebra and polynomial algebra.

scholarly journals Hopf algebra gauge theory on a ribbon graph

2021 ◽  
pp. 2150016
Author(s):  
Catherine Meusburger ◽  
Derek K. Wise
Keyword(s):  
Gauge Theory ◽  
Hopf Algebra ◽  
Hopf Algebras ◽  
Simons Theory ◽  
Ribbon Graph ◽  
Local Data ◽  
Tensor Power ◽  
Flat Connections ◽  
Algebra Of Functions ◽  
Embedded Graph

We generalize gauge theory on a graph so that the gauge group becomes a finite-dimensional ribbon Hopf algebra, the graph becomes a ribbon graph, and gauge-theoretic concepts such as connections, gauge transformations and observables are replaced by linearized analogs. Starting from physical considerations, we derive an axiomatic definition of Hopf algebra gauge theory, including locality conditions under which the theory for a general ribbon graph can be assembled from local data in the neighborhood of each vertex. For a vertex neighborhood with [Formula: see text] incoming edge ends, the algebra of non-commutative ‘functions’ of connections is dual to a two-sided twist deformation of the [Formula: see text]-fold tensor power of the gauge Hopf algebra. We show these algebras assemble to give an algebra of functions and gauge-invariant subalgebra of ‘observables’ that coincide with those obtained in the combinatorial quantization of Chern–Simons theory, thus providing an axiomatic derivation of the latter. We then discuss holonomy in a Hopf algebra gauge theory and show that for semisimple Hopf algebras this gives, for each path in the embedded graph, a map from connections into the gauge Hopf algebra, depending functorially on the path. Curvatures — holonomies around the faces canonically associated to the ribbon graph — then correspond to central elements of the algebra of observables, and define a set of commuting projectors onto the subalgebra of observables on flat connections. The algebras of observables for all connections or for flat connections are topological invariants, depending only on the topology, respectively, of the punctured or closed surface canonically obtained by gluing annuli or discs along edges of the ribbon graph.

TWISTED DRINFELD DOUBLES AND REPRESENTATIONS OF A HOPF ALGEBRA

2012 ◽  
Vol 11 (06) ◽  
pp. 1250118
Author(s):  
LI BAI ◽  
SHUANHONG WANG
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Direct Proof ◽  
Irreducible Representations ◽  
Quantum Double ◽  
Drinfeld Double ◽  
Braided Group ◽  
R Matrix ◽  
Coalgebra Structure

In this paper, we consider a class of non-commutative and non-cocommutative Hopf algebras Hp(α, q, m) and then show that these Hopf algebras can be realized as a quantum double of certain Hopf algebras with respect to Hopf skew pairings (Ap(q, m), Bp(q, m), τα). Furthermore, using the Hopf skew pairing with appropriate group homomorphisms: ϕ : π → Aut (Ap(q, m)) and ψ : π → Aut (Bp(q, m)), we construct a twisted Drinfeld double D(Ap(q, m), Bp(q, m), τ; ϕ, ψ) which is a Turaev [Formula: see text]-coalgebra, where the group [Formula: see text] is a twisted semi-direct square of a group π. Then we obtain its quasi-triangular Turaev [Formula: see text]-coalgebra structure. We also study irreducible representations of Hp(1, q, m) and construct a corresponding R-matrix. Finally, we introduce the notion of a left Yetter–Drinfeld category over a Turaev group coalgebra and show that such a category is a Turaev braided group category by a direct proof, without center construction. As an application, we consider the case of the quasi-triangular Turaev [Formula: see text]-coalgebra structure on our twisted Drinfeld double.

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