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Efficient construction of Drinfel'd doubles

1999
*
Proceedings of the 1999 international symposium on Symbolic and algebraic computation - ISSAC '99
*

In this paper, we present our initial implementation

doi:10.1145/309831.309856
dblp:conf/issac/BrunickGHS99
fatcat:o7iewhe26ngmjeqijru6eypsea
*of*the*Drinfel'd**double**construction*and report on performance issues. ... In particular, we develop the framework*of*encoding neutral algorithm analysis and apply the framework to the algorithms required in the*construction**of*the*Drinfel'd**double*. ... Acknowledgments The authors gratefully acknowledge the support*of*NSF grant CCR-9732068. ...##
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Page 162 of Mathematical Reviews Vol. , Issue 2003A
[page]

2003
*
Mathematical Reviews
*

This class

*of*Lie bialgebras is closed under the operations*of*taking the dual and the*Drinfel’d**double*. ... Let D(g) = g+g* be the*Drinfel’d**double**of*g, i*: g* — D(g) be the corresponding natural embedding and G’ be the subgroup*of*a*double*D(G) with Lie algebra Lie(G’) = i*(g*) C D(g). It was shown by M. ...##
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Page 3514 of Mathematical Reviews Vol. , Issue 97F
[page]

1997
*
Mathematical Reviews
*

Following

*Drinfel’d*the authors consider a category # whose objects are g-modules and whose morphisms Hom,g(U, W) = Hom(U, W )[A] are formal power series with co-*efficients*in Hom( U, W ). ... He uses the above decomposition to*construct*a neighborhood U*of*the identity in G. ...##
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Hopf Algebra Symmetries of an Integrable Hamiltonian for Anyonic Pairing

2012
*
Axioms
*

Rather, they are associated with the

doi:10.3390/axioms1020226
fatcat:jd2dw7phhnga5exjsbilwf5v2m
*Drinfel'd**doubles**of*dihedral group algebras D(D n ). ... Since the advent*of*Drinfel'd's*double**construction*, Hopf algebraic structures have been a centrepiece for many developments in the theory and analysis*of*integrable quantum systems. ... Acknowledgments This work was supported by the Australian Research Council through Discovery Project Topological properties*of*exactly solvable, two-dimensional quantum systems (DP110101414). ...##
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Towards a dual spin network basis for (3+1)d lattice gauge theories and topological phases

2018
*
Journal of High Energy Physics
*

We distinguish in particular a dual pair

doi:10.1007/jhep10(2018)023
fatcat:efb5sby3tffhzhxgsvary4rj2u
*of*boundary conditions, namely*of*electric type and*of*magnetic type. ... This can be used to consider a generalization*of*Ocneanu's tube algebra in order to reveal the algebraic structure*of*the excitations associated with certain 3d manifolds. ...*of*Research, Innovation and Science. ...##
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Twisted noncommutative equivariant cohomology: Weil and Cartan models
[article]

2009
*
arXiv
*
pre-print

The

arXiv:0706.3602v3
fatcat:apyympqtb5hntgd5dg36bnbn5u
*construction*is suggested by the noncommutative Weil algebra*of*Alekseev and Meinrenken; we show that one can implement a*Drinfel'd*twist*of*their models in order to take into account the noncommutativity ... We propose Weil and Cartan models for the equivariant cohomology*of*noncommutative spaces which carry a covariant action*of**Drinfel'd*twisted symmetries. ... This homomorphism can be*constructed*for a very ...##
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Fusion basis for lattice gauge theory and loop quantum gravity

2017
*
Journal of High Energy Physics
*

These excitations are classified by the irreducible representations

doi:10.1007/jhep02(2017)061
fatcat:5nt6pjej3zc3fppodiqscu3zia
*of*the*Drinfel'd**double**of*the gauge group, and can be readily "fused" together by studying the tensor product*of*such representations ... In a follow-up work, we will exploit this notion to provide a new definition*of*entanglement entropy for these theories. ... Research at Perimeter Institute is supported by the Government*of*Canada through the Department*of*Innovation, Science Open Access. ...##
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Page 6694 of Mathematical Reviews Vol. , Issue 91M
[page]

1991
*
Mathematical Reviews
*

91m:58012
The idea

*of*Woronowicz (and*Drinfel’d*) is that, since the cat- egory*of*groups is rigid under deformations, the quantum de- formations should instead be made in some dual category, e.g., functions ... The computational virtue*of*this*construction*is that identities can be formulated in terms*of*diagrams (like Feyn- man diagrams, and yet completely different). ...
We show that the Manin triple characterization

doi:10.1090/s1079-6762-98-00050-x
fatcat:eebqdef3rvhdrm36vb7elvatcq
*of*Lie bialgebras in terms*of*the*Drinfel'd**double*may be extended to arbitrary Poisson manifolds and indeed Lie bialgebroids by using*double*cotangent bundles ... This is achieved in terms*of*an abstract notion*of**double*Lie algebroid (where "*double*" is now used in the Ehresmann sense) which unifies many iterated*constructions*in differential geometry. ... It is in the same sense that we say that*Drinfel'd**doubles*and Ehresmann*doubles*are coextensive: Theorem 4.5 shows that the*Drinfel'd**double**of*a Lie bialgebroid has a*double*structure in the sense*of*...##
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A simplicial complex of Nichols algebras
[article]

2016
*
arXiv
*
pre-print

We translate the concept

arXiv:1503.08117v2
fatcat:kk7u53poyzf6fauafclsx6flia
*of*restriction*of*an arrangement in terms*of*Hopf algebras. ... As applications, some*of*these Nichols algebras provide Weyl groupoids which do not arise for Nichols algebras over finite groups and in fact we realize all root systems*of*finite Weyl groupoids*of*rank ... As a second application, we note that quantum groups u q (g) (and other pointed Hopf algebras [AY13] ) can be obtained from a*Drinfel'd**double**construction*. ...##
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Quantum algebras as quantizations of dual Poisson–Lie groups

2013
*
Journal of Physics A: Mathematical and Theoretical
*

The applicability

doi:10.1088/1751-8113/46/19/195203
fatcat:hvouwvimpvca3exc4x234byiw4
*of*this approach is explicitly demonstrated by*constructing*several instances*of*quantum deformations*of*physically relevant Lie algebras as sl(2,R), the (2+1) Anti de Sitter algebra ... A systematic computational approach for the explicit*construction**of*any quantum Hopf algebra (U_z(g),\Delta_z) starting from the Lie bialgebra (g,\delta) that gives the first-order deformation*of*the ... Also, in order to obtain*Double*Special Relativity models with non-vanishing cosmological constant [40, 41] the obtention*of*the*Drinfel'd*-Jimbo deformation for so (3, 2) and so(4, 1) by following ...##
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The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective
[article]

2020
*
arXiv
*
pre-print

The history

arXiv:2007.04969v3
fatcat:ip7dejn2t5e4vbuxnccoh2jfwm
*of*the geometry*of**Double*Field Theory is the history*of*string theorists' effort to tame higher geometric structures. ... In arXiv:1912.07089 [hep-th] we proposed that the global*doubled*space is not a manifold, but the total space*of*a bundle gerbe. ... However, as we will see, Born geometry is still extremely*efficient*in dealing with*doubled*group manifold and, in particular,*Drinfel'd**doubles*. ...##
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Doubled Aspects of Vaisman Algebroid and Gauge Symmetry in Double Field Theory
[article]

2019
*
arXiv
*
pre-print

We show that the Vaisman algebroid is obtained by an analogue

arXiv:1901.04777v1
fatcat:pgfpeohycfaehln2ealdxrqxmm
*of*the*Drinfel'd**double**of*Lie algebroids. ... Based on a geometric realization*of**doubled*space-time as a para-Hermitian manifold, we examine exterior algebras and a para-Dolbeault cohomology on DFT and discuss the structure*of*the*Drinfel'd**double*... Acknowledgments The work*of*S.S. is supported by the Japan Society for the Promotion*of*Science (JSPS) KAKENHI Grant Number JP17K14294. ...##
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Contractions on the Classical Double
[article]

1997
*
arXiv
*
pre-print

Lie algebra contractions on the classical

arXiv:q-alg/9702031v1
fatcat:dwpodoked5eydjg24tkoq4lcam
*Drinfel'd**Double**of*a given Lie bialgebra are introduced and compared to the usual Lie bialgebra contraction theory. ... The non-relativistic (Galilean) limit*of*a (1+1) Poincar\'e*Double*is used to illustrate the contraction process. ... allowed to*construct*quantum deformations*of*nonsemisimple algebras in a very*efficient*way [4] . ...##
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Page 8355 of Mathematical Reviews Vol. , Issue 2000m
[page]

2000
*
Mathematical Reviews
*

Brown and Weng Kee Wong, An algorithmic

*construction**of*optimal minimax designs for heteroscedastic linear models (103-114); N. ... Zalesskii, The profi- nite completion*of*certain torsion p-groups (113-123); Mikhail Grinenko, Birational correspondences*of*a*double*cone (125-133); Hermann Heineken, Groups with neighbourhood conditions ...
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