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Efficient construction of Drinfel'd doubles

Gerard P. Brunick, Edward L. Green, Lenwood S. Heath, Craig A. Struble
1999 Proceedings of the 1999 international symposium on Symbolic and algebraic computation - ISSAC '99  
In this paper, we present our initial implementation 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.  ... 
doi:10.1145/309831.309856 dblp:conf/issac/BrunickGHS99 fatcat:o7iewhe26ngmjeqijru6eypsea

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.  ... 

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.  ... 

Hopf Algebra Symmetries of an Integrable Hamiltonian for Anyonic Pairing

Jon Links
2012 Axioms  
Rather, they are associated with the 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).  ... 
doi:10.3390/axioms1020226 fatcat:jd2dw7phhnga5exjsbilwf5v2m

Towards a dual spin network basis for (3+1)d lattice gauge theories and topological phases

Clement Delcamp, Bianca Dittrich
2018 Journal of High Energy Physics  
We distinguish in particular a dual pair 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.  ... 
doi:10.1007/jhep10(2018)023 fatcat:efb5sby3tffhzhxgsvary4rj2u

Twisted noncommutative equivariant cohomology: Weil and Cartan models [article]

Lucio Cirio
2009 arXiv   pre-print
The 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  ... 
arXiv:0706.3602v3 fatcat:apyympqtb5hntgd5dg36bnbn5u

Fusion basis for lattice gauge theory and loop quantum gravity

Clement Delcamp, Bianca Dittrich, Aldo Riello
2017 Journal of High Energy Physics  
These excitations are classified by the irreducible representations 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.  ... 
doi:10.1007/jhep02(2017)061 fatcat:5nt6pjej3zc3fppodiqscu3zia

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).  ... 

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K. C. H. Mackenzie
2011 Electronic Research Announcements of the American Mathematical Society  
We show that the Manin triple characterization 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  ... 
doi:10.1090/s1079-6762-98-00050-x fatcat:eebqdef3rvhdrm36vb7elvatcq

A simplicial complex of Nichols algebras [article]

Michael Cuntz, Simon Lentner
2016 arXiv   pre-print
We translate the concept 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.  ... 
arXiv:1503.08117v2 fatcat:kk7u53poyzf6fauafclsx6flia

Quantum algebras as quantizations of dual Poisson–Lie groups

Ángel Ballesteros, Fabio Musso
2013 Journal of Physics A: Mathematical and Theoretical  
The applicability 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  ... 
doi:10.1088/1751-8113/46/19/195203 fatcat:hvouwvimpvca3exc4x234byiw4

The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective [article]

Luigi Alfonsi
2020 arXiv   pre-print
The history 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.  ... 
arXiv:2007.04969v3 fatcat:ip7dejn2t5e4vbuxnccoh2jfwm

Doubled Aspects of Vaisman Algebroid and Gauge Symmetry in Double Field Theory [article]

Haruka Mori, Shin Sasaki, Kenta Shiozawa
2019 arXiv   pre-print
We show that the Vaisman algebroid is obtained by an analogue 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.  ... 
arXiv:1901.04777v1 fatcat:pgfpeohycfaehln2ealdxrqxmm

Contractions on the Classical Double [article]

Angel Ballesteros, Mariano del Olmo
1997 arXiv   pre-print
Lie algebra contractions on the classical 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] .  ... 
arXiv:q-alg/9702031v1 fatcat:dwpodoked5eydjg24tkoq4lcam

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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