Hopf algebras in non-associative Lie theory

Mostovoy, Jacob; Pérez-Izquierdo, José M.; Shestakov, I. P.

doi:10.1007/s13373-013-0049-8

Cited by 35 publications

(20 citation statements)

References 48 publications

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“…Associative H-bialgebras are exactly associative Hopf algebras. For this reason nonassociative H-bialgebras are called also nonassociative Hopf algebras [8].…”

Section: Braided Algebras and Bialgebrasmentioning

confidence: 99%

“…A construction of universal enveloping algebras, which is very similar in their properties to usual cocommutative Hopf algebras, can be carried out for Bol algebras [9] and, more generally, for all Sabinin algebras [10]. The role of the nonassociative Hopf algebras in the fundamental questions of Lie theory such as integration was clarified in [7,8], where the authors describe the current understanding of the subject in view of the recent works, many of which use nonassociative Hopf algebras as the main tool.…”

Section: Introductionmentioning

confidence: 99%

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Free braided nonassociative Hopf algebras and Sabinin τ-algebras

Umirbaev

Kharchenko

2017

Journal of Algebra

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“…Associative H-bialgebras are exactly associative Hopf algebras. For this reason nonassociative H-bialgebras are called also nonassociative Hopf algebras [8].…”

Section: Braided Algebras and Bialgebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Free braided nonassociative Hopf algebras and Sabinin τ-algebras

Umirbaev

Kharchenko

2017

Journal of Algebra

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“…An analogous formula for non-associative algebras has appeared in the literature based on a particular definition of the exponential function so that e A e A = e 2A [37] (but see also [38] for an update of the recent developments in the subject). We are not going to worry about it here, 4 because any deviations from e A e B = exp(A + B + [A, B]/2 + · · · ) appear at the level of triple commutators or higher, which are not contributing to (3.11).…”

Section: -Cocycles In Lie Group Cohomologymentioning

confidence: 99%

3-cocycles, non-associative star-products and the magnetic paradigm of R-flux string vacua

Bakas

Lüst

2014

J. High Energ. Phys.

134

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Abstract:We consider the geometric and non-geometric faces of closed string vacua arising by T-duality from principal torus bundles with constant H-flux and pay attention to their double phase space description encompassing all toroidal coordinates, momenta and their dual on equal footing. We construct a star-product algebra on functions in phase space that is manifestly duality invariant and substitutes for canonical quantization. The 3-cocycles of the Abelian group of translations in double phase space are seen to account for non-associativity of the star-product. We also provide alternative cohomological descriptions of non-associativity and draw analogies with the quantization of point-particles in the field of a Dirac monopole or other distributions of magnetic charge. The magnetic field analogue of the R-flux string model is provided by a constant uniform distribution of magnetic charge in space and non-associativity manifests as breaking of angular symmetry. The Poincaré vector comes to rescue angular symmetry as well as associativity and also allow for quantization in terms of operators and Hilbert space only in the case of charged particles moving in the field of a single magnetic monopole.

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“…Addressing their integration to finite gauge transformations is a highly non-trivial question in the general setting. It evidently relates to the integration of Leibniz algebras [24][25][26], and, simultaneously, of semi-strict Lie 2-algebras [27][28][29] [33]. Both of these topics are subject to ongoing research in mathematics.…”

Section: Remarks On Finite Gauge Transformationsmentioning

confidence: 99%

Leibniz-Yang-Mills gauge theories and the 2-Higgs mechanism

Strobl

2019

Phys. Rev. D

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A quadratic Leibniz algebra (V, [·, ·], κ) gives rise to a canonical Yang-Mills type functional S over every space-time manifold. The gauge fields consist of 1-forms A taking values in V and 2-forms B with values in the subspace W ⊂ V generated by the symmetric part of the bracket. If the Leibniz bracket is anti-symmetric, the quadratic Leibniz algebra reduces to a quadratic Lie algebra, B ≡ 0, and S becomes identical to the usual Yang-Mills action functional. We describe this gauge theory for a general quadratic Leibniz algebra. We then prove its (classical and quantum) equivalence to a Yang-Mills theory for the Lie algebra g = V/W to which one couples massive 2-form fields living in a g-representation. Since in the original formulation the B-fields have their own gauge symmetry, this equivalence can be used as an elegant mass-generating mechanism for 2-form gauge fields, thus providing a "higher Higgs mechanism" for those fields.

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Hopf algebras in non-associative Lie theory

Abstract: We review the developments in the Lie theory for non-associative products from 2000 to date and describe the current understanding of the subject in view of the recent works, many of which use non-associative Hopf algebras as the main tool.

Cited by 35 publications

References 48 publications

Free braided nonassociative Hopf algebras and Sabinin τ-algebras

Free braided nonassociative Hopf algebras and Sabinin τ-algebras

3-cocycles, non-associative star-products and the magnetic paradigm of R-flux string vacua

Leibniz-Yang-Mills gauge theories and the 2-Higgs mechanism

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