Quantum groups as fundamental symmetries

Gresnigt, Niels G.

doi:10.22323/1.314.0665

Cited by 3 publications

(4 citation statements)

References 18 publications

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“…There exist several curious structural similarities between recent braid-and Hurwitz algebraic descriptions of a single generation of fermions. Clifford algebras that are isomorphic to complex numbers and quaternions admit precisely those braid group representations from which the Helon model is contructed [19]. Furthermore, the ribbon braids representing fermions in that model coincide exactly with the states that span the minimal left ideals of the adjoint algebra of the complex octonions, shown by Furey to describe one generation of leptons and quarks with unbroken SU (3) C and U (1) EM symmetry.…”

Section: Discussionmentioning

confidence: 93%

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Braided fermions from Hurwitz algebras

Gresnigt

2019

J. Phys.: Conf. Ser.

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Some curious structural similarities between a recent braid-and Hurwitz algebraic description of the unbroken internal symmetries for a single generations of Standard Model fermions were recently identified. The non-trivial braid groups that can be represented using the four normed division algebras are B2 and B c 3 , exactly those required to represent a single generation of fermions in terms of simple three strand ribbon braids. These braided fermion states can be identified with the basis states of the minimal left ideals of the Clifford algebra C (6), generated from the nested left actions of the complex octonions C ⊗ O on itself. That is, the ribbon spectrum can be related to octonion algebras. Some speculative ideas relating to ongoing research that attempts to construct a unified theory based on braid groups and Hurwitz algebras are discussed.

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Section: Discussionmentioning

confidence: 93%

“…Several curious structural similarities between the braid and Hurwitz algebra descriptions of SM fermions have recently been identified [19,20]. The complex numbers and quaternions contain representations of precisely the braid groups used to construct braided fermions in the Helon model.…”

Section: Introductionmentioning

confidence: 99%

Braided fermions from Hurwitz algebras

Gresnigt

2019

J. Phys.: Conf. Ser.

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“…More generally it is possible for any orientable braided 3-belt There are situations where writing a braided 3-belt in terms of braiding only rather than twisting only is advantageous. Together with the observation that Clifford algebras and normed division algebras contain representations of the circular Artin braid groups [7,8], it was recently shown that by removing the twists on ribbons in the Helon model braids, they can be written in a form such that they coincide with the basis states of the minimal left ideals of the complex octonions [9], which have been shown to transform as a single generation of leptons and quarks under the unbroken symmetries SU (3) c and U (1) em [10,11]. It is precisely because all the twisting can be exchanged for braiding that this identification between Helon braids and basis states of the minimal ideals of the complex octonions is possible.…”

Section: Introductionmentioning

confidence: 99%

Knotted boundaries and braid only form of braided belts

Gresnigt

2018

Preprint

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The Helon model identifies Standard Model quarks and leptons with certain framed braids joined together at both ends by a connecting node (disk). These surfaces with boundary are called braided 3-belts (or simply belts). Twisting and braiding of ribbons composing braided 3-belts are interchangeable, and it was shown in the literature that any braided 3-belt can be written in a pure twist form, specified by a vector of three multiples of half integers [a, b, c], a topological invariant. This paper identifies the set of braided 3-belts that can be written in a braid only form in which all twisting is eliminated. For these braids an algorithm to calculate the braid word is determined which allows the braid only word of every braided 3-belt to be written in a canonical form. It is furthermore demonstrated that the set of braided 3-belts do not form a group, due to a lack of isogeny.The conditions under which the boundary of a braided 3-belt is a knot are determined, and a formula for the Jones polynomial for knotted boundaries is derived. Considering knotted boundaries makes it possible to relate the Helon model to a model of quarks and leptons in terms of quantum trefoil knots, understood as representation of the quantum group SU q (2). Associating representations of a quantum group to the boundary of braided belts provides a possible means of developing the gauge symmetries of interacting braided belts in future work.

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“…For additional work related to this braid model and its connections with division algebras and Clifford algebras, the reader is referred to[12][13][14][15][16][17][18].…”

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confidence: 99%

The Standard Model particle content with complete gauge symmetries from the minimal ideals of two Clifford algebras

Gresnigt

2020

Eur. Phys. J. C

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Building upon previous works, it is shown that two minimal left ideals of the complex Clifford algebra C (6) and two minimal right ideals of C (4) transform as one generation of leptons and quarks under the gauge symmetry SU (3) C × U (1) E M and SU (2) L respectively. The SU (2) L weak symmetries are naturally chiral. Combining the C (6) and C (4) ideals, all the gauge symmetries of the Standard Model, together with its lepton and quark content for a single generation are represented. The combined ideals can be written as minimal left ideals of C (6) ⊗ C (4) ∼ = C (10) in a way that preserves individually the C (6) structure and C (4) structure of physical states. This resulting model captures many of the attractive features of the Georgi and Glashow SU (5) Grand Unified Theory without introducing proton decay or other unobserved processes. Such processes are naturally excluded because they do not preserve the underlying algebraic structure.

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Quantum groups as fundamental symmetries

Cited by 3 publications

References 18 publications

Braided fermions from Hurwitz algebras

Braided fermions from Hurwitz algebras

Knotted boundaries and braid only form of braided belts

The Standard Model particle content with complete gauge symmetries from the minimal ideals of two Clifford algebras

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