Real representations of C2-graded groups: The antilinear theory

Rumynin, Dmitriy; Taylor, James G.

doi:10.1016/j.laa.2020.09.040

Cited by 5 publications

(4 citation statements)

References 21 publications

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“…Further, we can introduce the group algebra R[G]: its image under ρ will be nothing but D. This useful point of view was used to describe Dyson's classification in [16].…”

Section: Aside: φ-Representation As Real-linear Representationsmentioning

confidence: 99%

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Dyson’s classification and real division superalgebras

Geiko

Moore

2021

J. High Energ. Phys.

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It is well-known that unitary irreducible representations of groups can be usefully classified in a 3-fold classification scheme: Real, Complex, Quaternionic. In 1962 Freeman Dyson pointed out that there is an analogous 10-fold classification of irreducible representations of groups involving both unitary and antiunitary operators. More recently, it was realized that there is also a 10-fold classification scheme involving superdivision algebras. Here we give a careful proof of the equivalence of these two 10-fold ways.

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“…Further, we can introduce the group algebra R[G]: its image under ρ will be nothing but D. This useful point of view was used to describe Dyson's classification in [16].…”

Section: Aside: φ-Representation As Real-linear Representationsmentioning

confidence: 99%

“…In [15] φrepresentations were studied in relation to modular theory of operator algebras. While completing our draft we learned of a very closely related paper by D. Rumynin and J. Taylor [16], which contains a modern review of Dyson's classification.…”

Section: Introduction and Conclusionmentioning

confidence: 99%

Dyson’s classification and real division superalgebras

Geiko

Moore

2021

J. High Energ. Phys.

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“…The non-trivial generator of Aut(U (1)) is the Galois action ω → ω −1 . Using φ, we can define a semi-direct product G ≡ U (1) ⋊ φ G. It is clear from definitions that any φ-representation (ρ, V R , I) of (G, φ) is equivalent to a real linear representation (ρ, V R ) of G. Further, we can introduce the group algebra R[ G]: its image under ρ will be nothing but D. This useful point of view was used to describe Dyson's classification in [16].…”

Section: Aside: φ-Representation As Real-linear Representationsmentioning

confidence: 99%

“…In [15] φ-representations were studied in relation to modular theory of operator algebras. While completing our draft we learned of a very closely related paper by D. Rumynin and J. Taylor [16] which contains Definition 2.1. Let G be a compact Z 2 -graded group, where the grading is given by a group homomorphism φ :…”

Section: Introduction and Conclusionmentioning

confidence: 99%

Dyson's Classification And Real Division Superalgebras

Geiko,

Moore

2020

Preprint

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It is well-known that unitary irreducible representations of groups can be usefully classified in a 3-fold classification scheme: Real, Complex, Quaternionic. In 1962 Freeman Dyson pointed out that there is an analogous 10-fold classification of irreducible representations of groups involving both unitary and antiunitary operators. More recently it was realized that there is also a 10-fold classification scheme involving superdivision algebras. Here we give a careful proof of the equivalence of these two 10-fold ways.

show abstract