Double-graded quantum superplane

Bruce, Andrew James; Duplij, Steven

doi:10.48550/arxiv.1910.12950

Cited by 8 publications

(14 citation statements)

References 31 publications

(41 reference statements)

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“…The split-quaternionic matrices (20) give a representation of the Z 2 ×Z 2 -graded superalgebra S10 ǫ=−1 . They are recovered, up to normalizing factors, from the matrices given in Appendix C after setting, for ǫ = −1, λ = 2, p = 1, q = 1.…”

Section: Z 2 × Z 2 -Graded Structures Of Quaternions and Split-quater...mentioning

confidence: 99%

“…We extend here the notion of Z 2 × Z 2 -graded superspace and covariant derivatives, originally introduced in [20] for the one-dimensional Z 2 × Z 2 -graded Poincaré superalgebra S10 ǫ=1 , to other cases of graded algebras and superalgebras listed in Tables 1 and 2, respectively.…”

Section: Graded Superspacesmentioning

confidence: 99%

“…It is nevertheless only recently that Z 2 × Z 2 -graded Lie superalgebras have been systematically investigated and applied to dynamical systems. It was recognized in [14,15] that they describe symmetries of Lévy-Leblond equations; furthermore the following constructions have been introduced: Z 2 × Z 2 -graded invariant worldline [16] and two-dimensional [17] sigma models, quantum mechanics [18,19], superspace [20]. The role of Z 2 × Z 2 -graded parastatistics has been clarified in [21] (see also [22,23] and references therein for earlier works).…”

Section: Introductionmentioning

confidence: 99%

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Classification of minimal ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie (super)algebras and some applications

Kuznetsova,

Toppan

2021

Preprint

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This paper presents the classification, over the fields of real and complex numbers, of the minimal Z 2 × Z 2 -graded Lie algebras and Lie superalgebras spanned by 4 generators and with no empty graded sector. The inequivalent graded Lie (super)algebras are obtained by solving the constraints imposed by the respective graded Jacobi identities.A motivation for this mathematical result is to sistematically investigate the properties of dynamical systems invariant under graded (super)algebras. Recent works only paid attention to the special case of the one-dimensional Z 2 × Z 2 -graded Poincaré superalgebra.As applications, we are able to extend certain constructions originally introduced for this special superalgebra to other listed Z 2 × Z 2 -graded (super)algebras. We mention, in particular, the notion of Z 2 × Z 2 -graded superspace and of invariant dynamical systems (both classical worldline sigma models and a quantum Hamiltonian).As a further byproduct we point out that, contrary to Z 2 × Z 2 -graded superalgebras, a theory invariant under a Z 2 × Z 2 -graded algebra implies the presence of ordinary bosons and three different types of exotic bosons, with exotic bosons of different types anticommuting among themselves.

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Section: Z 2 × Z 2 -Graded Structures Of Quaternions and Split-quater...mentioning

confidence: 99%

Section: Graded Superspacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Classification of minimal ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie (super)algebras and some applications

Kuznetsova,

Toppan

2021

Preprint

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“…Comment: the 13 inequivalent multiplication tables of the Z 2 × Z 2 -graded superdivision algebras are straightforwardly recovered from the matrix representations of the generators, given in (21) for the real series, (24) for the complex series and (27,28) for the quaternionic series. To save space they will not be reported here.…”

Section: The Quaternionic Z2xz2 Superdivision Algebrasmentioning

confidence: 99%

“…A model of Z 2 × Z 2 -graded invariant quantum mechanics was discussed in [14], while the systematic construction of Z 2 × Z 2 -graded classical mechanics was introduced in [15] and the canonical quantization of the models presented in [16]. Further developments include the construction of two-dimensional models [17][18][19], graded superconformal quantum mechanics [20], superspace [18,21,22], extensions and bosonization of double-graded supersymmetric quantum mechanics [23,24], and so forth. The construction of models with Z 2 × Z 2 -graded parabosons was discussed in [18].…”

Section: Introductionmentioning

confidence: 99%

Beyond the $10$-fold way: $13$ associative $Z_2\times Z_2$-graded superdivision algebras

Kuznetsova¹,

Toppan²

2021

Preprint

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The "10-fold way" refers to the combined classification of the 3 associative division algebras (of real, complex and quaternionic numbers) and of the 7, Z 2 -graded, superdivision algebras (in a superdivision algebra each homogeneous element is invertible). The connection of the 10-fold way with the periodic table of topological insulators and superconductors is well known. Motivated by the recent interest in Z 2 × Z 2 -graded physics (classical and quantum invariant models, parastatistics) we classify the associative Z 2 × Z 2graded superdivision algebras and show that 13 inequivalent cases have to be added to the 10-fold way. Our scheme is based on the "alphabetic presentation of Clifford algebras", here extended to graded superdivision algebras. The generators are expressed as equal-length words in a 4-letter alphabet (the letters encode a basis of invertible 2 × 2 real matrices and in each word the symbol of tensor product is skipped). The 13 inequivalent Z 2 × Z 2 -graded superdivision algebras are split into real series (4 subcases with 4 generators each), complex series (5 subcases with 8 generators) and quaternionic series (4 subcases with 16 generators).

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The Z2×Z2-graded general linear Lie superalgebra

Isaac

Stoilova

Jeugt

2020

Journal of Mathematical Physics

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We present a novel realisation of the Z 2 ×Z 2 -graded Lie superalgebra gl(m 1 , m 2 |n 1 , n 2 ) inside an algebraic extension of the enveloping algebra of the Z 2 -graded Lie superalgebra gl(m|n), with m = m 1 + m 2 and n = n 1 + n 2 . A consequence of this realisation is that the representations of gl(m|n) "lift up" to representations of gl(m 1 , m 2 |n 1 , n 2 ), with matrix elements differing only by a sign, which we are able to characterise concisely.

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Double-graded quantum superplane

Cited by 8 publications

References 31 publications

Classification of minimal ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie (super)algebras and some applications

Classification of minimal ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie (super)algebras and some applications

Beyond the $10$-fold way: $13$ associative $Z_2\times Z_2$-graded superdivision algebras

The Z2×Z2-graded general linear Lie superalgebra

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