Generalized quasispin for supergroups

Jarvis, Peter; Yang, Mei; Wybourne, B. G.

doi:10.1063/1.527566

Cited by 37 publications

(30 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…. , n. In each multiplet its component fields x i , ψ i , ξ i , z i transform according to (23). An invariant sigmamodel action S int , describing the motion of interacting multiplets, can be defined through the position…”

Section: Invariant Actions For the (2 2 0) Root Multipletmentioning

confidence: 99%

“…After the introduction [17][18][19][20] of Z 2 × Z 2 -graded Lie superalgebras, a long history of investigations of Z 2 × Z 2 (and higher) graded symmetries in physical problems, which recently attracted a renewed interest, began. Several works considered enlarged symmetries in various contexts such as extensions of spacetime symmetries (beyond ordinary de-Sitter and Poincaré algebras), supergravity theory, quasispin formalism, parastatistics and non-commutative geometry, see [21][22][23][24][25][26][27][28][29][30]. It was also recently revealed that the symmetries of the Lévy-Leblond equation, which is a nonrelativistic quantum mechanical wave equation for spin 1/2 particles, are given by a Z 2 × Z 2 -graded superalgebra [31,32].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

$${{\mathbb {Z}}}_2\times {{\mathbb {Z}}}_2$$-graded mechanics: the classical theory

Aizawa

Kuznetsova

Toppan³

2020

Eur. Phys. J. C

View full text Add to dashboard Cite

$${{\mathbb {Z}}}_2\times {{\mathbb {Z}}}_2$$Z2×Z2-graded mechanics admits four types of particles: ordinary bosons, two classes of fermions (fermions belonging to different classes commute among each other) and exotic bosons. In this paper we construct the basic $${{\mathbb {Z}}}_2\times {{\mathbb {Z}}}_2$$Z2×Z2-graded worldline multiplets (extending the cases of one-dimensional supersymmetry) and compute, based on a general scheme, their invariant classical actions and worldline sigma-models. The four basic multiplets contain two bosons and two fermions. They are (2, 2, 0), with two propagating bosons and two propagating fermions, $$(1,2,1)_{[00]}$$(1,2,1)[00] (the ordinary boson is propagating, while the exotic boson is an auxiliary field), $$(1,2,1)_{[11]}$$(1,2,1)[11] (the converse case, the exotic boson is propagating, while the ordinary boson is an auxiliary field) and, finally, (0, 2, 2) with two bosonic auxiliary fields. Classical actions invariant under the $${{\mathbb {Z}}}_2\times {\mathbb {Z}}_2$$Z2×Z2-graded superalgebra are constructed for both single multiplets and interacting multiplets. Furthermore, scale-invariant actions can possess a full $${{\mathbb {Z}}}_2\times {\mathbb {Z}}_2$$Z2×Z2-graded conformal invariance spanned by 10 generators and containing an sl(2) subalgebra.

show abstract

Section: Invariant Actions For the (2 2 0) Root Multipletmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

$${{\mathbb {Z}}}_2\times {{\mathbb {Z}}}_2$$-graded mechanics: the classical theory

Aizawa

Kuznetsova

Toppan³

2020

Eur. Phys. J. C

View full text Add to dashboard Cite

show abstract

“…The natural graded structure when both fermionic and bosonic oscillator modes are present, where closure on both linear and bilinear combinations is required, is indeed that of a Z 2 × Z 2 graded colour superalgebra. See[15] and references therein 8. Following the arguments of §2 above, in this case there are potentially 8 couplings, or 7 arbitrary coefficients up to normalisation (see §A.1 and (20) below).…”

mentioning

confidence: 99%

Polynomial super-gl(n) algebras

Jarvis¹,

Rudolph²

2003

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

We introduce a class of finite dimensional nonlinear superalgebras L = L 0 + L 1 providing gradings of L 0 = gl(n) ≃ sl(n) + gl(1). Odd generators close by anticommutation on polynomials (of degree > 1) in the gl(n) generators. Specifically, we investigate 'type I' super-gl(n) algebras, having odd generators transforming in a single irreducible representation of gl(n) together with its contragredient. Admissible structure constants are discussed in terms of available gl(n) couplings, and various special cases and candidate superalgebras are identified and exemplified via concrete oscillator constructions. For the case of the n-dimensional defining representation, with odd generators Q a , Q b , and even generators E a b , a, b = 1, . . . , n, a three parameter family of quadratic super-gl(n) algebras (deformations of sl(n/1)) is defined. In general, additional covariant Serre-type conditions are imposed, in order that the Jacobi identities be fulfilled. For these quadratic super-gl(n) algebras, the construction of Kac modules, and conditions for atypicality, are briefly considered. Applications in quantum field theory, including Hamiltonian lattice QCD and space-time supersymmetry, are discussed. † Alexander von Humboldt Fellowwhere there may be lower degree terms in the last line. Thus {Q α } form a tensor operator under the action of L 0 , and the anticommutator {Q α , Q β } transforms in the tensor product of the relevant representations of the even subalgebra.12 Further details of the generalisation of these Serre-type relations to the cases gl 2 (n/{1 3 } + {1 3 }) and gl 2 (n/{3} + {3}), as well as the present case, are given in §A.4.

show abstract

“…The color (super)algebras have been an objects of interest in mathematics for the last four decades, and many works such as classification of the algebras under some conditions, representations, cohomology and so on, have been done [5,6,7,8,9,10,11,12,13,14,15,16,17] (and references therein). On the other hand, physical applications are very limited [18,19,20,21,22,23,24,25] so one may conclude that color (super)algebras are not widely known in the physics community.…”

Section: Introductionmentioning

confidence: 99%

Z 2 × Z 2 generalizations of N=1 superconformal Galilei algebras and their representations

Aizawa

Isaac

Segar

2019

Journal of Mathematical Physics

View full text Add to dashboard Cite

We introduce two classes of novel color superalgebras of Z 2 × Z 2 grading. This is done by realizing members of each in the universal enveloping algebra of the N = 1 supersymmetric extension of the conformal Galilei algebra. This allows us to upgrade any representation of the super conformal Galilei algebras to a representation of the Z 2 × Z 2 graded algebra. As an example, boson-fermion Fock space representation of one class is given. We also provide a vector field realization of members of the other class by using a generalization of the Grassmann calculus to Z 2 × Z 2 graded setting.

show abstract

Generalized quasispin for supergroups

Cited by 37 publications

References 19 publications

$${{\mathbb {Z}}}_2\times {{\mathbb {Z}}}_2$$-graded mechanics: the classical theory

$${{\mathbb {Z}}}_2\times {{\mathbb {Z}}}_2$$-graded mechanics: the classical theory

Polynomial super-gl(n) algebras

Z 2 × Z 2 generalizations of N=1 superconformal Galilei algebras and their representations

Contact Info

Product

Resources

About