Comments of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>-supersymmetry in superfield formalism

Doi, S.; Aizawa, N.

doi:10.1016/j.nuclphysb.2021.115641

Cited by 12 publications

(9 citation statements)

References 23 publications

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“…This fact constrains possible form of Z 2 2 -supersymmetric actions. The N = 1 case has already been considered in [17]. Consideration of N = 2 is one of the works should be done.…”

Section: Discussionmentioning

confidence: 99%

“…Among others, Z n 2 -graded Lie superalgebras and symmetries generated by them attract some interest recently (Z n 2 := Z 2 ×• • •×Z 2 , n times). This is because such symmetries are found in the known quantum systems [5][6][7][8] and many simple systems having such symmetries are constructed [9][10][11][12][13][14][15][16][17][18][19]. It is also pointed out the existence of quantum mechanical observables which detect the particles obeying Z 2 2 -graded parastatistics [20,21].…”

Section: Introductionmentioning

confidence: 99%

“…Z n 2 -supermechanics has been studied only for n = 2 and N = 1 so far [14,15,17] where N counts the number of supercharges in each subspace of Z 2 2 -degree (1, 0) and (0, 1), see the equation (1.1). In the present work, we start exploring Z 2 2 -supermechanics of N = 2.…”

Section: Introductionmentioning

confidence: 99%

“…Then what we need is a Z 2 2 -SUSY transformation acting on Z 2 2 -commutative numbers [4] which are functions of time. Such transformation should be an irreducible representation (irrep) of the Z 2 2 -SUSY algebra, so we start our explore with an abstract representation theory of the Z 2 2 -SUSY algebra (see [17] for N = 1). In §2, representations of Z 2 2 -SUSY algebra induced from irreps of the Cartan subalgebra are presented.…”

Section: Introductionmentioning

confidence: 99%

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Irreducible representations of $\mathbb{Z}_2^2$-graded ${\cal N} =2$ supersymmetry algebra and $\mathbb{Z}_2^2$-graded supermechanics

Aizawa,

Doi

2022

Preprint

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Irreducible representations (irreps) of Z 2 2 -graded supersymmetry algebra of N = 2 are obtained by the method of induced representation and they are used to derive Z 2 2 -graded supersymmetric classical actions. The irreps are four dimensional for λ = 0 where λ is an eigenvalue of the Casimir element, and eight dimensional for λ = 0. The eight dimensional irreps reduce to four dimensional ones only when λ and an eigenvalue of Hamiltonian satisfy a particular relation. The reduced four dimensional irreps are used to define Z 2 2 -graded supersymmetry transformations and two types of classical actions invariant under the transformations are presented. It is shown that one of the Noether charges vanishes if all the variables of specific Z 2 2 -degree are auxiliary.

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“…This fact constrains possible form of Z 2 2 -supersymmetric actions. The N = 1 case has already been considered in [17]. Consideration of N = 2 is one of the works should be done.…”

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Irreducible representations of $\mathbb{Z}_2^2$-graded ${\cal N} =2$ supersymmetry algebra and $\mathbb{Z}_2^2$-graded supermechanics

Aizawa,

Doi

2022

Preprint

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“…Since the recognition of a Z 2 × Z 2 -graded Lie superalgebra underlying the symmetries of Lévy-Leblond equations [11,12], these Z 2 × Z 2 -graded algebras have experienced a revival in mathematical physics. They appeared in graded (quantum) mechanics and quantization [13][14][15][16][17][18], in Z 2 × Z 2graded two-dimensional models [19][20][21], in Z 2 × Z 2 -graded superspace formulations [22][23][24][25] and in particular in parastatistics [4,26] and in the description and application of other types of parabosons and parafermions [27,28].…”

Section: Introductionmentioning

confidence: 99%

Orthosymplectic Z2×Z2Z2×Z2 -graded Lie superalgebras and parastatistics

Stoilova,

der Jeugt

2024

J. Phys. A: Math. Theor.

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A Z2 x Z2-graded Lie superalgebra g is a Z2 x Z2-graded algebra with a bracket [·, ·] that satisfies certain graded versions of the symmetry and Jacobi identity. In particular, despite the common terminology, g is not a Lie superalgebra. We construct the most general orthosymplectic Z2 x Z2-graded Lie superalgebra osp(2m1+1, 2m2|2n1, 2n2) in terms of defining matrices. A special case of this algebra appeared already in work of Tolstoy in 2014. Our construction is based on the notion of graded supertranspose for a Z2 x Z2-graded matrix. Since the orthosymplectic Lie superalgebra osp(2m + 1|2n) is closely related to the definition of parabosons, parafermions and mixed parastatistics, we investigate here the new parastatistics relations following from osp(2m1+1, 2m2|2n1, 2n2). Some special cases are of particular interest, even when one is dealing with parabosons only.

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A Class of Representations of the Orthosymplectic Lie Superalgebras $$\mathcal{B}(n,n)$$ and $$\mathcal{B}(\infty ,\infty )$$

Stoilova

Jeugt

2020

Springer Proceedings in Mathematics &Amp; Statistics

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In 1982 Palev showed that the algebraic structure generated by the creation and annihilation operators of a system of m parafermions and n parabosons, satisfying the mutual parafermion relations, is the Lie superalgebra osp(2m + 1|2n). The "parastatistics Fock spaces" of order p of such systems are then certain lowest weight representations of osp(2m + 1|2n). We investigate now the situation when the number of parafermions and parabosons becomes infinite, which is of interest not only in a physics context but also from the mathematical point of view. In this contribution, we will discuss the various steps that are needed to understand the infinite-rank case. First, we will introduce appropriate bases and Dynkin diagrams for B(n, n) = osp(2n + 1|2n) that allow us to extend n → ∞. Then we will develop a new matrix form for B(n, n) = osp(2n + 1|2n), because the standard one is not appropriate for taking this limit. Following this, we construct a new Gelfand-Zetlin basis of the parastatistics Fock spaces in the finite rank case (in correspondence with this new matrix form). The new structures, related to a non-distinguished simple root system, allow the extension to n → ∞. This leads to the definition of the algebra B(∞, ∞) as a Lie superalgebra generated by an infinite number of creation and annihilation operators (subject to certain relations), or as an algebra of certain infinite-dimensional matrices. We study the parastatistics Fock spaces, as certain lowest weight representations of B(∞, ∞). In particular, we construct a basis consisting of well-described row-stable Gelfand-Zetlin patterns.

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Comments of Z22-supersymmetry in superfield formalism

Cited by 12 publications

References 23 publications

Irreducible representations of $\mathbb{Z}_2^2$-graded ${\cal N} =2$ supersymmetry algebra and $\mathbb{Z}_2^2$-graded supermechanics

Irreducible representations of $\mathbb{Z}_2^2$-graded ${\cal N} =2$ supersymmetry algebra and $\mathbb{Z}_2^2$-graded supermechanics

Orthosymplectic Z2×Z2Z2×Z2 -graded Lie superalgebras and parastatistics

A Class of Representations of the Orthosymplectic Lie Superalgebras $$\mathcal{B}(n,n)$$ and $$\mathcal{B}(\infty ,\infty )$$

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