arXiv subjects

N. Aizawa

Publications and source records attributed to N. Aizawa.

At least 19 recordsLinked to original sources

arXiv2026

An integrable $\mathbb{Z}_2^2$-graded extension of Camassa-Holm equation and its bi-Hamiltonian structure

By constructing Lax operators in the loop algebra of the $\mathbb{Z}_2^2$-graded extension of the Lie superalgebra $\mathfrak{osp}(1|2)$, we derive a $\mathbb{Z}_2^2$-graded extension of the Camassa-Holm equation. The resulting equation is an integrable nonlinear PDE for a system of four $\mathbb{Z}_2^2$-graded commutative functions, each associated with a distinct $\mathbb{Z}_2^2$-degree. We further show that the $\mathbb{Z}_2^2$-Camassa-Holm equation admits a bi-Hamiltonian structure. As a consequence, it possesses infinitely many conserved quantities, including one with non-trivial $\mathbb{Z}_2^2$-degree, which are mutually in involution with respect to the $\mathbb{Z}_2^2$-graded Poisson brackets.

nlin.SI↗

arXiv2026

An integrable hierarchy associated with loop extension of $\mathbb{Z}_2^2$-graded $\mathfrak{osp}(1|2)$

A hierarchy of $\mathbb{Z}_2^2$-graded integrable equations is constructed using the loop extension of the $\mathbb{Z}_2^2$-graded Lie superalgebra $\mathfrak{osp}(1|2)$. This hierarchy includes $\mathbb{Z}_2^2$-graded extensions of the Liouville, sinh-Gordon, cosh-Gordon, and, in particular, the mKdV equations. The $\mathbb{Z}_2^2$-graded KdV equation is also derived from the $\mathbb{Z}_2^2$-mKdV equation via the Miura transformation. We present explicit formulas for the conserved charges of the $\mathbb{Z}_2^2$-KdV and $\mathbb{Z}_2^2$-mKdV equations. A distinctive feature of these $\mathbb{Z}_2^2$-graded integrable systems is the existence of conserved charges with nontrivial grading.

math-ph↗

arXiv2026

Affine extensions of $\mathbb{Z}_2^2$-graded $osp(1|2)$ and Virasoro algebra

It is known that there are two inequivalent $\mathbb{Z}_2^2$-graded $osp(1|2)$ Lie superalgebras. Their affine extensions are investigated and it is shown that one of them admits two central elements, one is non-graded and the other is $(1,1)$-graded. The affine $\mathbb{Z}_2^2$-$osp(1|2)$ algebras are used by the Sugawara construction to study possible $\mathbb{Z}_2^2$-graded extensions of the Virasoro algebra. We obtain a $\mathbb{Z}_2^2$-graded Virasoro algebra with a non-trivially graded central element. Throughout the investigation, invariant bilinear forms on $\mathbb{Z}_2^2$-graded superalgebras play a crucial role, so a theory of invariant bilinear forms is also developed.

math-ph↗

arXiv2025

Universal weight systems from a minimal $\mathbb{Z}_2^2$-graded Lie algebra

Color Lie algebras, which were introduced by Ree, are a graded extension of Lie (super)algebras by an abelian group. We show that the color Lie algebras can be used to construct universal weight systems for knot invariants of of Vassiliev and Kontsevich. As a simple example, we take $\mathbb{Z}_2 \times \mathbb{Z}_2$ as the grading group and consider the four-dimensional color Lie algebra called $A1_ε$. The weight system constructed from $A1_ε$ is studied in some detail and some relations between the weights, such as the recurrence relation for chord diagrams, are derived. These relations show that the weight system from $A1_ε$ is a hybrid of those from $sl(2)$ and $gl(1|1)$.

math.GT↗

arXiv2024

Invariant Differential Operators for Non-Compact Lie Groups: the $Sp(n,1)$ Case

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebras $sp(n,1)$. Our choice of these algebras is motivated by the fact that they belong to a narrow class of algebras, which are of split rank one, of which class the other cases were studied, some long time ago. We concentrate on the case $n=2$. We give the main multiplets and the main reduced multiplets of indecomposable elementary representations for, including the necessary data for all relevant invariant differential operators. We also present explicit expressions for the singular vectors and for the intertwining differential operators.

math.RT↗

arXiv2024

$\mathcal{N}=2$ Double graded supersymmetric quantum mechanics via dimensional reduction

We present a novel $\mathcal{N} = 2 $ $\mathbb{Z}_2^2$-graded supersymmetric quantum mechanics ($\mathbb{Z}_2^2$-SQM) which has different features from those introduced so far. It is a two-dimensional (two-particle) system and is the first example of the quantum mechanical realization of an eight-dimensional irrep of the $\mathcal{N}=2$ $\mathbb{Z}_2^2$-supersymmetry algebra. The $\mathbb{Z}_2^2$-SQM is obtained by quantizing the one-dimensional classical system derived by dimensional reduction from the two-dimensional $\mathbb{Z}_2^2$-supersymmetric Lagrangian of $\mathcal{N}=1$, which we constructed in our previous work. The ground states of the $\mathbb{Z}_2^2$-SQM are also investigated.

math-ph↗

arXiv2023

Integration on minimal $\mathbb{Z}_2^2$-superspace and emergence of space

We investigate the possibilities of integration on the minimal $\mathbb{Z}_2^2$-superspace. Two definitions are taken from the works by Poncin and Schouten and we examine their generalizations. It is shown that these definitions impose some restrictions on the integrable functions. We then introduce a new definition of integral, which is inspired by our previous work, and show that the definition does not impose restrictions on the integrable functions. An interesting feature of this definition is the emergence of a spatial coordinate which means that the integral is defined on $\mathbb{R}^2$ despite the fact that the $(0,0)$ part of the minimal $\mathbb{Z}_2^2$-superspace is $\mathbb{R}. $

math-ph↗

arXiv2023

$\mathbb{Z}_2^2$-graded supersymmetry via superfield on minimal $\mathbb{Z}_2^2$-superspace

A superfield formalism for the minimal $\mathbb{Z}_2^2$-graded version of supersymmetry is developed. This is done by using the recently introduced definition of integration on the minimal $\mathbb{Z}_2^2$-superspace. It is shown that one may construct $\mathbb{Z}_2^2$-supersymmetric action by the procedure similar to the standard supersymmetry. However, the Lagrangian obtained has very general interaction terms, which give rise to a $\mathbb{Z}_2^2$-graded extension of many known theories defined in two-dimensional spacetime. As an illustration, we will give a $\mathbb{Z}_2^2$-extension of the sine-Gordon model different from the one already discussed in the literature.

math-ph↗

arXiv2023

New aspects of the Z$_{\textrm 2}$ $\times$ Z$_{\textrm 2}$-graded 1D superspace: induced strings and 2D relativistic models

A novel feature of the ${\mathbb Z}_2\times {\mathbb Z}_2$-graded supersymmetry which finds no counterpart in ordinary supersymmetry is the presence of $11$-graded exotic bosons (implied by the existence of two classes of parafermions). Their interpretation, both physical and mathematical, presents a challenge. The role of the "exotic bosonic coordinate" was not considered by previous works on the one-dimensional ${\mathbb Z}_2\times {\mathbb Z}_2$-graded superspace (which was restricted to produce point-particle models). By treating this coordinate at par with the other graded superspace coordinates new consequences are obtained. The graded superspace calculus of the ${\mathbb Z}_2\times {\mathbb Z}_2$-graded worldline super-Poincaré algebra induces two-dimensional ${\mathbb Z}_2\times {\mathbb Z}_2$-graded relativistic models; they are invariant under a new ${\mathbb Z}_2\times {\mathbb Z}_2$-graded $2D$ super-Poincaré algebra which differs from the previous two ${\mathbb Z}_2\times {\mathbb Z}_2$-graded $2D$ versions of super-Poincaré introduced in the literature. In this new superalgebra the second translation generator and the Lorentz boost are $11$-graded. Furthermore, if the exotic coordinate is compactified on a circle ${\bf S}^1$, a ${\mathbb Z}_2\times {\mathbb Z}_2$-graded closed string with periodic boundary conditions is derived. The analysis of the irreducibility conditions of the $2D$ supermultiplet implies that a larger $(β$-deformed, where $β\geq 0$ is a real parameter) class of point-particle models than the ones discussed so far in the literature (recovered at $β=0$) is obtained. While the spectrum of the $β=0$ point-particle models is degenerate (due to its relation with an ${\cal N}=2$ supersymmetry), this is no longer the case for the $β> 0$ models.

hep-th↗

arXiv2022

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

Irreducible representations (irreps) of $\mathbb{Z}_2^2$-graded supersymmetry algebra of ${\cal N}=2$ are obtained by the method of induced representation and they are used to derive $\mathbb{Z}_2^2$-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 $λ\neq 0.$ 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 $\mathbb{Z}_2^2$-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 $\mathbb{Z}_2^2$-degree are auxiliary.

math-ph↗

arXiv2022

Invariant differential operators for the Jacobi algebra $ {\cal G}_2$

In the present paper we construct explicitly the intertwining differential operators for the Jacobi algebra ${\cal G}_2.$ For the construction we use the singular vectors of the Verma modules over ${\cal G}_2$ which we have constructed earlier. We construct the function spaces on which the operators act. We display two versions of the left (representation) action and the right action. The latter is inserted in the singular vectors to provide the intertwining differential operators.

math.RT↗

arXiv2021

Comments of $\mathbb{Z}_2^2$-supersymmetry in superfield formalism

We investigate superfield formulation of the minimal $\mathbb{Z}_2^2$-supersymmetry. It is shown that the integrability on $\mathbb{Z}_2^2$-superspace guarantees the invariance of action. Then we present two superfields which carry distinct irreducible representations of the $\mathbb{Z}_2^2$-supersymmetry algebra. One of them gives integrable Lagrangian and the other does not. We also show that integrable superfields with different $\mathbb{Z}_2^2$-degree also carry irreducible representations and they give invariant actions. To perform this analysis, the representation theory of the minimal $\mathbb{Z}_2^2$-supersymmetry algebra is studied in some detail.

math-ph↗

arXiv2021

Classification of the Reducible Verma Modules over the Jacobi Algebra $ {\cal G}_2$

In the present paper we study the representations of the Jacobi algebra. More concretely, we define, analogously to the case of semi-simple Lie algebras, the Verma modules over the Jacobi algebra ${\cal G}_2$. We study their reducibility and give explicit construction of the reducible Verma modules exhibiting the corresponding singular vectors. Using this information we give a complete classification of the reducible Verma modules. More than this we exhibit their interrelation of embeddings between these modules. These embeddings are illustrated by diagrams of the embedding patterns so that each reducible Verma module appears in one such diagram.

math.RT↗

arXiv2021

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

In the previous paper arXiv:2003.06470 we introduced the notion of ${\mathbb Z}_2\times{\mathbb Z}_2$-graded classical mechanics and presented a general framework to construct, in the Lagrangian setting, the worldline sigma models invariant under a ${\mathbb Z}_2\times{\mathbb Z}_2$-graded superalgebra. In this work we discuss at first the classical Hamiltonian formulation of some of these models and later present their canonical quantization. As the simplest application of the construction we recover the ${\mathbb Z}_2\times{\mathbb Z}_2$-graded quantum Hamiltonian introduced by Bruce and Duplij in arXiv:1904.06975. We prove that this is the first example of a large class of ${\mathbb Z}_2\times{\mathbb Z}_2$-graded quantum models. We derive in particular interacting multiparticle quantum Hamiltonians given by Hermitian, matrix, differential operators. The interacting terms appear as non-diagonal entries in the matrices. The construction of the Noether charges, both classical and quantum, is presented. A comprehensive discussion of the different ${\mathbb Z}_2\times{\mathbb Z}_2$-graded symmetries possessed by the quantum Hamiltonians is given.

hep-th↗

arXiv2021

A classification of lowest weight irreducible modules over $\mathbb{Z}_2^2$-graded extension of $osp(1|2)$

We investigate representations of the $\mathbb{Z}_2^2$-graded extension of $osp(1|2)$ which is the spectrum generating algebra of the recently introduced $\mathbb{Z}_2^2$-graded version of superconformal mechanics. The main result is a classification of irreducible lowest weight modules of the $\mathbb{Z}_2^2$-graded extension of $osp(1|2)$. This is done via introduction of Verma modules and its maximal invariant submodule generated by singular vectors. Explicit formula of all singular vectors are also presented.

math-ph↗

arXiv2021

$\cal N$-Extension of duble-graded supersymmetric and superconformal quantum mechanics

In the recent paper, Bruce and Duplij introduced a double-graded version of supersymmetric quantum mechanics (SQM). It is an extension of Lie superalgebraic nature of ${\cal N}=1$ SQM to a $\mathbb{Z}_2^2$-graded superalgebra. In this work, we propose an extension of Bruce-Duplij model to higher values of $\cal N.$ Furthermore, it is shown that our construction of double-graded SQM is a special case of the method which converts a given Lie superalgebra to a $\mathbb{Z}_2^2$-graded superalgebra. By employing this method one may convert a model of superconformal mechanics to its double-graded version. The simplest example of ${\cal N}=1$ double-graded superconformal mechanics is studied in some detail.

math-ph↗

arXiv2020

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

${\mathbb Z}_2\times {\mathbb Z}_2$-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 ${\mathbb Z}_2\times {\mathbb Z}_2$-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, $(1,2,1)_{[00]}$ (the ordinary boson is propagating, while the exotic boson is an auxiliary field), $(1,2,1)_{[11]}$ (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 ${\mathbb Z}_2\times {\mathbb Z}_2$-graded superalgebra are constructed for both single multiplets and interacting multiplets. Furthermore, scale-invariant actions can possess a full ${\mathbb Z}_2\times {\mathbb Z}_2$-graded conformal invariance spanned by $10$ generators and containing an $sl(2)$ subalgebra.

hep-th↗

arXiv2020

$\mathbb{Z}_2 \times \mathbb{Z}_2$ generalizations of infinite dimensional Lie superalgebra of conformal type with complete classification of central extensions

We introduce a class of novel $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded color superalgebras of infinite dimension. It is done by realizing each member of the class in the universal enveloping algebra of a Lie superalgebra which is a module extension of the Virasoro algebra. Then the complete classification of central extensions of the $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded color superalgebras is presented. It turns out that infinitely many members of the class have non-trivial extensions. We also demonstrate that the color superalgebras (with and without central extensions) have adjoint and superadjoint operations.

math-ph↗