ℤ³₂-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics
Quantum mechanical systems whose symmetry is given by the ℤ³₂-graded version of superconformal algebra are introduced. This is done by finding a realization of a ℤ³₂-graded Lie superalgebra in terms of a standard Lie superalgebra and the Clifford algebra. The realization allows us to map many models...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2021 |
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Інститут математики НАН України
2021
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| Цитувати: | ℤ³₂-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics. Shunya Doi and Naruhiko Aizawa. SIGMA 17 (2021), 071, 14 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859904392531017728 |
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| author | Doi, Shunya Aizawa, Naruhiko |
| author_facet | Doi, Shunya Aizawa, Naruhiko |
| citation_txt | ℤ³₂-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics. Shunya Doi and Naruhiko Aizawa. SIGMA 17 (2021), 071, 14 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Quantum mechanical systems whose symmetry is given by the ℤ³₂-graded version of superconformal algebra are introduced. This is done by finding a realization of a ℤ³₂-graded Lie superalgebra in terms of a standard Lie superalgebra and the Clifford algebra. The realization allows us to map many models of superconformal quantum mechanics (SCQM) to their ℤ³₂-graded extensions. It is observed that for the simplest SCQM with (1|2) symmetry, there exist two inequivalent ℤ³₂-graded extensions. Applying the standard prescription of conformal quantum mechanics, the spectrum of the SCQMs with the ℤ³₂-graded (1|2) symmetry is analyzed. It is shown that many models of SCQM can be extended to a ℤⁿ₂-graded setting.
|
| first_indexed | 2026-03-17T10:24:10Z |
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| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 071, 14 pages
Z3
2-Graded Extensions of Lie Superalgebras
and Superconformal Quantum Mechanics
Shunya DOI and Naruhiko AIZAWA
Department of Physical Science, Osaka Prefecture University,
Nakamozu Campus, Sakai, Osaka 599-8531, Japan
E-mail: s s.doi@p.s.osakafu-u.ac.jp, aizawa@p.s.osakafu-u.ac.jp
Received March 24, 2021, in final form July 14, 2021; Published online July 20, 2021
https://doi.org/10.3842/SIGMA.2021.071
Abstract. Quantum mechanical systems whose symmetry is given by Z3
2-graded version of
superconformal algebra are introduced. This is done by finding a realization of a Z3
2-graded
Lie superalgebra in terms of a standard Lie superalgebra and the Clifford algebra. The
realization allows us to map many models of superconformal quantum mechanics (SCQM)
to their Z3
2-graded extensions. It is observed that for the simplest SCQM with osp(1|2)
symmetry there exist two inequivalent Z3
2-graded extensions. Applying the standard pre-
scription of conformal quantum mechanics, spectrum of the SCQMs with the Z3
2-graded
osp(1|2) symmetry is analyzed. It is shown that many models of SCQM can be extended to
Zn
2 -graded setting.
Key words: graded Lie superalgebras; superconformal mechanics
2020 Mathematics Subject Classification: 17B75; 17B81; 81R12
1 Introduction
In the recent works [2, 3, 14], Zn2 -graded extensions of supersymmetric quantum mechanics
(SQM) were introduced and their properties were investigated (Zn2 denotes the direct product
of n copies of the Abelian group Z2). They are a quantum mechanical realization of Zn2 -graded
version of supersymmetry algebra introduced by Bruce [12] (see also [18]), i.e., the Hamiltonian
is a matrix differential operator acting on a Zn2 -graded Hilbert space and the symmetry is given
by a Zn2 -graded Lie superalgebra. As a Zn2 -graded Lie superalgebra (see Appendix for definition)
is an extension of Lie superalgebra to more complex grading structure [23, 24, 25, 26], the Zn2 -
graded SQM is a natural generalization of the standard SQM. It is observed in [2, 3] that the
Zn2 -graded SQM is constructed by a combination of the standard SQM and Clifford algebras.
In fact, it is known that a tensor product of a Clifford algebra and a standard Lie superalgebra
realizes a Zn2 -graded Lie superalgebra [1, 25]. Such realization is not unique since for a given
Lie superalgebra there exist some distinct ways of tensoring Clifford algebras. Usually, the
distinct tensoring produces inequivalent Zn2 -graded extensions of the Lie superalgebra. However,
it can happen that those Zn2 -graded extensions are identical and the different tensoring produces
inequivalent representations of a single Zn2 -graded Lie superalgebra. This is the case of Zn2 -
graded SQM studied in [2] where tensor product of a standard SQM and a sequence of Clifford
algebras gives inequivalent representations of the Zn2 -graded version of supersymmetry algebra.
The realizations in [2] are restricted to the standard SQM and it is not clear that it can
be applicable to other Lie superalgebras. On the other hand, the realization presented in [3] is
applied to a larger class of Lie superalgebras though it produce only Z2
2-graded extensions. Thus
one can use it to define a Z2
2-graded extension of superconformal quantum mechanics (SCQM).
It is shown that by this realization many models of the standard SCQM are mapped to their
Z2
2-graded extension. The simplest case, Z2
2-graded osp(1|2) SCQM, is investigated in some
mailto:s_s.doi@p.s.osakafu-u.ac.jp
mailto:aizawa@p.s.osakafu-u.ac.jp
https://doi.org/10.3842/SIGMA.2021.071
2 S. Doi and N. Aizawa
detail [3] and an abstract representation theory of the Z2
2-graded osp(1|2) is developed in [10]
where the richness of irreducible representations of the Z2
2-graded osp(1|2) is observed.
As a continuation of the works on quantum mechanical realizations of Zn2 -graded Lie su-
peralgebras, in the present work we explore Zn2 -graded version of SCQM and present models
of Z3
2-graded SCQM explicitly. Although we focus on Z3
2-graded SCQM, models of Zn2 -graded
SCQM for any n are also introduced. In fact, our result is more general since we start with a new
way of mapping a standard Lie superalgebra to its higher graded version. This means that higher
graded extensions of physically relevant algebras such as super-Poincaré, super-Schrödinger etc
are also obtained in our formalism.
The present work is motivated by the recent renewed interest in Zn2 -graded superalgebras in
physics and mathematics. In physics side, they give a new symmetry different from the ones
generated by Lie algebras and superalgebras. Here we mention only some of them. It was found
that symmetries of some differential equations such as Lévy-Leblond equation (non-relativistic
Dirac equation) are generated by Z2
2-graded Lie superalgebras [5, 6]. Some supersymmetric
classical theories are extended to Z2
2-graded setting [8, 9, 13]. It is shown that non-trivial physics
can be detected in the multiparticle sector of the Z2
2-graded SQM [28]. Z2
2-Graded version of
spacetime symmetries are proposed by several authors, e.g., [27]. In mathematics side, Zn2 -
graded supergeometry which is an extension of supergeometry on supermanifolds, is studied
extensively, see, e.g., [22]. More exhaustive list of references of physical and mathematical
aspects of Zn2 -graded Lie superalgebras is found in [10].
We organise this paper as follows: We start Section 2 with the definition of Z3
2-graded Lie
superalgebra. Then we review briefly the results of [2] on Zn2 -graded SQM. An emphasis is put
on the fact that there exists a sequence of inequivalent models of Zn2 -graded SQM for a given
standard SQM because we also consider the sequence of Z3
2-graded SCQM in this work. In
Section 3 the first member of the sequence (there are three members), Cl(4) model, is presented.
We give a realization of a Z3
2-graded Lie superalgebra in terms of the Clifford algebra Cl(4) and
a standard Lie superalgebra. This realization is applied to osp(1|2) SCQM, then we obtain its
Z3
2-graded extension. The spectrum of Z3
2-graded osp(1|2) SCQM is investigated by employing
the standard procedure of conformal quantum mechanics. In Section 4 other two members of
the sequence, for which Cl(6) is used, are considered and it will be shown that one of them is
irrelevant as it does not give an irreducible representation of Z3
2-graded osp(1|2). For the relevant
one which defines an another Z3
2-graded extension of osp(1|2), the same analysis as Cl(4) case
is repeated. We close the paper with some remarks in Section 5.
2 Preliminaries
2.1 Z3
2-graded Lie superalgebras
We define the Z3
2-graded Lie superalgebra according to [24, 25]. Let ~a = (a1, a2, a3), ~b =
(b1, b2, b3) be elements of Z3
2. Here we regard an element of Z3
2 as a three-dimensional vector
and their sum and inner product are computed in modulus 2
~a+~b = (a1 + b1, a2 + b2, a3 + b3), ~a ·~b =
3∑
k=1
akbk.
We also introduce the parity of ~a defined by
|~a| :=
3∑
k=1
ak mod 2.
Z3
2-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics 3
Consider a complex vector space g consisting of eight subspaces each of which is labelled by an
element of Z3
2:
g = g(0,0,0) ⊕ g(0,0,1) ⊕ g(0,1,0) ⊕ g(1,0,0) ⊕ g(0,1,1) ⊕ g(1,0,1) ⊕ g(1,1,0) ⊕ g(1,1,1).
The vector space g is refereed to as a Z3
2-graded Lie superalgebra if its elements are closed in
commutator or anticommutator which is chosen according to the following rule
JX~a, X~bK :=
{
[X~a, X~b], ~a ·~b = 0,
{X~a, X~b}, ~a ·~b = 1,
X~a ∈ g~a, X~b ∈ g~b. (2.1)
We use J·, ·K as a unified notation of commutator and anticommutator. See appendix for more
rigorous definition of Zn2 -graded Lie superalgebras.
2.2 A sequence of Zn
2 -graded SQM
We review the results of [2] briefly as the present work is an algebraic generalization of them. The
Zn2 -graded SQM is defined as a realization of Zn2 -graded supersymmetry algebra in a Zn2 -graded
Hilbert space. The Zn2 -graded supersymmetry algebra consists of one Hamiltonian, 2n−1 super-
charges of parity 1 and 2n−2
(
2n−1 − 1
)
central elements of parity 0.
It was shown that a tensor product of N = 1 standard SQM and a complex irreducible
representation (irrep) of the Clifford algebra Cl(2m) can give the realization which define a model
of the Zn2 -graded SQM. The N = 1 standard SQM is generated by one supercharge Q and its
defining relations are given by
{Q,Q} = 2H, [H,Q] = 0.
Both Q and H are 2× 2 matrix differential operators acting on Z2-graded Hilbert space.
The Clifford algebra Cl(2m) is generated by γj (j = 1, 2, . . . , 2m) subject to the conditions
{γj , γk} = 2δjk.
The Hermitian complex irrep of Cl(2m) is 2m-dimensional and given explicitly as follows [15, 20]
γ1 = σ⊗m1 , γj = σ
⊗(m−j+1)
1 ⊗ σ3 ⊗ I⊗(j−2)2 , 2 ≤ j ≤ m,
γ̃j := γj+m = σ
⊗(m−j)
1 ⊗ σ2 ⊗ I⊗(j−1)2 , 1 ≤ j ≤ m, (2.2)
where σk is the Pauli matrix and I2 denotes the 2 × 2 identity matrix. Therefore, a model of
Zn2 -graded SQM is a set of 2m+1-dimensional matrix differential operators.
For a fixed value of n, one may have a sequence of inequivalent models of Zn2 -graded SQM
by tensoring the standard SQM and the following sequence of the Clifford algebra
Cl(2(n− 1)), Cl(2n), Cl(2(n+ 1)), . . . , Cl(2n − 2). (2.3)
For instance, we have five distinct models of Z4
2-graded SQM from the Clifford algebras
Cl(6), Cl(8), Cl(10), Cl(12), Cl(14).
The difference in the models is the number of linearly independent central elements. The Zn2 -
graded supersymmetry algebra has a lot of central elements. Some of the central elements are
realized as dependent operators unless the Clifford algebra of the maximal dimension in the
above sequence is used. Lower the dimension of the Clifford algebra, more central elements are
realized as dependent operators.
In the next two sections, we show the existence of a sequence of realizations of Z3
2-graded Lie
superalgebra and by which one may introduce models of Z3
2-graded SCQM.
4 S. Doi and N. Aizawa
3 Cl(4) model of Z3
2-graded SCQM
In this and the following sections, we deal with Z3
2-graded Lie superalgebras and Z3
2-graded
SCQM. Setting n = 3 in (2.3), we see that the sequence has only two Clifford algebras: Cl(2(n−
1)) = Cl(4), Cl(2n) = Cl(2n − 2) = Cl(6). However, realizations for Cl(2n) and Cl(2n − 2)
considered in [2] are not identical. We thus explore three cases, one for Cl(4) and two for Cl(6).
It will then turn out that, contrary to Zn2 -graded SQM, we have two inequivalent Z3
2-graded
extensions of osp(1|2). In this section, we focus on Cl(4).
In the sequel, we denote a standard (Z2-graded) Lie superalgebra by s and its even and odd
subspaces by s0 and s1, respectively. We use a Hermitian representation of s to realize a Z3
2-
graded Lie superalgebra. Recalling that |~a| = 0 or 1 for ~a ∈ Z3
2, we denote a Hermitian matrix
representing an element of s|~a| by X|~a| and suppose its size is 2m× 2m.
3.1 Cl(4) realization of Z3
2-graded Lie superalgebra
The irrep (2.2) for Cl(4) is given by
γ1 = σ1 ⊗ σ1, γ2 = σ1 ⊗ σ3, γ3 = σ1 ⊗ σ2, γ4 = σ2 ⊗ I2.
Let Γ be a matrix subject to
[X0,Γ] = 0, {X1,Γ} = 0, Γ2 = I2m, ∀X|~a| ∈ s|~a|. (3.1)
Then the matrices defined by
XXX~a = if(~a)γa11 γ
a2
2 ⊗X|~a|Γ
a1+a2 , f(~a) := a1a2 + |~a|(a1 + a2) mod 2 (3.2)
are Hermitian and define a Z3
2-graded Lie superalgebra. More explicitly, XXX~a is given by
XXX(0,0,0) = I4 ⊗X0, XXX(1,0,0) = iγ1 ⊗X1Γ, XXX(0,1,0) = iγ2 ⊗X1Γ,
XXX(0,0,1) = I4 ⊗X1, XXX(1,1,1) = iγ1γ2 ⊗X1, XXX(1,1,0) = iγ1γ2 ⊗X0,
XXX(1,0,1) = γ1 ⊗X0Γ, XXX(0,1,1) = γ2 ⊗X0Γ.
It is immediate to see XXX~a is Hermitian
(XXX~a)
† = (−i)f(~a)γa22 γ
a1
1 ⊗ Γa1+a2X|~a| = (−1)f(~a)+a1a2+|~a|(a1+a2)XXX~a = XXX~a.
To verify the Z3
2-graded Lie superalgebra structure, we need to prove the closure in (anti)com-
mutator and graded Jacobi relations (A.1). This will be done by showing that the Z3
2-graded
commutators and Jacobi relations are reduced to those for the Lie superalgebra s. It is not
difficult to see the (anti)commutators (see (A.2)) are computed as
JXXX~a,XXX~b
K = X~aX~b − (−1)~a·
~bX~bX~a
= (−1)a2b1+(a1+a2)|~b|if(~a)+f(
~b)γa1+b11 γa2+b22 ⊗ 〈X|~a|, X|~b|〉Γ
a1+a2+b1+b2 , (3.3)
where
〈X|~a|, X|~b|〉 := X|~a|X|~b| − (−1)|~a||
~b|X|~b|X|~a|
is the (anti)commutator of the Lie superalgebra s. Writing the (anti)commutation relations of s
in the form
〈X|~a|, X|~b|〉 = i1−|~a||
~b|X|~a|+|~b| = i1−|~a||
~b|X|~a+~b|,
Z3
2-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics 5
(3.3) yields
JXXX~a,XXX~b
K = (−1)a2b1+(a1+a2)|~b|i1+f(~a)+f(
~b)−f(~a+~b)−|~a||~b|XXX
~a+~b
with
XXX
~a+~b
= if(~a+
~b)γa1+b11 γa2+b22 ⊗X|~a+~b|Γ
a1+a2+b1+b2 .
Therefore, JXXX~a,XXX~b
K ∈ g
~a+~b
, i.e., closure of Z3
2-graded (anti)commutator has been proved.
By the similar computation one may see
(−1)~a·~cJXXX~a, JXXX~b
,XXX~cKK =(−1)κif(~a)+f(
~b)+f(~c)γa1+b1+c11 γa2+b2+c22
⊗ (−1)|~a||~c|〈X|~a|, 〈X|~b|, X|~c|〉〉Γ
∑2
k=1(ak+bk+ck),
where
κ :=
2∑
k=1
(akbk + bkck + ckak) + a1b2 + b1c2 + c1a2
+ a1b3 + b1c3 + c1a3 + a2b3 + b2c3 + c2a3.
κ is invariant under the cyclic permutation of a, b, c. This shows that the graded Jacobi relations
are reduced to those for s. It follows that the graded Jacobi identity holds true and the Z3
2-graded
Lie superalgebra structure has been proved.
The realization (3.2) is able to generalize to a realization of Zn2 -graded Lie superalgebras by s
and Cl(2(n− 1)):
XXX~a = if(~a)
n−1∏
j=1
γ
aj
j ⊗X|~a|Γ
∑n−1
k=1 ak , (3.4)
f(~a) =
n−2∑
k=1
ak
n−1∏
l=k+1
al + |~a|
n−1∑
l=1
al mod 2.
One may prove this in the same way as Z3
2-graded Lie superalgebras so we do not present the
proof.
3.2 Cl(4) model of Z3
2-graded osp(1|2) SCQM
As shown in Section 3.1, any Lie superalgebra satisfying the condition (3.1) can be promoted
to a Z3
2-graded superalgebra. If one starts with a matrix differential operator realization of
a superconformal algebra, i.e., a model of SCQM, then one may obtain its Z3
2-graded version.
Many models of SCQM have been obtained so far (see, e.g., [17, 19, 21]). Some of the models,
e.g., the ones in [4, 7], satisfy the condition (3.1) so that we may have the Z3
2-graded SCQM of
N = 2, 4, 8 and so on.
Here we analyse the simplest example of Z3
2-graded SCQM obtained from the osp(1|2) su-
perconformal algebra. Let us consider the following realization of osp(1|2) which is a N = 1
SCQM:
Q =
1√
2
(
σ1p− σ2
β
x
)
, S =
x√
2
σ1,
H =
1
2
(
p2 +
β2
x2
)
I2 +
β
2x2
σ3, D = −1
4
{x, p}I2, K =
x2
2
I2, (3.5)
6 S. Doi and N. Aizawa
where β is a coupling constant. The non-vanishing relations of osp(1|2) read as follows
[D,K] = iK, [H,K] = 2iD, [D,H] = −iH,
{Q,Q} = 2H, {S, S} = 2K, {Q,S} = −2D,
[D,Q] = − i
2
Q, [D,S] =
i
2
S, [Q,K] = −iS,
[S,H] = iQ.
One may immediately see that Γ = σ3 satisfies the condition (3.1). Thus by (3.2) we obtain
twenty operators: the diagonal degree (0, 0, 0) operators are given by
HHH000 = I4 ⊗H, DDD000 = I4 ⊗D, KKK000 = I4 ⊗K. (3.6)
Here and in the following sections we use a simplified notation XXXa1a2a3 := XXX(a1,a2,a3). The oper-
ator HHH000 is the Hamiltonian of the model and these three operators form the one-dimensional
conformal algebra so(1, 2). The other parity even operators, which are not diagonal, are given
by
XXX110 = iγ1γ2 ⊗X, XXX101 = γ1 ⊗Xσ3, XXX011 = γ2 ⊗Xσ3, X = H, D, K,
and the parity odd ones are given by
QQQ100 = iγ1 ⊗Qσ3, SSS100 = iγ1 ⊗ Sσ3,
QQQ010 = iγ2 ⊗Qσ3, SSS010 = iγ2 ⊗ Sσ3,
QQQ001 = I4 ⊗Q, SSS001 = I4 ⊗ S,
QQQ111 = iγ1γ2 ⊗Q, SSS111 = iγ1γ2 ⊗ S. (3.7)
(Anti)commutator of these operators are closed and define a Z3
2-graded extension of osp(1|2)
which we denote simply by G1. We note that dimG1 = 20.
For the range of β where the potential is repulsive, the Hamiltonian H in (3.5) has continuous
spectrum. It is known that the eigenfunctions of H with the positive eigenvalue are plane
wave normalizable, however, the zero energy state is not even plane wave normalizable [16].
This property is inherited to the Hamiltonian HHH000 of the Z3
2-graded SCQM (3.7). In order to
analyse the syetem (3.7) we follow the standard prescription of conformal mechanics. That is,
the eigenspace of HHH000 is not taken as the Hilbert space of the theory. Instead, the eigenspace
of an operator which is a linear combination of HHH000 and KKK000 is chosen as the Hilbert space.
We thus introduce the following operators
RRR~a = HHH~a +KKK~a, LLL±~a =
1
2
(KKK~a −HHH~a)± iDDD~a,
aaa~a = SSS~a + iQQQ~a, aaa†~a = SSS~a − iQQQ~a.
The diagonal operator RRR000 is the new Hamiltonian and it has discrete eigenvalues due to the
oscillator potential (see (3.5) and (3.6)). The eigenspace of RRR000 is H = L2(R) ⊗ C8 which is
taken to be the Hilbert space of the model. The space H has a vector space decomposition
according to the Z3
2-degree
H =
⊕
~a∈Z3
2
H~a.
The operators aaa~a, aaa
†
~a, LLL
±
~a generate the spectrum of RRR000
[RRR000, aaa~a] = −aaa~a,
[
RRR000, aaa
†
~a
]
= −aaa†~a,
[
RRR000,LLL
±
~a
]
= ±2LLL±~a , (3.8){
aaa~a, aaa
†
~a
}
= 2RRR000,
Z3
2-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics 7
and the operators aaa~a, aaa
†
~a, together with F defined below, form a Klein deformed oscillator algebra[
aaa~a, aaa
†
~a
]
= I8 − 2βF, F := I4 ⊗ σ3,
F 2 = I8, {F,aaa~a} =
{
F,aaa†~a
}
= 0.
It follows that
RRR000 = aaa†~aaaa~a +
1
2
(I8 − 2βF ).
Thus the ground state is obtained by
aaa~aΨ(x) = 0,
where
Ψ(x) = (ψ000(x), ψ001(x), ψ110(x), ψ111(x), ψ011(x), ψ010(x), ψ101(x), ψ100(x))T ∈H .
For all aaa~a this condition is reduced to the set of relations for the components of Ψ(x)(
∂x + x− β
x
)
ψ~a(x) = 0, ~a = (0, 0, 1), (1, 1, 1), (0, 1, 0), (1, 0, 0), (3.9)(
∂x + x+
β
x
)
ψ~a(x) = 0, ~a = (0, 0, 0), (1, 1, 0), (0, 1, 1), (1, 0, 1). (3.10)
The solution of these equations are given by ψ~a(x) = x±βe−x
2/2 and the normalizability of
the functions are studied in detail in [4]. For β > 1 (repulsive potential) only one of them is
normalizable so that the ground state is either
xβe−x
2/2(0, C1, 0, C2, 0, C3, 0, C4)
T
with the energy 1
2(1 + 2β) or
x−βe−x
2/2(C1, 0, C2, 0, C3, 0, C4, 0)T
with the energy 1
2(1− 2β) where Ci is a constant. Thus the ground state is four-fold degenerate
and belongs to either parity odd or even subspaces of H .
The excited states with various Z3
2-degree are obtained by repeated application of aaa†~a on the
ground state and one may see from (3.8) that the operator RRR(0,0,0) has equally spaced spectrum.
We remark that no need to consider the action of LLL+
~a because of the relation
{
aaa†~a, aaa
†
~a
}
= 4LLL+
~a .
The excited state is also four-fold degenerate. This is verified as follows. Let φ~a ∈ H~a and
φ~b ∈ H~b
be eigenfunctions of RRR000 with the same eigenvalue. Then aaa~cφ~a equals to aaa~dφ~b up to
a constant multiple if ~a + ~c = ~b + ~d. For instance, it is not difficult to see the following two
functions in H(1,0,1) are identical up to a constant
aaa100ψ001, ψ001 = xβe−x
2/2(0, C1, 0, 0, 0, 0, 0, 0),
aaa010ψ111, ψ111 = xβe−x
2/2(0, 0, 0, C2, 0, 0, 0, 0).
This Cl(4) model of Z3
2-graded SCQM gives irreps of G1. Recalling that the order of Z3
2 is∣∣Z3
2
∣∣ = 8, the minimal dimension of non-trivial irreps of G1 in a Z3
2-graded vector space is also
eight which is the dimension of the Cl(4) model.
As mentioned at the end of Section 3.1, we have a realization of Zn2 -graded Lie superalgebras
by s and Cl(2(n−1)) where the condition (3.1) is required. The condition is satisfied for osp(1|2)
SCQM (3.5) as we have seen. It is also satisfied for other models of SCQM. For instance,
Γ = σ3, σ3 ⊗ I2 and
(
I8 0
0 I8
)
for the osp(2|2), D(2, 1;α) and F (4) models, respectively (see [4, 7]
for the models). These models are promoted to their Zn2 -graded version by using (3.4). Thus,
there exists various Cl(2(n− 1)) models of Zn2 -graded SCQM and we may analyze its properties
in a manner similar to this section.
8 S. Doi and N. Aizawa
4 Cl(6) models of Z3
2-graded SCQM
In this section, we explore the models of Z3
2-graded SCQM obtained via Cl(6) in a way similar
to the ones via Cl(4). As already mentioned, we investigate two realizations of Z3
2-graded Lie
superalgebras via Cl(2n) and Cl(2n − 2). These two Clifford algebras are degenerate for n = 3,
however the way of realizing the Z3
2-graded Lie superalgebras are not identical.
The irrep (2.2) for Cl(6), which is common for Cl(2n) and Cl(2n − 2), is given by
γ1 = σ1 ⊗ σ1 ⊗ σ1, γ2 = σ1 ⊗ σ1 ⊗ σ3, γ3 = σ1 ⊗ σ3 ⊗ I2,
γ4 = σ1 ⊗ σ1 ⊗ σ2, γ5 = σ1 ⊗ σ2 ⊗ I2, γ6 = σ2 ⊗ I2 ⊗ I2.
4.1 Cl(2n) model
The realization of Zn2 -graded Lie superalgebra in terms of Cl(2n) and a ordinary Lie superal-
gebra s is given in [1]. Thus we are able to use the result to investigate a model of Z3
2-graded
SCQM. For n = 3, the realization of Z3
2-graded Lie superalgebra by Xa ∈ sa reads as follows
XXX000 = I8 ⊗X0,
XXX100 = γ1 ⊗X1, XXX010 = γ2 ⊗X1, XXX001 = γ3 ⊗X1, XXX111 = iγ1γ2γ3 ⊗X1,
XXX110 = iγ1γ2 ⊗X0, XXX101 = iγ1γ3 ⊗X0, XXX011 = iγ2γ3 ⊗X0.
Contrast to Cl(4), there is no condition like (3.1) so that any models of SCQM can be extended
to Z3
2-grading by this realization. We consider again the osp(1|2) model (3.5) as the simplest
example. The Z3
2-graded SCQM so obtained is the set of matrix differential operators
XXX000 = I8 ⊗X,
QQQ100 = γ1 ⊗Q, SSS100 = γ1 ⊗ S,
QQQ010 = γ2 ⊗Q, SSS010 = γ2 ⊗ S,
QQQ001 = γ3 ⊗Q, SSS001 = γ3 ⊗ S,
QQQ111 = iγ1γ2γ3 ⊗Q, SSS111 = iγ1γ2γ3 ⊗ S,
XXX110 = iγ1γ2 ⊗X, XXX101 = iγ1γ3 ⊗X, XXX011 = iγ2γ3 ⊗X (4.1)
with X = H,D,K.
It is not difficult to see that these twenty operators form an closed algebra whose (anti)com-
mutation relations are identical to the ones for Cl(4) model (3.7). Namely, the operators in (4.1)
give 16-dimensional representation of G1. However, this is a reducible representation of G1. To
see this, let F (R) be a space of complex valued functions on a real line and H = F (R) ⊗ C16.
The operators (4.1) act on H and it is readily seen from the explicit form of the operators that
the following subspaces H1 and H2 are invariant under the action of (4.1):
H = H1 ⊕H2,
H1 = (ψ000, 0, ψ110, 0, ψ011, 0, ψ101, 0, 0, ψ001, 0, ψ111, 0, ψ010, 0, ψ100)
T,
H2 = (0, ψ000, 0, ψ110, 0, ψ011, 0, ψ101, ψ001, 0, ψ111, 0, ψ010, 0, ψ100, 0)T.
This shows that the operators (4.1) are a reducible representation of G1.
Although the combination of Cl(2n) and osp(1|2) are not physically relevant for n = 3. This
does not implies other models such as D(2, 1;α), F (4) are irrelevant, either. At least, one may
see the existence of various models of Zn2 -graded SCQM. Because the realization given in [1]
does not require any further conditions like (3.1) so than any models of standard SCQM can be
Z3
2-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics 9
promoted to their Zn2 -graded counterparts. Physical relevance of those models should be studied
case by case.
As already mentioned, some inequivalent realizations of Zn2 -graded Lie superalgebras in terms
of the Clifford algebra and a standard Lie superalgebra are known. However, the present example
elucidates not all such realizations are suitable for physical applications. We need to find an
appropriate one to discuss physical problems. Therefore, general study of realizations of Zn2 -
graded Lie superalgebra by ordinary superalgebras is an important research problem.
4.2 Cl(2n − 2) model
A non-trivial Cl(6) model is obtained by an analogy of Zn2 -extension of SQM considered in [2].
First, we introduce the following ordering into the parity odd elements of Z3
2:
~a0 = (1, 1, 1), ~a1 = (1, 0, 0), ~a2 = (0, 1, 0), ~a3 = (0, 0, 1).
Define the following Hermitian matrices
XXX(0,0,0) = I8 ⊗X0,
XXXµ = γµ ⊗X1,
XXXµν = i1−~aµ·~aνγµγν ⊗X0, µ < ν,
XXXµνρ = iγµγνγρ ⊗X1, µ < ν < ρ,
XXX0123 = iγ1γ2γ3 ⊗X0, (4.2)
where the Greek indices run from 0 to 3 and γ0 = I8. The suffix (0, 0, 0) of XXX(0,0,0) denotes
its Z3
2-degree where the original notation is restored to avoid confusion. The Z3
2-degree of the
matrices with the Greek indices is determined as follows
deg(XXXµ) = ~aµ, deg(XXXµν) = ~aµ + ~aν ,
deg(XXXµνρ) = ~aµ + ~aν + ~aρ, deg(XXX0123) = (0, 0, 0).
With this assignment of Z3
2-degree the matrix operators of (4.2) define a Z3
2-graded Lie superal-
gebra. This is verified by observing that the Z3
2-graded (anti)commutators are reduced to those
for a standard superalgebra. In order to see this we write the relations of the superalgebra s as
follows
[X0, Y0] = iZ0, [X0, Y1] = iZ1, {X1, Y1} = W0.
By definition JXXX(0,0,0),YYY K = [XXX(0,0,0),YYY ] for any YYY . One may compute the commutator as
follows
[XXX(0,0,0),YYY (0,0,0)] = I8 ⊗ [X0, Y0] = iZZZ(0,0,0),
[XXX(0,0,0),YYY µ] = γµ ⊗ [X0, Y1] = iZZZµ,
[XXX(0,0,0),YYY µν ] = i1−~aµ·~aνγµγν ⊗ [X0, Y0] = iZZZµν ,
[XXX(0,0,0),YYY µνρ] = iγµγνγρ ⊗ [X0, Y1] = iZZZµνρ,
[XXX(0,0,0),YYY 0123] = iγ1γ2γ3 ⊗ [X0, Y0] = iZZZ0123.
Similarly for XXX0123
[XXX0123,YYY µ] = iγ1γ2γ3γµ ⊗ [X0, Y1] =
∑
ν,ρ,σ
fµνρσZZZνρσ,
10 S. Doi and N. Aizawa
[XXX0123,YYY µν ] = −i−~aµ·~aνγ1γ2γ3γµγν ⊗ [X0, Y0] = i
∑
ρ,σ
gµνρσZZZρσ,
[XXX0123,YYY µνρ] = −γ1γ2γ3γµγνγρ ⊗ [X0, Y1] = i
∑
σ
hµνρσZZZσ,
[XXX0123,YYY 0123] = I8 ⊗ [X0, Y0] = iZZZ(0,0,0),
where the structure constants are given as follows
f0123 = f1023 = f3012 = 1, f2013 = −1,
g0123 = g0312 = g1203 = g2301 = 1, g0213 = g1302 = −1,
h0123 = h0231 = h1230 = 1, h0132 = −1,
and the others are zero.
Other (anti)commutation relations are more involved ((2.1), see also (3.3)):
JXXXµ,YYY νK = XXXµYYY ν − (−1)~aµ·~aνYYY νXXXµ = γµγν ⊗ {X1, Y1} = i−1+~aµ·~aνWWWµν ,
JXXXµ,YYY νρK = XXXµYYY νρ − (−1)~aµ·(~aν+~aρ)YYY νρXXXµ = i1−~aν ·~aργµγνγρ ⊗ [X1, Y0] = i1−~aν ·~aρWWWµνρ,
JXXXµ,YYY νρσK = XXXµYYY νρσ − (−1)~aµ·(~aν+~aρ+~aσ)YYY νρσXXXµ
= iγµγνγργσ ⊗ {X1, Y1} = WWWµνρσ, (4.3)
where we introduced W1 ∈ s1 by [X1, Y0] = iW1 and WWWµνρ = iγµγνγρ ⊗W1. The indices of WWW
on the right-hand side of (4.3) do not always respect the restriction given in (4.2). Such WWW is
converted to the one in (4.2) by the following relations:
WWWµµ = WWW (0,0,0), WWWµν = (−1)1−~aµ·~aνWWW νµ, µ 6= ν,
WWWµµρ = iWWW ρ, WWWµνµ = −iWWW ν ,
WWWµµνρ = WWWµνρµ = (−i)~aν ·~aρWWW νρ, WWWµνµρ = −(−i)~aν ·~aρWWW νρ,
WWW νµµρ = i~aν ·~aρWWW νρ, WWW νµρµ = −i~aν ·~aρWWW νρ,
and if all the indices are different value, then
WWWµνρ = (−1)1−~aµ·~aνWWW νµρ = (−1)~aµ·(~aν+~aρ)WWW νρµ,
WWWµνρσ = (−1)1−~aµ·~aνWWW νµρσ = (−1)~aµ·(~aν+~aρ)WWW νρµσ = −(−1)~aµ·(~aν+~aρ+~aσ)WWW νρσµ. (4.4)
We further need to check the closure of multi-index matrices
JXXXµν ,YYY ρσK = XXXµνYYY ρσ − (−1)(~aµ+~aν)·(~aρ+~aσ)YYY ρσXXXµν
= −i−~aµ·~aν−~aρ·~aσγµγνγργσ ⊗ [X0, Y0] = −i−~aµ·~aν−~aρ·~aσZZZµνρσ, (4.5)
where ZZZµνρσ is understood as in (4.4) and (4.5),
JXXXµν ,YYY ρστ K = XXXµνYYY ρστ − (−1)(~aµ+~aν)·(~aρ+~aσ+~aτ )YYY ρστXXXµν
= −i−~aµ·~aνγµγνγργσγτ ⊗ [X0, Y1] = −i1−~aµ·~aνγµγνγργσγτ ⊗ Z0. (4.6)
There are five gamma matrices in this case so that one or two pairs of identical gamma matrices
exist. When there exist one pair of identical matrices, say γν = γτ , (4.6) equals to ZZZµρσ up to
a constant factor. When there exist two pair of identical matrices, say γµ = γρ and γν = γσ,
(4.6) equals to ZZZτ up to a constant multiple. In this way, we see the closure of (4.6). Similarly,
there exist identical gamma matrices in the following (anti)commutator
JXXXµνρ,YYY στλK = XXXµνρYYY στλ − (−1)(~aµ+~aν+~aρ)·(~aσ+~aτ+~aλ)YYY στλXXXµνρ
= −γµγνγργσγτγλ ⊗ {X1, Y1} = −γµγνγργσγτγλ ⊗W0. (4.7)
Z3
2-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics 11
A special subcase of this is three pairs of identical gamma matrices
{XXXµνρ,XXXµνρ} = I8 ⊗W0 = WWW (0,0,0).
Except the special case, there always exist two pairs of identical gamma matrices. Therefore,
(4.7) equals to WWWµν up to a constant factor. We thus have proved the closure of (anti)commu-
tators.
We observed that the Z3
2-graded (anti)commutator is reduced to the one of superalgebra. It
follows from this fact that a Z3
2-graded Jacobi relation is also reduced to the one of superalgebra.
Therefore, it is straightforward to verify that (4.2) satisfies the Z3
2-graded Jacobi relations.
Now we are able to use (4.2) to construct a Cl(6) model of Z3
2-graded SCQM. Taking s as
any model of SCQM, (4.2) produces a corresponding model of Z3
2-graded SCQM. As a simplest
example, here we take s = osp(1|2) given in (3.5). Then (4.2) gives us 40 operators which close
in a Z3
2-graded extension of osp(1|2). We denote this algebra simply by G2:
XXX(0,0,0) = I8 ⊗X,
QQQµ = γµ ⊗Q, SSSµ = γµ ⊗ S,
XXXµν = i1−~aµ·~aνγµγν ⊗X, µ < ν,
QQQµνρ = iγµγνγρ ⊗Q, SSSµνρ = iγµγνγρ ⊗ S, µ < ν < ρ,
XXX0123 = iγ1γ2γ3 ⊗X, X = H,D,K. (4.8)
The number of operator is double of the Cl(4) model discussed in Section 3.2 where we have 20
operators. The reason of this difference is same as the Zn2 -graded SQM considered in [2] and it
is best seen in the next example:
[QQQ1,QQQ2] = 2γ1γ2 ⊗H = −2iHHH12, {QQQ0,QQQ3} = 2γ3 ⊗H = 2HHH03.
These are the relations of Cl(6) model and deg(HHH12) = deg(HHH03) = (1, 1, 0). As γ1γ2 6= γ3,
HHH12 and HHH03 are linearly independent. The corresponding relations in the Cl(4) model are
[QQQ100,QQQ010] = 2γ1γ2 ⊗H, {QQQ111,QQQ001} = 2iγ1γ2 ⊗H.
Obviously, the operators on the right hand side are not linearly independent. Namely, in
the Cl(4) model degeneracy of operators, which are linearly independent operators in Cl(6)
model, happens and the number of the operators are reduced.
By using the explicit form of (4.8), it is not difficult to see that the space H = F (R)⊗ C16
is not decoupled into two subspaces by the action of G2. This is a sharp contrast to the model
in Section 4.1 and suggests the Z3
2-graded SCQM (4.8) gives an irreducible representation of G2.
More precise analysis of irreducible representation of G2 will be done in a way similar to [10] but
it is beyond the scope of the present work.
Let us briefly analyse the spectrum of the Cl(6) model by employing the standard prescription
of conformal mechanics. That is, we define the operators
RRR(0,0,0) = HHH(0,0,0) +KKK(0,0,0), LLL±(0,0,0) =
1
2
(KKK(0,0,0) −HHH(0,0,0))± iDDD(0,0,0),
aaaµ = SSSµ + iQQQµ, aaa†µ = SSSµ − iQQQµ,
aaaµνρ = SSSµνρ + iQQQµνρ, aaa†µνρ = SSSµνρ − iQQQµνρ,
and take the eigenspace of RRR(0,0,0), which is L2(R) ⊗ C16, as the Hilbert space of the theory.
There exist twice many creation and annihilation operators than Cl(4) model[
RRR(0,0,0), aaaµ
]
= −aaaµ,
[
RRR(0,0,0), aaa
†
µ
]
= −aaa†µ,[
RRR(0,0,0), aaaµνρ
]
= −aaaµνρ,
[
RRR(0,0,0), aaa
†
µνρ
]
= −aaa†µνρ,[
RRR(0,0,0),LLL
±
(0,0,0)
]
= ±2LLL±(0,0,0).
12 S. Doi and N. Aizawa
These creation and annihilation operators satisfy the relations similar to the Cl(4) model{
aaaµ, aaa
†
µ
}
=
{
aaaµνρ, aaa
†
µνρ
}
= 2RRR(0,0,0),{
aaa†µ, aaa
†
µ
}
=
{
aaa†µνρ, aaa
†
µνρ
}
= 2LLL+
(0,0,0), {aaaµ, aaaµ} = {aaaµνρ, aaaµνρ} = 2LLL−(0,0,0).
Furthermore, aaaµ and aaa†µ satisfy a Klein deformed oscillator algebra[
aaaµ, aaa
†
µ
]
=
[
aaaµνρ, aaa
†
µνρ
]
= I16 − 2βF, F := I8 ⊗ σ3,
{F,aaaµ} =
{
F,aaa†µ
}
= {F,aaaµνρ} =
{
F,aaa†µνρ
}
= 0, F 2 = I16.
With these relations one may have
RRR(0,0,0) = aaa†µaaaµ +
1
2
(I16 − 2βF ) = aaa†µνρaaaµνρ +
1
2
(I16 − 2βF ).
Thus the ground state Ψ(x) or RRR(0,0,0) is defined by
aaaµΨ(x) = aaaµνρΨ(x) = 0. (4.9)
We write Ψ(x) in components
Ψ(x) = (ψ000, ψ111, ψ110, ψ001, ψ011, ψ100, ψ101, ψ010, ψ110, ψ001, ψ000,
ψ111, ψ101, ψ010, ψ011, ψ100)
T.
Then the condition (4.9) gives the same relations as Cl(4) model (3.9) and (3.10). It follows
that the ground state of Cl(6) model is eight-fold degenerate. The excited states are obtained
by repeated application of aaa†µ and aaa†µνρ on the ground state. Repeating the argument same as
the Cl(4) model, one may see that the spectrum of RRR(0,0,0) is equally spacing and the excited
state has eight-fold degeneracy.
As an abstract Lie algebra we regard G2 is inequivalent to G1 as dimG2 = 40 > dimG1.
5 Concluding remarks
We showed that many models of SCQM are able to extend to Zn2 -graded setting by the use of
the Clifford algebras Cl(2(n− 1)) and Cl(2n). It was also shown the existence of a sequence of
models of Z3
2-graded osp(1|2) SCQM and we analyzed the spectrum of the models. Most likely,
for a given model of standard SCQM there would exists a sequence of models of Zn2 -graded SCQM
produced via the sequence of the Clifford algebras (2.3). However, full analysis of Zn2 -graded
SCQM will require lengthy computation and invention of better notations (especially for the
models via higher-dimensional Clifford algebras) which make the presentation simpler and more
readable. Therefore, we are planning to present them in a separate publication. We convince
that the present analysis of Z3
2-graded SCQM provides all the essentials of Zn2 -graded extensions.
Although the existence of Zn2 -graded SCQM has been established, its physical implications
and how much it differs from the standard SCQM are not clear yet. To have better understanding
of Z3
2 and higher graded SCQM, there would be some more works to be done. For example,
one may consider multiparticle extensions of the models presented in this paper. As showed in
Z2
2-graded SQM, difference from the standard SQM becomes clear when a multiparticle model
is considered. A multiparticle extension may be done in a way similar to [28].
The second example is classical theories of Z3
2-graded SCQM which reproduce the models of
this work upon quantization. Such classical theories will shed a new light on Z3
2-graded SCQM
and they have their own interest. For the simpler grading by Z2
2, D-module presentation and su-
perfield approach to the classical theory of Z2
2-graded SQM are discussed in the literature [8, 13].
Z3
2-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics 13
It is a very interesting but challenging problem to generalize these to Zn2 -graded (n ≥ 3) set-
ting since integration on Zn2 -graded (n ≥ 3) supermanifolds has not been established yet [22].
Nonlinear realization is a widely used approach to superconformal mechanics, see, e.g., [11, 17].
Z3
2-graded extension of nonlinear realization will be possible and it will give some geometrical
understanding of Z3
2-graded SCQM.
A Definition of Zn
2 -graded Lie superalgebra
In this appendix we give a rigorous definition of Zn2 -graded Lie superalgebra [23, 24, 25]. Let g
be a vector space over R or C and ~a = (a1, a2, . . . , an) an element of Zn2 . Suppose that g is
a direct sum of graded subspaces labelled by ~a
g =
⊕
~a
g~a.
Homogeneous elements of g~a are denoted by X~a, Y~a, . . . . If g admits a bilinear operation (the
general Lie bracket), denoted by J·, ·K, satisfying the identities
JX~a, Y~bK ∈ g
~a+~b
, JX~a, Y~bK = −(−1)~a·
~bJY~b, X~aK,
(−1)~a·~cJX~a, JY~b, Z~cKK + (−1)
~b·~aJY~b, JZ~c, X~aKK + (−1)~c·
~bJZ~c, JX~a, Y~bKK = 0, (A.1)
where
~a+~b = (a1 + b1, a2 + b2, . . . ) ∈ Zn2 , ~a ·~b =
n∑
k=1
akbk,
then g is referred to as a Zn2 -graded Lie superalgebra. The relation (A.1) is called the Zn2 -graded
Jacobi relation.
We take g to be contained in its enveloping algebra, via the identification
JX~a, Y~bK = X~aY~b − (−1)~a·
~bY~bX~a, (A.2)
where an expression such as X~aY~b is understood to denote the associative product on the en-
veloping algebra. In other words, by definition, in the enveloping algebra the general Lie bracket
J·, ·K for homogeneous elements coincides with either a commutator or anticommutator.
This is a natural generalization of Lie superalgebra which is defined on Z2-grading structure.
Namely, the vector ~a is one-dimensional
g = g(0) ⊕ g(1)
with ~a+~b = (a+ b), ~a ·~b = ab.
Acknowledgements
The authors would like to thank the referees for the valuable comments for improvement of this
paper.
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1 Introduction
2 Preliminaries
2.1 Z23-graded Lie superalgebras
2.2 A sequence of Z2n-graded SQM
3 Cl(4) model of Z23-graded SCQM
3.1 Cl(4) realization of Z23-graded Lie superalgebra
3.2 Cl(4) model of Z23-graded osp(1|2) SCQM
4 Cl(6) models of Z23-graded SCQM
4.1 Cl(2n) model
4.2 Cl(2n-2) model
5 Concluding remarks
A Definition of Z2n-graded Lie superalgebra
References
|
| id | nasplib_isofts_kiev_ua-123456789-211352 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-17T10:24:10Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Doi, Shunya Aizawa, Naruhiko 2025-12-30T15:54:09Z 2021 ℤ³₂-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics. Shunya Doi and Naruhiko Aizawa. SIGMA 17 (2021), 071, 14 pages 1815-0659 2020 Mathematics Subject Classification: 17B75; 17B81; 81R12 arXiv:2103.10638 https://nasplib.isofts.kiev.ua/handle/123456789/211352 https://doi.org/10.3842/SIGMA.2021.071 Quantum mechanical systems whose symmetry is given by the ℤ³₂-graded version of superconformal algebra are introduced. This is done by finding a realization of a ℤ³₂-graded Lie superalgebra in terms of a standard Lie superalgebra and the Clifford algebra. The realization allows us to map many models of superconformal quantum mechanics (SCQM) to their ℤ³₂-graded extensions. It is observed that for the simplest SCQM with (1|2) symmetry, there exist two inequivalent ℤ³₂-graded extensions. Applying the standard prescription of conformal quantum mechanics, the spectrum of the SCQMs with the ℤ³₂-graded (1|2) symmetry is analyzed. It is shown that many models of SCQM can be extended to a ℤⁿ₂-graded setting. The authors would like to thank the referees for their valuable comments on the improvement of this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications ℤ³₂-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics Article published earlier |
| spellingShingle | ℤ³₂-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics Doi, Shunya Aizawa, Naruhiko |
| title | ℤ³₂-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics |
| title_full | ℤ³₂-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics |
| title_fullStr | ℤ³₂-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics |
| title_full_unstemmed | ℤ³₂-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics |
| title_short | ℤ³₂-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics |
| title_sort | ℤ³₂-graded extensions of lie superalgebras and superconformal quantum mechanics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211352 |
| work_keys_str_mv | AT doishunya z32gradedextensionsofliesuperalgebrasandsuperconformalquantummechanics AT aizawanaruhiko z32gradedextensionsofliesuperalgebrasandsuperconformalquantummechanics |