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N. Aizawa et al.
abelian group Z 2 and Lie algebras with this kind of grading are referred to as color
(super)algebras in the literatures. Soon after this work, Bruce and Duplij presented
a model of Z 2
2 -graded SQM based on the reduction of the Z 2
2 -graded super-Poincaré
algebra of [3] to (0 + 1)-dimension [4] (references for other works on extension of
SQM are found in [4], too). We call the model discussed in [4] Bruce–Duplij model.
In the present work, we interpret the Bruce–Duplij model from more general
perspective. Namely, we provide a realization of Z 2
2 -graded color superalgebra by
ordinary Lie superalgebra. Then it can be seen that Bruce–Duplij model is a special
case of this realization. Moreover, the realization allows us further generalizations
of SQM. The Bruce–Duplij model may be regarded as a Z 2
2 -graded version of N =
1 SQM. By using the realization, one may easily construct models of Z 2
2 -graded
version of SQM with higher values of N . It is also possible to include conformal
invariance, since one may apply the realization to many models of superconformal
mechanics (SCM).
We remark that color superalgebras attract some physical interests in connection
with symmetries of non-relativistic Dirac equation (Lévy-Leblond equation) and
parastatistics [5, 6]. The present work, as well as [4], provides a new example of
deep connection of such algebras and physics.
This paper is organized as follows: In the next section, we give a definition of
Z n
2 -graded color superalgebra and an algebraic basis of the Bruce–Duplij model. In
Sect. 3, it is shown that if a matrix Lie superalgebra satisfies a certain condition, then
one may obtain a Z 2
2 -graded color superalgebra with the same structure constants.
This result is used to extend the result in [4] to extended supersymmetry and
conformal supersymmetry. We also show in Sect. 4 that it is possible to construct
Z 3
2 -graded SQM from the ordinary SQM as we did for Z 2
2 -graded case.
2 Z n
2
-Graded Color Superalgebra and Z 2
2
-Graded SQM
We start with the definition of Z n
2 -graded color superalgebra. Let g be a vector space
over C or R which is a direct sum of 2 n subspaces labelled by an element of the
group Z n
2 :
g =
α∈Z n
2
g α .
(1)
Regarding an element α = (α 1 , α 2 , . . . , α n ) of Z n
2 as an n dimensional vector, we
define an inner product of two elements of Z n
2 by
α · β =
n
i=1
α i β i .
(2)
N. Aizawa et al.
abelian group Z 2 and Lie algebras with this kind of grading are referred to as color
(super)algebras in the literatures. Soon after this work, Bruce and Duplij presented
a model of Z 2
2 -graded SQM based on the reduction of the Z 2
2 -graded super-Poincaré
algebra of [3] to (0 + 1)-dimension [4] (references for other works on extension of
SQM are found in [4], too). We call the model discussed in [4] Bruce–Duplij model.
In the present work, we interpret the Bruce–Duplij model from more general
perspective. Namely, we provide a realization of Z 2
2 -graded color superalgebra by
ordinary Lie superalgebra. Then it can be seen that Bruce–Duplij model is a special
case of this realization. Moreover, the realization allows us further generalizations
of SQM. The Bruce–Duplij model may be regarded as a Z 2
2 -graded version of N =
1 SQM. By using the realization, one may easily construct models of Z 2
2 -graded
version of SQM with higher values of N . It is also possible to include conformal
invariance, since one may apply the realization to many models of superconformal
mechanics (SCM).
We remark that color superalgebras attract some physical interests in connection
with symmetries of non-relativistic Dirac equation (Lévy-Leblond equation) and
parastatistics [5, 6]. The present work, as well as [4], provides a new example of
deep connection of such algebras and physics.
This paper is organized as follows: In the next section, we give a definition of
Z n
2 -graded color superalgebra and an algebraic basis of the Bruce–Duplij model. In
Sect. 3, it is shown that if a matrix Lie superalgebra satisfies a certain condition, then
one may obtain a Z 2
2 -graded color superalgebra with the same structure constants.
This result is used to extend the result in [4] to extended supersymmetry and
conformal supersymmetry. We also show in Sect. 4 that it is possible to construct
Z 3
2 -graded SQM from the ordinary SQM as we did for Z 2
2 -graded case.
2 Z n
2
-Graded Color Superalgebra and Z 2
2
-Graded SQM
We start with the definition of Z n
2 -graded color superalgebra. Let g be a vector space
over C or R which is a direct sum of 2 n subspaces labelled by an element of the
group Z n
2 :
g =
α∈Z n
2
g α .
(1)
Regarding an element α = (α 1 , α 2 , . . . , α n ) of Z n
2 as an n dimensional vector, we
define an inner product of two elements of Z n
2 by
α · β =
n
i=1
α i β i .
(2)