Research Article  Open Access
Invariant Operators of FiveDimensional Nonconjugate Subalgebras of the Lie Algebra of the Poincaré Group P(1,4)
Abstract
We have classified all fivedimensional nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4) into classes of isomorphic subalgebras. Using this classification, we have constructed invariant operators (generalized Casimir operators) for all fivedimensional nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4) and presented them in the explicit form.
1. Introduction
At present, there are many papers devoted to the methods for construction and various applications of invariant operators (generalized Casimir operators) of the Lie algebras to the theory of representations of the Lie groups (and their Lie algebras), theory of special functions, theoretical and mathematical physics, and the theory of differential equations. The details can be found in [1–27] and references therein.
The generalized Poincaré group P(1,4) is a group of rotations and translations of the fivedimensional Minkowski space M(1,4). This group is applied to solve various problems of theoretical and mathematical physics (see, e.g., [28–30]). Invariant operators of the Lie algebra of the Poincaré group P(1,4) have been constructed by Fushchich and Krivskiy [4, 5, 28, 31]. Those operators are used for the classification of the irreducible representations of the Lie algebra of the Poincaré group P(1,4) and for the construction of P(1,4)invariant differential equations.
The subgroup structure of the group P(1,4) has been studied in [32–36]. One of the nontrivial consequences of the description of the nonconjugate subalgebras of the Lie algebra of the group P(1,4) is that the Lie algebra of the group P(1,4) contains, as subalgebras, the Lie algebra of the Poincaré group P(1,3) and the Lie algebra of the extended Galilei group G(1,3) [37], that is, it naturally unites the Lie algebras of the symmetry groups of relativistic and nonrelativistic physics.
In [38, 39], invariant operators for some nonconjugate subalgebras of the Lie algebra of the group P(1,4) have been constructed. The description of invariant operators of eightdimensional nonconjugate subalgebras of the Lie algebra of the group P(1,4) can be found in [40].
Invariant operators for all nonconjugate subalgebras of dimension ≤4 of the Lie algebra of the group P(1,4) have been constructed in [41, 42].
The aim of the paper is to construct the invariant operators of all fivedimensional nonconjugate subalgebras of the Lie algebra of the group P(1,4).
The outline of this paper is as follows. In Section 2, we present the brief information about the methods for calculating invariant operators. In Section 3, we define the Lie algebra of the Poincaré group P(1,4). In Section 4, we present the results of the classification of all fivedimensional decomposable nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4) into isomorphism classes as well as their invariant operators. Section 5 is devoted to the presentation of the results of the classification of all fivedimensional indecomposable nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4) into isomorphism classes as well as their invariant operators. The conclusions and results are discussed in Section 6.
2. About Methods for Calculating Invariant Operators
A method for calculating invariant operators of Lie algebras goes back to the original work of Lie; it has been discussed in detail in a paper of Patera et al. [7]. The method consists in reducing this problem to that of solving a set of linear firstorder partial differential equations.
In the same work, the method has been applied for calculating invariant operators of all real Lie algebras of dimension less or equal five as well as real nilpotent algebras of dimension six. A short version of this method as well as the application for construction of invariant operators of the Lie algebra of the Poincaré group P(1,3) can be found in the paper by Patera et al. [8]. According to Patera et al. [7, 8] we also distinguish between Casimir operators (polynomials in the basic operators of the Lie algebra), rational invariants (rational functions of the basic operators of the Lie algebra), and general invariants (irrational and transcendental functions of the basic operators of the Lie algebra).
Recently, Boyko et al. [20] have proposed a new purely algebraic algorithm for computation of invariant operators (generalized Casimir operators) of Lie algebras. It uses the Cartan method of moving frames and the knowledge of the group of inner automorphisms of each Lie algebra. In particular, the algorithm has been applied to the computation of invariant operators for real lowdimensional Lie algebras, finitedimensional solvable Lie algebras restricted only by a required structure of the nilradical, the class of triangular algebras, the class of solvable triangular Lie algebras with one nilindependent diagonal element, solvable Lie algebras with triangular nilradicals, and diagonal nilindependent elements, and so forth. The details can be found in Boyko et al. [20–24]. The discussion of a purely algebraic algorithm for the computation of invariant operators of Lie algebras by means of moving frames as well as the extension of the exploitation of Cartan's method in the FelsOlver version can be found in the paper of Boyko et al. [25].
In order to construct invariant operators for fivedimensional nonconjugate subalgebras of the Lie algebra of the group P(1,4) we have done the following steps.(i)Based on the complete classification of real structures of Lie algebras of dimension ≤5 obtained by Mubarakzyanov in [43, 44], we classify all fivedimensional nonconjugate subalgebras of the Lie algebra of the group P(1,4) into classes of isomorphic subalgebras.
In order to select nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4) from the classification of fivedimensional Lie algebras provided by Mubarakzyanov we first choose any nonconjugate subalgebra of the Lie algebra of the group P(1,4) and investigate for which subalgebra from the Mubarakzyanov classification (or subalgebra from some Mubarakzyanov’s class) this subalgebra is isomorphic. In order to realize it we directly use the following theorem.
Theorem 1 (see [45]). If the structural constants of the Lie algebraare equal to the structural constants of the Lie algebracorrespondingly, then these Lie algebras are isomorphic. Inversely, if the Lie algebrasandare isomorphic, then in these algebras there exist such bases in which their structural constants will be equal, correspondingly.
Next, we choose any other subalgebra from the remaining nonconjugate subalgebras of the Lie algebra of the group P(1,4) and do with it the same, and so on. We do the same with all nonconjugate subalgebras of the Lie algebra of the group P(1,4). In the result we obtain all classes of isomorphic fivedimensional subalgebras of the Lie algebra of the group P(1,4).(ii)We use invariant operators for all real Lie algebras of dimension ≤5 constructed by Patera et al. [7] for the construction of invariant operators for all fivedimensional nonconjugate subalgebras of the Lie algebra of the group P(1,4).
In order to present the results obtained, we consider the Lie algebra of the group P(1,4).
3. The Lie Algebra of the Poincaré Group P(1,4)
The Lie algebra of the group P(1,4) is given by 15 basic elements,,and,that satisfy the commutation relations where,is the metric tensor with componentsandif. Here and below,.
We pass fromandto the following linear combinations:
Definition 2. We say that two subalgebras of the Lie algebrawhich are map to each other by the group of inner automorphisms of the Lie algebraare conjugate.
In order to describe nonconjugate subalgebras of the Lie algebra of the group P(1,4), we have used a method proposed by Patera et al. [46].
In the paper, we use the complete list of nonconjugate (up to P(1,4)conjugation) subalgebras of the Lie algebra of the group P(1,4) given in [47].
4. Invariant Operators of FiveDimensional Decomposable Nonconjugate Subalgebras of the Lie Algebra of the Poincaré Group P(1,4)
In the paper, the symboldenotes theth Lie algebra of dimensionandis a continuous parameter for the algebra. It should be indicated that the notationcorresponds to those used in the paper by Patera et al. [7]. In what follows, for the given specific Lie algebra, we write only nonzero commutation relations [7, 44].
Definition 3. We say that Lie algebra is decomposable if it is the direct sum of algebras of lower dimension.
Let us consider two Lie algebrasand.
Definition 4. We say that the Lie algebrais direct sum of Lie algebrasandif it consists of the vector spaceof the pairs,,, satisfying the commutation relation
We present results for fivedimensional decomposable nonconjugate subalgebras of the Lie algebra of the group P(1,4).
4.1. Lie Algebras of the Type
Consider.
Since the Lie algebras of the typeare Abelian, the invariant operators of these algebras are their basis elements.
4.2. Lie Algebras of the Type
The nonzero commutation relation for algebrahas the following form: The nonconjugate subalgebra of the typeof the Lie algebra of the group P(1,4) can be written as It is known that the invariant operators for Lie algebras of the typeare invariant operators of the subalgebrasand(see, e.g., Patera et al. [7]). The Lie algebras of the typedo not have invariant operators according to Patera et al. [7, 8]. Each Lie algebra of the typehas one invariant operator, which is its basis element. Therefore, the invariant operators for Lie algebra of the typeare basis elements of subalgebras,, and.
4.3. Lie Algebras of the Type
The nonzero commutation relation for algebrahas the following form: There exist nine fivedimensional nonconjugate subalgebras of the Lie algebra of the group P(1,4) which are isomorphic to algebra of the type. Two of them depend on parameters and hence constitute continua of subalgebras.
For all nonconjugate subalgebras invariant operators are Casimir operators.
The nonconjugate subalgebras of the typeof the Lie algebra of the group P(1,4) and their invariant operators are given in Table 1.

4.4. Lie Algebras of the Type
The nonzero commutation relations for algebrahave the following form: There exists only one class of fivedimensional nonconjugate subalgebras of the Lie algebra of the group P(1,4) which are isomorphic to subalgebra of the type.
Among invariant operators of nonconjugate subalgebras there are Casimir operators, and general invariants.
The nonconjugate subalgebras of the typeof the Lie algebra of the group P(1,4) and their invariant operators are given in Table 2.

4.5. Lie Algebras of the Type
The nonzero commutation relations for algebrahave the following form: There exists only one fivedimensional nonconjugate subalgebra of the Lie algebra of the group P(1,4) which is isomorphic to subalgebra of the type.
Among invariant operators of nonconjugate subalgebra there are Casimir operators and rational invariant.
The nonconjugate subalgebra of the typeof the Lie algebra of the group P(1,4) and its invariant operators are given in Table 3.

4.6. Lie Algebras of the Type
The nonzero commutation relations for algebrahave the following form: There exist three fivedimensional nonconjugate subalgebras of the Lie algebra of the group P(1,4) which are isomorphic to subalgebra of the type. One of them depends on parameters and hence constitute continua of subalgebras.
For all nonconjugate subalgebras invariant operators are Casimir operators.
The nonconjugate subalgebras of the typeof the Lie algebra of the group P(1,4) and their invariant operators are given in Table 4.

4.7. Lie Algebras of the Type
The nonzero commutation relations for algebrahave the following form: There exist thirteen fivedimensional nonconjugate subalgebras of the Lie algebra of the group P(1,4) which are isomorphic to subalgebra of the type. Three of them depend on parameters and hence constitute continua of subalgebras.
For all nonconjugate subalgebras invariant operators are Casimir operators.
The nonconjugate subalgebras of the typeof the Lie algebra of the group P(1,4) and their invariant operators are given in Table 5.

4.8. Lie Algebras of the Type
The nonzero commutation relations for algebrahave the following form: The nonzero commutation relations for algebrahave the following form: There exist five fivedimensional nonconjugate subalgebras of the Lie algebra of the group P(1,4) which are isomorphic to subalgebra of the type. Three of them depend on parameters and hence constitute continua of subalgebras.
For all nonconjugate subalgebras invariant operators are Casimir operators. In this case all nonconjugate subalgebras have the same invariant operator.
The nonconjugate subalgebras of the typeof the Lie algebra of the group P(1,4) and their invariant operators are given in Table 6.

4.9. Lie Algebras of the Type
The nonzero commutation relations for algebrahave the following form: There exists only one fivedimensional nonconjugate subalgebra of the Lie algebra of the group P(1,4) which is isomorphic to subalgebra of the type.
For this nonconjugate subalgebra invariant operators are Casimir operators.
The nonconjugate subalgebra of the typeof the Lie algebra of the group P(1,4) and its invariant operators are given in Table 7.

4.10. Lie Algebras of the Type
The nonzero commutation relations for algebrahave the following form: There exist only two fivedimensional nonconjugate subalgebras of the Lie algebra of the group P(1,4) which are isomorphic to subalgebra of the type.
For all nonconjugate subalgebras invariant operators are Casimir operators.
The nonconjugate subalgebras of the typeof the Lie algebra of the group P(1,4) and their invariant operators are given in Table 8.

4.11. Lie Algebras of the Type
The nonzero commutation relations for algebrahave the following form: The nonzero commutation relations for algebrahave the following form: There exist only one fivedimensional nonconjugate subalgebra of the Lie algebra of the group P(1,4) which is isomorphic to subalgebra of the type.
For this nonconjugate subalgebra invariant operator is Casimir operator.
The nonconjugate subalgebra of the typeof the Lie algebra of the group P(1,4) and its invariant operator are given in Table 9.

4.12. Lie Algebras of the Type
The nonzero commutation relations for algebrahave the following form: There exist five fivedimensional nonconjugate subalgebras of the Lie algebra of the group P(1,4) which are isomorphic to subalgebra of the type. One of them depends on parameters and hence constitute continua of subalgebras.
For all nonconjugate subalgebras invariant operators are Casimir operators.
The nonconjugate subalgebras of the typeof the Lie algebra of the group P(1,4) and their invariant operators are given in Table 10.

4.13. Lie Algebras of the Type
The nonzero commutation relations for algebrahave the following form: There exists only one class of fivedimensional nonconjugate subalgebras of the Lie algebra of the group P(1,4) which are isomorphic to subalgebras of the type.
Among invariant operators of nonconjugate subalgebras there are Casimir operator, rational invariant and general invariants.
The nonconjugate subalgebras of the typeof the Lie algebra of the group P(1,4) and their invariant operators are given in Table 11.

4.14. Lie Algebras of the Type
The nonzero commutation relations for algebrahave the following form: There exists only one fivedimensional nonconjugate subalgebra of the Lie algebra of the group P(1,4) which is isomorphic to subalgebra of the type.
Among invariant operators of nonconjugate subalgebra there are Casimir operator and rational invariants.
The nonconjugate subalgebra of the typeof the Lie algebra of the group P(1,4) and its invariant operators are given in Table 12.

4.15. Lie Algebras of the Type
The nonzero commutation relations for algebrahave the following form: There exist only two classes of fivedimensional nonconjugate subalgebras of the Lie algebra of the group P(1,4) which are isomorphic to subalgebras of the type.
Among invariant operators of nonconjugate subalgebras there are Casimir operators, rational invariant and general invariants.
The nonconjugate subalgebras of the typeof the Lie algebra of the group P(1,4) and their invariant operators are given in Table 13.

4.16. Lie Algebras of the Type
The nonzero commutation relations for algebrahave the following form: There exist five fivedimensional nonconjugate subalgebras of the Lie algebra of the group P(1,4) which are isomorphic to algebra of the type. Three of them depend on parameters and hence constitute continua of subalgebras.
For all nonconjugate subalgebras invariant operators are Casimir operators.
The nonconjugate subalgebras of the typeof the Lie algebra of the group P(1,4) and their invariant operators are given in Table 14.

4.17. Lie Algebras of the Type
The nonzero commutation relations for algebrahave the following form: There exist two fivedimensional nonconjugate subalgebras of the Lie algebra of the group P(1,4) which are isomorphic to subalgebra of the type. One of them depends on parameters and hence constitute continua of subalgebras.
For all nonconjugate subalgebras invariant operators are Casimir operators.
The nonconjugate subalgebras of the typeof the Lie algebra of the group P(1,4) and their invariant operators are given in Table 15.
