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Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Brehova 7, 110 00 Prague, Czech Republic

- Corresponding Author:
- Goce Chadzitaskos

Department of Physics

Faculty of Nuclear Sciences and Physical Engineering

Czech Technical University, Brehova 7

110 00 Prague, Czech Republic

**E-mail:**[email protected]

**Received Date:** September 22, 2011; **Accepted Date:** January 04, 2012; **Published Date:** January 06, 2012

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

A theory of grading preserving contractions of representations of Lie algebras has been developed. In this theory, grading of the given Lie algebra is characterized by two sets of parameters satisfying a derived set of equations. Here we introduce a list of resulting 3-dimensional representations for the Z3-grading of the sl(2) Lie algebra.

Contraction, the concept named by Inönü and Wigner [3], is suitable for unification of theories, studying relations between symmetries, etc. The first papers about this concept were published in 1950’s by Segal [6]. The next papers parallel to Inönü and Wigner came from Saletan [5] and Doebner and Melsheimer [2]. In this paper, contractions of representations are studied.

The problem is the following: starting with a given Lie algebra characterized by its commutation relations, a question arises: which Lie algebras can be obtained using graded contractions? This is a natural way of obtaining Lie algebras, starting with a fixed one. Moreover, for n-dimensional representations of the initial Lie algebra, using another tool of the subject, namely the contraction of representations, we can produce new representations of Lie algebras. Using both concepts simultaneously, we get representations of Lie algebras which arise as the graded contractions.

The contribution of this paper is the investigation of the Z3-graded contraction of 3-dimensional representation of the complex Lie algebra sl(2). In Sections 2 and 3, graded contractions are defined and basic features of sl(2) are described. We study which Lie algebras can be obtained by contractions of sl(2). In Section 4, we introduce the definition of graded contractions of representations of Lie algebras. It is a tool for construction of new Lie algebras from a given Lie algebra. Section 5 describes all contractions of 3-dimensional representations of Z3-graded sl(2). For this purpose, results published in [1,4] will be applied.

**Graded contractions of Lie algebras**

Graded contractions of Lie algebras are a tool for the production of new Lie algebras from the original Lie algebra. In this section, we introduce the definition of this term.

**Definition 1.** Let n ∈ N and let g be a Lie algebra. A grading is the decomposition [1]

where the commutators satisfy

and the addition of indices is taken modulo n. Then the algebra g is Zn-graded.

Definition 2. The graded contraction of the given Lie algebra g is the Lie algebra g^{ε} which is isomorphic to the
given Lie algebra g as a linear space, and with commutation relations preserving the grading [gj ,gk] ⊆ gj+k:

(2.1)

The contraction parameters εjk should satisfy the equations

(2.2)

We define the contraction matrix ε as

From (2.1), it follows that εjk = εkj (i.e., the matrix is symmetric). Taking into account (2.2), we get in the case of Z3-graded contractions the equations

Then we have (up to equivalence of matrices corresponding to the change of bases of the subspaces g1 and g2) these possible forms of the contraction matrix [4]:

(2.3)

The Lie algebra sl(2) is one of the main kinds of the simple Lie algebras—sl(n+1) ≡ An. The dimension is given
by the number of its generators, and it is n^{2} +2n (i.e. the dimension of sl(2) is 3). We denote its generators h, e,f.
They satisfy the commutation relations

(3.1)

In the case of the Z3-grading, we have the decomposition

where

(3.2)

We apply the matrices from (2.3) on (2.1) and use the notation of (3.2), then we have only 4 distinct possibilities how the commutation relations (3.1) can be changed:

Starting with the definition of graded contractions of representations, we find different sets of equivalent contractions of Lie algebras and representations for the Z3-grading, and we ask which Lie algebras correspond to the representations given by the Z3-graded contractions of the 3-dimensional representations of the Lie algebra sl(2).

**Definition 3.** Let g be a Zp-graded Lie algebra, where p ∈ N, and let V be a g-module. Then assume that the action
of Zp-graded g on V respects the grading on

(4.1)

where Vj , j = 0, . . . , q−1, defines the corresponding Zq-grading of V and the addition of indices is modulo q. If one denotes the contracted action T(g) on V as

(4.2)

where ψ_{jm} are the contraction parameters, then

(4.3)

where |m>, m = 0, . . . , q−1, are the base vectors of the subspace V_{m}, ε and ε_{jk} denote the contraction of g and the
corresponding parameters. The relations (4.3) are definitely satisfied when

(4.4)

One uses the notation V^{ψ} for V considered as graded T(g^{ε})-module, where ψ satisfies (4.4). V^{ψ} is then called the
contraction of the representation V with respect to the contraction ε.

We write the contraction parameters into the matrix as

(4.5)

In the case of contractions of Z3-graded representations of Z3-graded Lie algebra, the system of equations (4.4) gives the following relations:

(4.6)

where m ∈ {0,1,2}. From this it follows that (4.6) leads to the system of 18 equations, where for the simplification we suppose that either ψjk =0 or ψjk =1 for arbitrary j,k. Now we denote the set of the right-hand sides of (4.6). We divide this set into six parts, each containing a triplet of numbers corresponding to a concrete set of equations in (4.6):

There is a possibility of the cyclic permutation of columns of the contraction matrix ψ and each of these permutations corresponds to the cyclic permutation of numbers in each of the mentioned triplets. Starting with a given contraction matrix ψ, we denoteψ’, respectively ψ“, the contractions matrices, where all the columns are shifted one position to right, respectively to left.

The possible forms of depend on a pair of contraction matrices (ε,ψ), where ε was assigned in (2.3) and both ε and ψ can be found in the list of possible contractions of representations published in [4] which is also written in the appendix of this paper. We can distinguish these 12 possibilities:

Now we investigate how the contractions in the previous list transform the 3-dimensional representations of sl(2). First, we study the cases which coincide with the representations of the Abelian algebra c. These cases immediately provide the solution, because the Abelian algebra cannot be decomposed into anything simpler. In Section 3, we see that the Abelian algebra c coincides with the contraction matrices εIV , εV I, and εV II. This means that we know the solution for the cases (6), (9), (10), (11), and (12) in the previous list.

In order to study the other cases, we start with the 3-dimensional representation of sl(2). All irreducible 3- dimensional representations of sl(2) are isomorphic with the representation T given by matrices

(4.7)

Now we consider the remaining cases using this representation T, the corresponding contraction matrices ψ from [4] introduced in the list of all possible contractions of 3-dimensional representations and the contraction matrices ψ’,ψ“ arising from ψ by the cyclic permutation of columns. According to the properties of commutation relations of the matrices corresponding to the resulting representations, we distinguish representations corresponding to Lie algebras e(2), c, l, and l’, where l’ and l differ in exchanging the commutation relations for e and f:

We explain the procedure on the first case, corresponding to

First, we have to find the grading of the g-module corresponding to (4.1). It appears that the only suitable grading has the form

(4.8)

Now we take the representations T from (4.7) and we multiply the columns of the particular matrices (from right to left in agreement with the grading (4.8)) by the elements of the rows of the matrix ψ: the first row in ψ corresponds to T(h), the second row corresponds to T(e), the third row corresponds to T(f). By this way, we get a new representation

its commutation relations correspond to the Lie algebra l. Next, we compose the matrices ψ’ and ψ“ which arise by the permutation of the columns of ψ and get

Now we repeat the same procedure as we did for ψ and get the matrices corresponding to representations of e(2) for both ψ’ andψ“.

The results for all the cases are listed in **Table 1**, where for each case in the previous list, the resulting representation
is written.

The presented results of the contractions of 3-dimensional representations of sl(2) are applicable in some mathematical areas.

One of them is the area of universal enveloping algebras. The problem, depicted in the following diagram, is
the following: we investigate if an adjoint representation ρ represented in a given universal enveloping algebra U(g)
associated with a Lie algebra g can be simplified by a contraction ψ to the corresponding adjoint representation ρε
represented in another universal enveloping algebra U(g^{ε}) associated with a Lie algebra g which is a contraction of
g. It can be shown that in the case g ≡ sl(2), this task is solvable for the 3-dimensional representations, but for the
higher dimensions it is unfeasible. The main problem consists in the need of the same base for the matrices of both
representations. The form of the matrices corresponding to the representation of the contracted g in the given base
is too difficult to be obtained with the help of the contraction.

The support by the Ministry of Education of Czech Republic (project MSM6840770039) is acknowledged as well as support by the Grant Agency of the Czech Technical University in Prague, Grant no. SGS11/132/OHK4/2T/14. The authors are grateful to the referees for constructive remarks which helped to improve the presentation.

- Montigny M, Patera J (1991) Discrete and continuous graded contractions of Lie algebras and superalgebras. J Phys A 24: 525–547.
- Doebner HD, Melsheimer O (1967) On a class of generalized group contractions. Nuovo Cimento A 49: 306–311.
- Inonu E, Wigner EP (1953) On the contraction of groups and their representations. Proc Nat Acad Sci USA 39: 510–524.
- Moody RV, Patera J (1991) Discrete and continuous graded contractions of representations of Lie algebras. J Phys A 24: 2227–2257.
- Saletan EJ (1961) Contraction of Lie groups. J Mathematical Phys 2: 1–21.
- Segal E (1951) A class of operator algebras which are determined by groups. Duke Math J 18: 221–265.

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