Contractions of Certain Lie Algebras in the Context of the DLF-Theory

Contractions of the Lie algebras d = u(2), f = u(1 ,1) to the oscillator Lie algebra l are realized via the adjoint action of SU(2,2) when d, l, f are viewed as subalgebras of su(2,2). Here D, L, F are the corresponding (four-dimensional) real Lie groups endowed with bi-invariant metrics of Lorentzian signature. Similar contractions of (seven-dimensional) isometry Lie algebras iso(D), iso(F) to iso(L) are determined. The group SU(2,2) acts on each of the D, L, F by conformal transformation which is a core feature of the DLF-theory. Also, d and f are contracted to T, S-abelian subalgebras, generating parallel translations, T, and proper conformal transformations, S (from the decomposition of su(2,2) as a graded algebra T + Ω + S, where Ω is the extended Lorentz Lie algebra of dimension 7).


Introduction
As noticed by the first author (see [1,2]), there are precisely three four-dimensional non-abelian Lie algebras that admit a non-degenerate invariant bilinear form of Lorentzian signature: the oscillator Lie algebra l, defined by the following commutation relations in a certain basis l 1 , l 2 , l 3 , l 4 : the Lie algebra d = u(2) generated by vectors X 0 , X 1 , X 2 , X 3 with the following commutation relations:

[ ] [ ] [ ]
the Lie algebra ( ) Relations (1.2) are the same as the ones in [3].Certain Lie groups (corresponding to the Lie algebras d, l, f), endowed with bi-invariant metrics of Lorentzian signature, provide so-called homogeneous solutions to Einstein's equations of General Relativity Theory, GRT.We denote these solutions by the corresponding capital letters.They have been studied in [4].In GRT literature (see [5]), D is known as the perfect fluid, F is a tachyonic fluid, and that L is a very special case of plane waves.
Namely, it is an isotropic electromagnetic field determined by a covariantly constant light-like vector (see [6]).Groups defined by the oscillator Lie algebra l are also studied in detail on pp.409-414 of [7].
Relations (1.1), (1.2), and (1.3) have been used in [8,9] where the basics of the DLF-approach have been presented (by A. L.).This DLF-theory can be briefly characterized as the LF-modification of Segal's Chronometry (the latter one is based on D, see [3,9] and references therein).
It is known (see [9]) that these three space-times are conformally flat, and that in each case the isometry group of the corresponding Lorentzian manifold is of dimension 7.Each of d, l, f (as well as a four-dimensional abelian Lie algebra) can be realized as a subalgebra of the conformal algebra ( ) and these imbeddings will be specified below.Conventions about ( ) where s is a diagonal matrix: .
Also, commutation relations relative to a certain basis in su(2,2) are (3.1) of our Section 3.
The general subject of contractions (and deformations) of Lie algebras is important in physics (see, for example, Section IV.7 of [10]).It provides an explanation of one physical model being a limiting case of another one.In [10], it was the case of Newtonian world versus Minkowski space-time M. The findings of the present article (together with what has been already published by the first author) form the necessary base for investigation of similar relationships between space-times M, D, L, F.
The subject is also of interest from a pure mathematical point of view.Regarding contractions of Lie algebras, we follow [11].

Lie-Theoretical Contractions of d and f
Namely, we use the name Lie-theoretical contraction for the method defined by [11, p.2, (3)], where the commutation relations of a contracted Lie algebra are given by [ ] ( ) ( ) ( ) Here .q p x δ = + When q is not equal to p, U q is a non-singular linear transformation of the original Lie algebra.When q = p, the inverse of the linear transformation U p does not exist.
Theorem 2.1.There is a Lie-theoretical contraction of ( ) to the oscillator Lie algebra l.Proof.Consider the following vectors in d: One can verify that for any non-zero value of α, the algebra generated by L 1 , L 2 , L 3 , L 4 , is isomorphic to d, since the Equations (2.1) are uniquely solvable for X i .
The commutation relations for L 1 , L 2 , L 3 , L 4 , are as follows: It can be easily seen that as α goes to zero, the commutation relations become (1.1), that is, of the oscillator Lie algebra l.
Remark 2.1.Alternatively, we can choose a different set of vectors in d: It defines a Lie algebra isomorphic to d (when β is not zero).The commutation relations converge to those of the oscillator algebra when β goes to zero.Theorem 2.2.There is a Lie-theoretical contraction of ( ) to the oscillator Lie algebra l.Proof.We apply similar contraction procedures to f as we did in the case of d.Let ( ) ( )( ) Then, as α goes to 0, the commutation relations for L 1 , L 2 , L 3 , L 4 , become (1.1), that is, of the oscillator algebra l.Clearly, for any nonzero α, the Lie algebra generated by L 1 , L 2 , L 3 , L 4 , is isomorphic to f. Remark 2.2.As noticed in [12], there always exists a (trivial) Lie-theoretical contraction of any Lie algebra to an abelian algebra.

Realization in su(2,2)
Our current goal is to realize the above-mentioned contractions of the Lie algebras d and f through the adjoint action of the group SU(2,2) on its Lie algebra su (2,2) of which all d, f and l are subalgebras.We will call such contractions the su(2,2)-inner contractions of the corresponding Lie algebras.
More specifically, we will conjugate the generators of d and f with elements of the maximal abelian subgroup A of ( )  [8,9]) that ( ) acts on each of D, L, F. Joint consideration of the three worlds is the key feature of the DLF-theory.
In [13, p.135], the generic element a of the two-dimensional maximal abelian subalgebra A of ( ) where s, t are real parameters.It thus can be written in terms of the generators L ij of ( ) 03 .s t All fifteen matrices L ij (where , 1, 0, , 4 i j = −  ) forming a (standard) basis of ( ) 2, 2 su can be found in [14].Relative to this basis, the The following imbedding of ( ) Set t = 0 in the above system and rewrite the equations as: ,0  ,0 , As s goes to infinity, the commutation relations become (1.1), which means that the outcome of the contraction is a subalgebra of ( ) , isomorphic to the oscillator algebra l.Namely, basic matrices ) 1 2 , of new subalgebra are the limits of the matrices 1 2 3 4 , , , .l l l l Alternatively, we can set s = 0 in system (3.3), and after a similar procedure, we get a limiting Lie algebra spanned by It is also isomorphic to the oscillator Lie algebra, since the commutation relations are (1.1).Finally, setting s = t in system (3.2), and repeating the procedure above, realizes a contraction of d to an abelian subalgebra of ( ) If we make s = t go to negative infinity, then the resulting abelian Lie algebra S is generated by This finishes the proof of Theorem 3.1.Remark 3.3.The (above) two subalgebras are known (in that order) as the Lie algebra of translations and the Lie algebra of "proper conformal transformations" of the Minkowski space-time: see [15], where su(2,2) is written as a graded algebra T + Ω + S with T, S being the two abelian algebras above, and Ω being the Lorentz Lie algebra extended by (infinitesimal) dilatations.The above choice of generators for T and for S has been made in Table V of [3].
In the remaining part of this section we arrange for similar procedures, as in the above, but for the case of ( ) We realize it as a ( ) This choice of the basis, with commutation relations (1.3), has been made in [8].Theorem 3.2.There exists a ( ) -inner contraction of this subalgebra f to a subalgebra of ( ) 2, 2 su isomorphic to the oscillator Lie algebra l.
Proof.Choose one other basis for f: Conjugating these vectors with ( ) , a s t as in the proof of Theorem 3.1, we get the following: If we set t = 0 and perform a contraction analogous to that for d, we get in the limit as s goes to infinity, a subalgebra of ( ) This subalgebra is isomorphic to the oscillator Lie algebra since the commutation relations are the same as (1.1).

Contractions of the Isometry Lie Algebras
As mentioned in the Introduction, the isometry groups of the Lorentzian manifolds, corresponding to the Lie algebras d, f, and l are of dimension 7.All three of these 7-dimensional Lie algebras, iso(D), iso(F), and iso(L) can be viewed as subalgebras of ( ) .Considerable part of observations in this section has been known before, and we indicate a few references (see below).However, bringing together different methods and applying them to each of our main objects of study (space-times D, L, F), make the content of the section to a new ingredient of the mathematical presentation of the DLF-theory.
According to [3], the following elements form a basis for ( ) The two other isometry Lie algebras will be described below.
Theorem 4.2.The following is true for the Lie algebra iso(F): Theorem 4.3.The centralizer of l 1 is a nine-dimensional Lie subalgebra of ( ) . More specifically, it is a two-dimensional nilpotent extension of iso(L).
Proof of the theorem 4.1.It is a straightforward exercise, based on (3.1), to verify that the second equality in the above (4.1)holds.Regarding the first equality, it immediately follows from (4.0) that one of the two ( ) , , , X X X and the other is generated by { } The center {R} (4.1) is generated by 0 L −10 from (4.0).
Rather than to present other details from [3], one can use results from [16] to show that the basis (4.0) determines the Lie algebra iso(D).These results are based on the notion of a symmetric quadruple (k, q, p, B), where k is a finite dimensional real Lie algebra, q is a subalgebra of k, p is a non-zero, q-invariant, complementary vector subspace to q in k.B is a non-degenerate, q-invariant symmetric bilinear form on p. Also, [p, p] = q, and q contains no nonzero ideals of k.A certain involutive automorphism h of k is instrumental since q is its λ = 1 eigenspace, whereas p is its λ = −1 eigenspace.
It is shown in [16] that a simply connected pseudo-Riemannian symmetric space determines (up to an isomorphism) a symmetric quadruple.Given a symmetric quadruple, there exists a corresponding simply connected pseudo-Riemannian symmetric space.In our case iso(D) = k is reductive.To finish this alternative proof of equality (4.1), it is enough to present the corresponding symmetric quadruple (a Lorentzian one, due to the signature ( ) , , , + − − − of the form B on p).To do so, introduce the following linear bijection h of the Lie algebra k: h is an identical transformation on the subalgebra q spanned by { } , and h is negative 1 on the four-dimensional vector space p generated by Clearly, q is isomorphic to su (2).
It is a straightforward exercise to verify that h is an involutive isomorphism of k.The form B on p is introduced as a "pull-back" of the following invariant Lorentzian form on ( ) : vectors X 0 , X 1 , X 2 , X 3 are orthonormal, their scalar squares being 1, −1, −1, −1, respectively.This finishes the proof of theorem 4.1.
Let us now apply that last approach in the proof of theorem 4.2.The center R in (4.2) is generated by L 34 .One of the ( ) Let us show that the seven-dimensional subalgebra h spanned by , , , , , , ,  H H H H H H is the Lie algebra of the (reductive) Lie group Iso(F).The latter group is a subgroup of ( ) . Introduce the following linear bijection t of the Lie algebra h: t is an identity transformation on the subalgebra q generated by ), and t is negative 1 on the four-dimensional vector space p generated by It is a straightforward exercise to verify that t is an involutive isomorphism of h.The form B on p is introduced as a "pull-back" of the following invariant Lorentzian form on ( ) orthonormal, their scalar squares being 1, −1, −1, −1, respectively.The corresponding Lorentzian quadruple is now (h,q,p,B).
Again, it is a straightforward exercise, based on (3.1), to verify that (4.2) holds, and that the above mentioned identity component of Iso(F) is the one of the block-diagonal subgroup of ( )
Proof of Theorem 4.3.It is known that the Lie algebra iso(L) is solvable and is of dimension 7.As a homogeneous symmetric Lorentzian manifold, L has been studied in [6].The first author got to know the oscillator Lie group L from [17].The L's important property to admit a non-degenerate bi-invariant metric has only been noticed in early 80s: [1,2,18].
In paragraph 3 of [16] solvable Lorentzian quadruples are described in detail.That description includes a vector space w which is Euclidean in our case (the metric on w is chosen as a negative definite one).Also, a symmetric bilinear form A on w is part of the description.It is stated in [16] that if (k, q, p, B, w, A) is a solvable quadruple associated with a Lorentzian symmetric space L, then the full isometry Lie algebra iso(L) is the canonical semi-direct product of k with the algebra of skew-symmetric linear maps of w which commute with A.
Here A is a linear operator associated with the form A. The operator A is defined by the formula ( ) ( ) The approach of [6] was based on a choice of four left-invariant vector fields on the oscillator Lie group L (their commutation relations are (1.1) from the Introduction).In (1.1), the vector field l 1 (which is also e 1 of (4.4) below) is both left-, and right-invariant.There are three more linearly independent right-invariant vector fields (they commute with left-invariant vector fields on L).Overall, in the approach of [6], we get the following table , can be chosen as follows (notice that this choice is different from the one in our Section 3): Starting with (4.4), introduce w with an orthonormal basis 5 3 2 6 , e e e e − + .Relative to this basis, the negative definite form .,. , to be of use below, is given by the diagonal matrix 0 .
Introduce w * with a reciprocal basis 5 3 6 2 , e e e e + − .This involves a choice of a linear bijection between w and w * , with w * being the image of w, etc. Choose the form A given by the matrix ,0 ,0 , , In the limit as s goes to infinity, the seven matrices form the following Lie subalgebra, isomorphic to iso(L):  We can contract

( )
h iso F = in, essentially, the same way we did the contraction of ( ).This last Lie algebra is (isomorphic to) iso(L).
The generic element of the corresponding group A can then be written as ( ) above.What we observe here is a special case of the contraction (2.1) with

1 .
As it follows from (1.1), (1.2), (1.3), (3.1),(3.4),and (3.5), the centers of d, f, and of l are of dimension one, and they are generated by L −10 , by L 34 , and by The following is true for the Lie algebra iso(D):

4 )
It determines the structure of the full isometry Lie algebra of the space-time in question.The seven generators,
to obtain a basis for iso(F), and extend the contraction, defined earlier for f, to this algebra.Namely, conjugate a t s introduced in Section 3. Setting t = 0 and letting s go to infinity, we get a Lie algebra generated the vectors ,