STRUCTURE OF CONTRACTED LIE ALGEBRASBy

Bull. Kyushu Inst. Tech.

(M. &N. S.) No. 25, 1978, pp. 1-7

STRUCTURE OF CONTRACTED LIE ALGEBRAS

By

Fumitake MiMuRA and Akira IKusHiMA

(Received Sept. 5, 1977)

1. Introduction

Classical mechanics is a limiting case of relativistic mechanics. Hence a problem arises how the Lie groups or its Lie algebras in the former are limiting cases of those in the latter. A significance of contraction is to give a group theoretical or an algebraic mean- ing to the relations between classical and relativistic mechanics under a limiting procedure.

E. in6nU and E. P. Wigner [1] first undertook such a limiting problem and dealt with the method of contraction (in6nU-Wigner contraction). On a generalization of In6nU- Wigner contraction, E. Saletan [2] discussed the underlying mathematical structure of contraction (Saletan contraction). Succeeded to his method, M. Levy-Nahas [3] in- troduced the more singular contraction (Levy-Nahas contraction).

This paper is concerned with a Lie algebraic structure of Saletan and Levy-Nahas contractions. So in 2, on a brief review of their works, a generalization is made of the result by R. Gilmore [4], in which there is no discussion for Levy-Nahas contraction.

In 3, there js given some algebraic properties of contracted Lie algebras such as nilpotency or solvability. Finally in 4, the contracted Lie products are rewritten, there the decom- position of underlying vector space of Lie algebra does not appear directly on the products.

This follows from a simplification of contracted Lie prcducts in the previous paper [5].

2. Method of contraction

For a given Lie algebra S over the underlying vector space S, the discussion begins with a family {U,} of linear transformations on S, nonsingular for t40 and singular for t =O, of the following form:

U,=:tnU+tn+iE (n ==O, 1, 2,...),

where U is a singular matrix and E is the unit matrix. The contraction S("+i) is then defined, if exist, by the limiting products:

[a, b](n"i)=:lim U;i[U,a, U,b] .

t.o

This limiting products exist if and only if

Un[Ua, Ub],,=Un"([Ua, b].+[a, Ub]rv --- U[a, b]N), (1)

and in this case the limiting products are as follows for n=O (Saletan contraction: [2, pp. 3-4])

[a, b](i) = U-'[Ua, Ub].+[Ua, b]N+[a, Ub]N-U[a, b]N, (2)

and as follows for n ). 1 (Levy-Nahas contraction: [3, pp. 1218-1219])

[a, b](n+i)==(-1)n-iUn-'{[Ua, Ub].-U([Ua, b]N+[a, Ub]N-U[a, b]N)}• (3) Here the subscription R and N of Lie products denote respectively the components of U-invariant subspaces SR and Siv of S defined in a way that U is nonsingular on SR and njlpotent on SN that is UMS== SR and UMSN=O where m=dim S.

Under this situation we can state the following theorem which is a generalization of the result by R. Gilmore [4, pp. 466-467]. His result is in the case: n=O (Saletan contraction).

THEoREM 1. The contraction S("'i) (nl.l:O) exists ifand only if

Un+r+s[a, b]N-Un+r[a, Usb]N== Un+s[Ura, b]N-Un[Ura, Usb]N, (4)

where the integers are rl.lr 1 and sl.lt 1.

PRooF. The following method is due to R. Gilmore, The equation (1) is written as Un+2[a, b].-- Un+i[Ua, b],, == Un'i[a, Ub]N- U"[Ua, Ub]iv,

so, operating the powers Um-2 (m ). 2) on both sides, the equation is obtained :

Un+m[a, b]N-Un+m-i[Ua, b]N=Un+m-i[a, Ub]N-Un+m-2[Ua, Ub]N.

From which, by putting m= r+s (r+sl2), it follows that

Un+r+s[a, b]N-Un+r+s-i[Ua, b]N == Un+r+s-t[a, Ub]N-Un+r+S-2[Ua, Ub]iv, and moreover, by putting m=:r+s-1 (r+sl3) and interchanging the element b by Ub,

it follows that

Un+r+s-1[a, Ub]N-Un+r+s-2[Ua, Ub]N=:Un+r+s-2[a, U2b]N-Un+r+s"3[Ua, U2b]N.

Proceeding in this way, it follows inductively that

un+r+s[a, b]N-- Un+r+s-1[Ua, b]N=Un+r+s-2[a, U2b]lv-Un+r+s-3[Ua, U2b]lv i

=Un+r[a, Usb]N-- Un+rml[Ua, Usb]N.

Again, this leads inductively to

Structure of Contracted Lie Algebras 3

Un+r+s[a, b]N- Un+r[a, Usb]lv= Un+r+s-1[Ua, b]N-- Un+r-1[Ua, Usb]N =Un+s[Ura, b]N-- Un[Ura, USb]N.

Hence (1) proceeds to (4), Conversely, (l) is obtained by putting r=s=l in (4). There- fore the proof is completed.

REMARK. For sufficiently large r, since U' annihilates SN, the equation js obtained by putting s =1 in (4):

un+i[Ura, b]iv= Un[Ura, Ub]N.

From which, interchanging a by (U-')'a if a is in SR, it follows that

Un+i[a, b].== Un[a, Ub]N. (5)

Hence, ifa is in SR, equation (1) and (5) are equivalent. This was given in [5, p, 9] fora sjmplification of the limiting products.

3. Structure of (S(n+i)

First we shall consider the Saletan contraction (!i(i). In this case, when (5(i) is a

contraction of S, U becomes a Lie algebra homomorphism from S(i) into S [3, p. 1218],

i. e.

U[a, b](')== [Ua, Ub] .

So that the kernel of the homomorphism U is an ideal in S('), Let denote this ideal by S•

THEoREM 2. The ideal S.} in (S(t) is nilpotent.

PRooF. First observe that, if b6Si)==kerU that is Ub==O, the equation (4) yields by putting n=O and s== 1:

U[U'a, b].== U"'[a, b]N (6)

Let a, be SO ii! S5. Then [a, b] (i) == - U[a, b]N [see (2)], and so, since Sb is an ideal in S(`), it follows that

S5i i!i [SpO, S5](i)c US. n S5.

Let aiGS5i, beS5. Then, since ai is written as ai=Ua where Uai=:O, the equation is valid [see (2)]:[ai,b](i)=--U[Ua,b]N. From which, since (6) yields by putting r=:1: U[Ua, b]N= U2[a, b]iv, it follows that

s)2.[Si, .S](i)c U2SN fl S..S.

Proceeding in this way, with a help of equation (6), step by step: r=2, 3,..., it follows that ssr=- [S5r-i, S5](i)c UrSN n Si}.

This completes the proof, since U is nilpotent on SN.

Ifa Lie algebra is nilpotent, then it is always solvable. Hence, the ideal S in S(') is solvable. This was first given by E. Saletan [2, p. 6; 4, p. 468].

Recall now that U is a Lie algebra homomorphism from S(') into S. So that, since the kernel S of U is a solvable ideal, the fundamental property for solvable Lie algebra [6, p. 24] yields the following theorem.

THEoREM 3. The contraction S(i) is solvable if and only if US(') is sotvable in S.

PRooF. Let S(i) be solvable. Then US(i) is solvable, since it is a homomorphic image of solvable Lie algebra S(i), Conversely let U(S(`) be solvable. Then the quotient algebra S(i)IS. by the ideal S=ker U is solvable, since it is isomorphic to US(').

Therefore, since S is solvable, S-(i) is solvable. This completes the proof.

Since US(i) is a subalgebra in S [2, p. 5; 4, p. 467], it is solvable if S is solvable.

This gives the following theorem.

THEoREM 4. The contraction S(i) is solvable ilfS is solvable.

Next we shall consider the Levy-Nahas contraction S("'i)(nll). In this case the circumstance differs from the Saletan contraction, that is, U annihilates the products [a, b](n") [3, p. 1219], i.e.

U[a, b](n+i)=O.

This means that [a, b]("+`) belongs to the kernel S of U. So particularly that S is an ideal in S(n+D.

TEoREM 5. The contraction S(n'i) (n l.1 1) is nilpotent.

PRooF. First observe that equation (4) yields by putting s== 1 :

Un+i[Ura, b]N-- Un[Ura, Ub]N = Un+r+i[a, b]N-- Un+r[a, Ub]N, (7) which is used jn the following inductive method similarly as in the proof of Theorem 2.

Let a,bESO!!S. Then [a, b](n"') belongs to Un-iSN [see (3)]. From which, since

[a, b](n'i) belongs to the kernel S of U, it follows that

Structure of Contracted Lie Algebras 5

sl Ei [sO, bi](n+1) c Un-ISN n Si).

Let aieSi, beS. Then, since ai is written as ai= U"-ia where Uai=O, the equation

is valid [see (3)]:

[a,, b](n+i) =(-- 1)n-i(Un+i[Un-ia, b]iv-Un[Un-ia, Ub]N) . From which, since (7) yields by putting r :n-1:

Un+i[Un-ia, b].-- Un[Un-ia, Ub]iv == U2n[a, b]N-U2n-i[a, Ub]N, (8)

it follows that

S2 lii [Sl, S](n+1)c U2n-ISN n S).

Therefore by using of (8), step by step: r =2n-1, 3n-- 1,..., it follows that Sr iEi [Sr-1, S](n+1) c Urn-ISN n S.

This completes the proof, since U is nilpotent on SN.

4. Lie products of S(n+i)

First of all, we shall keep in mind that, in the limiting products [a, b](n'i) (n).O),

U can be replaced by Vsuch as Vis identity on SR and equal to U on SN [5, pp. 11-12].

Indeed, a map F, such as F is equal to U on SR and identity on SN, is an isomorphism from the original products to the replaced one. This replacement yields the following theorem.

THEoREM 6. The contraction S(') exists if and only if V[a, b]V)==[Va, Vb], and S(i) is isomorphic to the algebra defined by [a, b]e'):

[a, b]e" =[Va, Vb]+(E-V)([Va, b]+[a, Vb]-V[a, b]).

PRooF. The -underlying vector space S decomposes into a direct sum of subspaces SR and SN. So from V[a, b]V)=[Va, Vb], since Vis identity on SR, it follows that

[Va, Vb].=V[Va, Vb].+ V(E- V)([Va, b].+[a, Vb]N-V[a, b]N) .

Which leads inductively to

[Va, Vb]iv -- V([Va, b]N+[a, Vb]iv-V[a, b]iv) == V[Va, Vb] -- V2([Va, b]N+ [a, Vb]N --- V[a, b]N) l

,.= vr[Va, Vb] - Vr'i([Va, b]N+ [a, Vb]N -- V[a, b]N) •

So that, since Vis nilpotent on SN, the equation is valid :

[Va, Vb].== V([Va, h].+[a, Vb].-V[a, b],v), (9)

Hence the contraction S(') exists [see (1)]. In this case, by using of (9), the products [a, bW') are rewritten as

[a, b]V) = [Va, Vb].+[Va, Vb].+(E- V)([Va, b].+[a, Vb]N-V[a, b]N) =[Va, Vb].+[Va, b]iv+[a, Vb]N-V[a, b]N,

which completes the proof.

By means of a simjlar method, the following theorem can be proved.

THEoREM 7. The contraction (!5("'i) (ni.lrl) exists if and only if V[a, b]Vn'i)=O, and S("") is isomorphic to the algebra defined by [a, b]t"'i):

[a, b] Vn+i) =( --- 1)n-i{(Vnm i - Vn) [Va, Vb] -- (Vn - Vn+ i) ([ Va, b]

+ [a, Vb] - Y[a, b])} .

REMARK. It is an interesting fact that the products [a, b]V') is obtained by putting n=Oin [a, b]V"'i) and regarding V-' (not exist on SN) as zero. In terms of [a, b]Vi), the products [a, b]e"'i) are expressed as [see 3, p. 1219]

[a, b] en+i) =(- 1)n-i Vn'i([ Va, Vb] - V[a, b] ei )) , and hence V[a, b]f"")=O is equivalent to

Vn+i[a, b]V) == V"[Va, Vb] .

In which, V[a, b]Vi)==[Va, Vb] is obtained by putting n=O.

Acknowledgement

One of the authors (F. Mimura) would like to express his deep thanks to professor T. N6no for his constant guides and encouragements in the course of the work.

References

[1] E. IN6NO and E. P. WiGNER, On the contraction of groups and their representations, Proc. Nat.

Akad. Sci. 39 (1953), 511-542.

[2] E. SALETAN, Contraction ofLiegroups, J, Mathematical Phys.2(1961), 1-21.

[3] M. LEvy-NAHAs, Deformation and contraction ofLie algebras, J. Mathematical Phys. 8 (1967), 1211-1222.

[4] R. GiLMoRE, Lie groups, Lie algebras, and some of their applications, John Wiley & Sons, New

York, 1974.

Structure of Contracted Lie Algebras 7

[5] F, MiMuRA, Contractions ofLie algebras, Bull, Kyushu Inst, Tech. Math. Natur, Sci. 19 (1972), 1-14.

[6] N.JAcoBsoN, Liealgebras, Interscience,NewYork,1962.

Department of Mathematics Kyushu Institute of Technology and

Kyushu Junior Cotlege

of Science and Engineering