Contractions of Lie algebras: Generalized Inönü–Wigner contractions versus graded contractions

Contractions of Lie algebras: Generalized Inönü–Wigner contractionsversus graded contractionsEvelyn WeimarWoods Citation: J. Math. Phys. 36, 4519 (1995); doi: 10.1063/1.530905 View online: http://dx.doi.org/10.1063/1.530905 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v36/i8 Published by the AIP Publishing LLC. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

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Contractions of Lie algebras: Generalized In6nikWigner contractions versus graded contractions

Evelyn Weimar-Woods Freie Universitiit Berlin, Fachbereich Mathematik und Informatik, I. Mathematisches Institut, A&m&lee 2-6, D-14195 Berlin, Germany

(Received 29 April 1994; accepted for publication 12 May 1995)

Recently, a new notion of contraction has been introduced, the purely algebraically defined graded contractions that fall into two disjoint classes: continuous and discrete. Our main result is that all continuous graded contractions can be realized by generalized Inonii-Wigner contractions. Furthermore, we give a complete charac- terization of the discrete graded contractions via certain higher-order identities inherent to the grading group. 0 I995 American Institute of Physics.

CONTENTS I. INTRODUCTION ........................................................ 4519 II. TWO CONCEPTS FOR CONTRACTIONS OF LIE ALGEBRAS. ............... 4520

A. Generalized In%i-Wigner contractions. .................................... 4520 B. 4521Gradedcontractions ................................................. 4521

III. GAMMA MATRICES .................................................... 4523 IV. GAMMA MATRICES WITHOUT ZEROS. .................................. 4527 V. GAMMA MATRICES WITH ZEROS. ...................................... 4530

A. 4531Higher-order identities. .............................................. 4530 B. 4535Minimal sets of zeros. ............................................... 4531

VI. MAIN THEOREMS FOR GAMMA MATRICES WITH ZEROS. ................. 4532 VII. CONSTRUCTION OF GAMMA MATRICES WITHOUT ZEROS THAT ARE

RELATED TO GAMMA MATRICES WITH ZEROS. .......................... 4535 APPENDIX A: HIGHER-ORDER IDENTITIES. ................................. 4538 APPENDIX B: COMPLEX G-GRADED CONTRACTIONS OF COMPLEX LIE ALGEBRAS ............................................................... 4542 APPENDIX C: REAL G-GRADED CONTRACTIONS OF REAL LIE ALGEBRAS ... 4543 APPENDIX D: EXAMPLE ................................................... 4544 REFERENCES ............................................................. 4548

I. INTRODUCTION

In this paper Lie algebra means a complex Lie algebra (except for Appendix C). Segal’ introduced the idea of a contraction of Lie algebras in 1951. When two physical

theories (like relativistic and classical mechanics) are linked by a limiting process (the velocity of light going to infinity), then there should exist a corresponding limiting process between the two underlying invariance groups (the Poincare and Galilean groups). Inonii and Wigne? introduced in 1953 what is now called a simple InGnu-Wigner contraction for Lie algebras. Saletan gave a general definition of contractions of Lie algebras.

Generalized Inonii-Wigner contractions were introduced4 in order to deal better with appli- cations such as the contraction of representations of Lie algebras.5 They seem to be a very natural class, and it is an open question whether every contraction is equivalent to a generalized Inonii- Wigner contraction.4 A positive answer to this question would be nice from the viewpoint of physics, since the generalized Irkktii-Wigner contractions are easy to interpret and to work with. See Sec. II A for a discussion of this (analytical) approach.

Q 1995 American Institute of Physics 4519


4520 Evelyn Weimar-Woods: Contractions of Lie algebras

In 1991, the purely algebraic notion of graded contractions6’7 appeared. The structure constants of a graded Lie algebra are multiplied by complex constants, which are chosen to give again a Lie algebra with the same grading. These graded contractions fall into two disjoint classes: continuous and discrete. See Sec. II B for details.

It seems unlikely that the discrete graded contractions will play any role in physical applica- tions. (Indeed they are not contractions in the original sense, and it is therefore questionable whether they should even be called contractions.) Our main result is that a continuous graded contraction is equivalent to a generalized In&ii-Wigner contraction. (This result is not entirely obvious. A generalized Inonii-Wigner contraction for a Z&-graded Lie algebra would be expected to have N parameters, whereas a graded contraction is described by N2 constants. However, our results show that, in the continuous case, at most N of them are independent). We also introduce the notion of “higher-order identities” (see Sec. V A) and use them to provide a complete criterion for a graded contraction to be discrete.

We use concepts from linear spaces like linear independence, basis, and basis expansion, which we reformulate for the nonlinear problem at hand. These tools are fully effective only for graded contractions without zeros. But they allow us to construct graded contractions without zeros, which are closely linked to a given contraction with zeros (see Sec. VII).

We prove our results first for graded contractions with non-negative entries, which we call Gamma matrices. From a physical point of view, these are the most interesting. We show later in Appendix B that every complex graded contraction is equivalent to a Gamma matrix. The proofs are carried out for the grading group Z, only, but they can be extended in a straightforward way to any finite Abelian grading group. Appendix C contains a brief account of the situation for real Lie algebras, which is somewhat more complicated. In particular, the main theorem holds only for Z, with N= 2 or N odd. The paper is organized in the following way. Section II contains the basic known results. Section III contains some results about Gamma matrices, including the result that every Gamma matrix without zeros is trivial because it is equivalent to the identity contraction. Section IV contains more detailed results about Gamma matrices without zeros, which are needed for our results about Gamma matrices with zeros. In particular, the concept of a basis is introduced. In Sec. V we deal with Gamma matrices with zeros. We show that the concept of a basis breaks down, and we introduce the notions of “higher-order identities” and “minimal sets of zeros.” In Sec. VI the main results are stated. Section VII contains some technical theorems needed in Sec. VI. The concept of “null dimension” is introduced here. In Appendix A we deal with the higher-order identities. All higher-order identities for ZN, Nc 8 are given. In Appendix B we show that all results for non-negative graded contractions hold for complex graded contractions. In Appendix C we deal with real Lie algebras. In Appendix D we show how the general theory works in an illustrative example.

II. TWO CONCEPTS FOR CONTRACTIONS OF LIE ALGEBRAS

Let V be a complex, finite-, or infinite-dimensional vector space. Let L = (V,p) be a Lie algebra with Lie multiplication ,u: VX V + V. In both notions of a contraction discussed here, one leaves V alone and changes the Lie multiplication according to certain rules, which results in a contracted Lie algebra L’=. (V,p’) which is, in general, more Abelian than L. The interesting contractions will lie between the two trivial ones L’=L, i.e., p’=p, and L’ Abelian, i.e., p’=O.

A. Generalized In&ii-Wigner contractions

The analytic concept of contractions can be described by a continuous family of homomorphisms,‘*4

U(E):V-+V, EE[O,l], U(l)=l,

which are nonsingular for 00 and singular for E=O.

(2.1)

J. Math. Phys., Vol. 36, No. 8, August 1995


Evelyn Weimar-Woods: Contractions of Lie algebras 4521

The new Lie bracket on V,

~~(a,b)=U-‘(E)~(U(E)u,U(E)b), U,bEV, E>O, (2.2)

corresponds to a change of basis given by U(E), and leads to the Lie algebra L,= (V,p.,) isomorphic to L. If

. ~‘(a&)= limpu,(u,b) (2.3) E+O

exists for all u, b E V, we call L’ = (V,p’) the contraction of L by U(E). L’ is, in general, not isomorphic to L.

For a generalized Inonii-Wigner contraction,4 U(E) has a simple form, namely

v= : v(j), M>l, j=O

(2.4)

with

OGno<n~<-**<nM, TzjEW.

The necessary and sufficient condition for Eq. (2.4) to define a contraction is

with

The contracted Lie algebra L’ = ( V,,u ‘) is given by

with

(2.5)

(2.6)

where all surviving structure constants are the same as for L. The special case no= 0 and M= 1 is called a simple Inonii-Wigner contraction2 [A contraction is a very special case of a Lie algebra deformation (although from some

viewpoints, they are opposite notions). However, the present theory of deformations is not relevant for the natural questions one encounters with contractions. See, e.g., Onishchik and Vinberg.8]

B. Graded contractions

V is again the direct sum of subspaces,

but this time the index j runs through a finite Abelian group G. To simplify the proofs, we restrict ourselves to




G=Z,,,={O,1,2...iV- l}, N~N={1,2,3 ,... }, (2.7)

the additive group of Z modulo Iv’. The generalization to arbitrary finite Abelian groups (which are known to be direct sums of primary cyclic groups’) is straightforward. Let L = ( V, p) be graded by G, i.e.,

(2.8) .

[This is, in general, a stronger restriction than Eq. (2.5), where the Lie product can lie in more than one subspace.]

A G-graded contraction of L is defined by de Montigny, Patera, Moody et al.‘,’ according to

~‘(Vj,Vk)=Yjk~U(Vj,Vk)CYjkVj+k, j&EC, (2.9)

where the complex matrix

is symmetric,

Yjk=Ykj, j,kEG, (2.10)

and satisfies

YjkYi,jik= YjlYk,j+l= YklYj,ktl* .iJdE G. (2.11)

This is the so-called generic case, where ,u’ is a Lie multiplication for all L graded by G. Example I: The special change of basis,

Vj+ajVj, O#ajE~=, jEG, (2.12)

produces a Lie algebra L’ = (V,p’) isomorphic to L, which corresponds to

uj”k Yjk’-

aj+k’ j,kEG. (2.13)

Equations (10) and (11) are trivially satisfied. [In this case, the 2-cocycle y is a coboundary.7] Definition I: If Eq. (2.13) holds, we will say

“y has the a form.”

Remark I: The elementwise product of two G-graded contractions y1 and y2, i.e.,

(2.14)

is again a G-graded contraction. This property will play a crucial role in Sets. V, VI, and VII. If two G-graded contractions only differ by a special change of basis, we call them

equivalent.7 Dejinition 2: Two G-graded contractions y and y’ ure called equivalent ifthere exist O#ajEC,

j E G, such that

I -'i"k Yjk-- Yjk 7

ajtk j,k E G. (2.15)

Remark 2: A G-graded contraction having the a form, i.e.,




uj"k Yjk=-

uj+k’ O#ajEC, j,kEG,

is equivalent to the (trivial) identity contraction yik = 1. Dejinition 3: A G-graded contraction y, yjk E c, is called continuous if there exists a continu-

ous family of G-graded contractions y(6), O<&l, with

Yjk(l)=l, o#Yjk(E)EC, E>O,

such that

Yjk= lim Yjk( 6). C+O

Otherwise y is called discrete. Remark 3: To prove y is continuous, it is enough to show that y is equivalent to a continuous

graded contraction, since we can always define the desired family Yjk( E), such that

yjk,f f ! lim Yjk( e) = lim Yjk( E) . uj+k 6-0 e-0

A first condition for continuity is easily found. Theorem 1: If y is continuous, then

YOO’YOk, kEG.

Proof: Equations (2.10) and (2.11) yield, for j = I= 0,

i?k = 3/00 YOk *

Therefore, every y without zeros must satisfy yao= yak, k E G. The theorem now follows easily.

III. GAMMA MATRICES

Now we restrict ourselves-without loss of generality (see Appendix B)-to G-graded contractions y, with

yjk>o, (3.0

and (because of Theorem 2.1) with

YOO=7/Okr kE.G.

We prove for these so-called “Gamma matrices” (see Definition 1) a generalized a form (Theorem 1). It follows that all Gamma matrices without zeros have the a form (see Definition 2.1), so that they are equivalent to the identity contraction.

Definition I: A matrix y=(yjR)‘is culled a Gamma matrix for G 8 for all j, k, 1~ G,

yjk>ov Ykj= Yjk 9 3/00= YOkr

and the “Gamma equations, ”

YjkYl,j+k= YjlYk,j+i= YklYj,k+l,




hold. We denote both the set of equations dejining a Gamma matrix, and the set of all Gamma matrices for G, by [G].

Remark I: The system of Gamma equations is invariant under

i.e., replacing all indices by their negative values transforms a Gamma equation into a Gamma equation. After taking symmetry and the condition yoo= yOk into account,

N(N- 1) 2 +1,

non-negative elements remain for a Gamma matrix y~[Z,l, namely

Y= lyoo yll z: ::j fIJ,

which have to satisfy the Gamma equations. In determining the Gamma equations for Z,v, it is useful to note that there are three types of Gamma equations: (i) the normal ones for j, k, 1 all different; (ii) the short ones for j = k # I,

YjjY1,2j= YjlYj,j+l;

and (iii) the long ones with k = -j (resulting from ylo = yoo),

The Gamma equations for YE [a,], N< 6 are




N= 2 No equation

N=3 y11~22=~12~00

N=4 ylly22= y12y13

y33y22= y13y23

yOOyl3=ylly23=y12y33

yOOy22=yl2y23

N=5 ylly22=y12y13

ylly23=y13y14

y22y14=y12y23

y22y44=y12y24

Y33Yll=Y13Y34

Y33Y14= Y34Y23

Y44Y23=Y14Y24

y44y33=y24y34

yOOy14=ylly24=y12y34=y13y44

yOOy23= y12y33= y13y24= y22y34

N=6 ylly22= y12y13

ylly23=y13y14

ylly24= y14y15

y22y14= y12y23

y22y34=y23y25

y22y45= y12y25

y44y12=y14y45

y44y23=y14y34

y44 y25 = y34 y45

y55y24= ylSy25

ySSy34=y25y35

Y55Y44= Y35Y45

y12y33=y13y24=y15y23

Y13Y44=Y14Y35=YllY34

Y23Y55=Y13Y25=Y22Y35

Y34Y15=Y24Y35=Y33Y45

yOOyl5=ylly25=y12y35=y13y45=y14y55

yOOy24=y12y34=y22y44=y23y45=y25y14

yOOy33=y13y34=y23y35*

3 elements yjk and 3/j,k’(j,k,j’,k’ E G) of a Gamma matrix are called G ‘uct appears in at least one Gamma equation. Otherwise, they are called G




Remark 2: The pair Yjj and Yj,zj is G incompatible. Apart from this exception, Yjk and Yj'k' are G compatible if

j+keG’,k’} or j’+k’sfi,k}.

Example I: For ~E[ZN], N even, the N(N+2)/8 elements,

are pairwise G incompatible. Theorem 1: Given a Gamma matrix y~[Z,l. The N numbers,

aj20, jezN,

defined by

N-l

'?=a 'Yj,ktljt kE&,

which yields

a0= y00,

satisfy

aj'o~yjk'o, fOrSOIl%? k

aj>Oeyjk>O, Vk,

Yjkaj+k=ajak

Proofi (i) The definition of a? is independent of k since

N-l N-l

n Yj,k+lj= YjkYj,k+jYj,k+2j”‘Yj,k+(N-l)j= PO Yj,lj t

because

YjkYj,k+j= Yjj3/k,2j 7

Yk,(N- l)jYj,k+(N- l)j= Yj,(N- l)jYk.Nj 9

and since

Nj=O and YOk'YOj.

(3.2)

(3.3)

(3.4)

(ii) Equation (3.2) yields




N-l N-l

?$ay+k= u YjkYj+k,l(j+k)= PO ~j,l(j+k)3/k,(l+l)j+lk

= ‘YjOYkj’Yj,j+kYkyk,2j+kYj.2ji-2kYk33j+2k ” * Yj,(N- l)j+(N- l)kYk,Nj+(N- 1)k

NN =ajak,

because

?‘(N- l)j,(N- l)kYj,(N- l)j+(N- l)k= ‘Yj,(N- l)jY(N- l)k,Nj ?

Y(N- l)k,NjYk,Nj+(N- l)k= 7/k,(N- l)kYNj,Nk 3

and since

Nj=Nk=O and yOO=yOk.

Taking the unique non-negative Nth root of this equation yields Eq. (3.3). An immediate consequence of theorem 1 is the following. Corollary I: Given a Gamma matrix y~[Z,l without zeros, i.e.,

Yjk’O, j,kEZN.

cl

Then y has the a form, which means that all elements of y can be turned into ones by a special change of basis.

Proof: Theorem 1 yields

ajak Yjk=-

aj+k’ j&E&.

with aj>O defined by Eq. (2). The special change of basis is (see Example 2.1)

1 Vj-- Vj f

ai cl

IV. GAMMA MATRICES WITHOUT ZEROS

Although Gamma matrices without zeros are equivalent to the identity contraction and hence trivial (Corollary 3.1), we study them here in more detail since they will be useful in the proofs of our main results about Gamma matrices with zeros. We introduce the notion of a basis for Gamma matrices without zeros, and give a systematic procedure for constructing a basis.

[The construction of certain Gamma matrices without zeros in Sec. VII will require the use of an appropriate basis.]




We know from Corollary 3.1 that a Gamma matrix YE[ZN] without zeros is completely defined by N arbitrary positive numbers,

aO,al ?...raN-1,

according to

yjk=Yk. aj+k

(4.1)

[On the other hand, if we use Eq. (4.1) as an ansatz for the ai’s, we recover Rq. (3.2) so that the aj’s are unique for a given y.]

Sometimes it will be advantageous to use, instead of the N ai’s, a “basis” of N appropriate individual matrix elements Yjk (see Definition 1 below) to define the Gamma matrix as a whole.

De$nition I: A set of matrix elements,

{ySilsjEZNXZN;i=1,2 )..., &f},

is called independent iJ for every arbitrary choice yS, = ci > 0, a Gamma matrix y~[Zp,]

without zeros exists whose elements ySi have the assigned values. A maximal set of independent matrix elements { ySi} is called a basis.

Remark I: The concept of a basis depends only on the group G involved. The construction of a basis is straightforward. We go through all Gamma equations (cf. [G] ,

one equation connects four different matrix elements) one by one in any order and declare the individual matrix elements to be dependent or tentatively independent according to the following rule.

At the start we call all elements tentatively independent. If all four elements in the Gamma equation under consideration are labeled tentatively inde-

pendent, we choose one arbitrarily and change its status to dependent. If dependent elements are involved, we express all of them in terms of tentatively independent

elements (see Remark 2 below). If this transforms the equation at hand into a trivial identity, no changes are necessary. If this yields a nontrivial equation, then choose arbitrarily one of the tentatively independent elements and change its status to dependent.

At the end of this construction, the tentatively independent elements are a maximal set of, in fact, independent elements, and hence a basis.

Remark 2: It follows directly from the structure of the Gamma equations that at any stage of this construction a dependent element Yjk can be written as

(4.2)

where { ySili = 1,2,..., M} are the tentatively independent elements. Corollary 1: Every basis for Gamma matrices without zeros for &, has N elements. Proofi Let {ysilsi E zN x ZN; i= 1,2 ,...,M} be a basis. Basically the proof is that, since the

equations relating the aj and the ysysi are straightforward, the number of parameters cannot change. In detail, the argument is as follows. Write down the h4 equations,

ajak Yjk’-

aji-k’ (j,k) E{sili= 1,2 ,..., M}.




Solve the first equation for one of the aj’s that appear, and replace this specific aj (by the obtained expression) in all following equations. Continue this procedure. At each step, there will be at least one Uj to solve for, since otherwise ysi would be expressed in terms of ysj, j= 1,2* ** i - 1, which contradicts independence. Hence MG N. Since all dependent Yjk are quotients of the basis elements, it follows that all Uj'S must be determined from these M equations, hence MaN.

There is another proof avoiding the a form. We show first that every basis has the same number of elements. This can be done by a

replacement principle similar in spirit to the corresponding replacement principle for different sets of independent vectors spanning a vector space. Then we only have to count the elements of one special basis, such as the “natural” basis in Example 1 below. Cl

Corollary 2: As the explicit construction of a basis shows, all other matrix elements yjk follow uniquely from the elements { ysili = 1 ,Z,..., N} of a basis. These dependent elements are of the form (cf. Remark 2)

(4.3)

[This basis expansion can always be rewritten without negative powers by multiplying Eq. (4.3) with y,yni whenever ni<O.]

The powers n, ni will be unique for every Yjk up to trivial multiples n ’ = mn, n; = mlli (m EN), since otherwise we could deduce a dependence between the independent elements of the basis.

Equation (4.3) is valid for ysi, too, with n = ni= 1, nj= 0 (j # i). Corollary 3: It follows immediately from Corollary 2, that a set of M matrix elements

(?I$ = 1,2,...,M s N} is independent if and only if no relation of the form

y;;y:;.+;=1, 0=;;5 ni, IZi E 2, i= 1

holds between them. Example I:

{Yll ;Yl2;**** Yl,N- 1; YlN= YOOI~

is a “natural” basis for Z, . The dependent elements are

%kYl,k+l”’ ‘Yjk =

Yl,k+j-2’ Yl,k+j-1

YllY12”‘Yl,j-1 ’ jsk, j#O,l.

The aj’s follow according to

ao= YOO?

4= YOOYllYl2”’ Yl,N- 1,

a: ak=

YllYl2”‘%,k-1’ k=2,3 ,..., N- 1.

(4.4)




V. GAMMA MATRICES WITH ZEROS

A. Higher-order identities

If we allow the elements of a Gamma matrix to vanish, new features appear. The concept of a basis breaks down (see Sec. V B), and the repetitive application of Gamma equations produces a few group-dependent equations of higher order (see below) that are satisfied automatically by every Gamma matrix without zeros, but which can be violated by a Gamma matrix with zeros. Therefore these “higher-order identities” (see Definition 1 below) are another condition besides yoo= yak for a Gamma matrix with zeros to be continuous (see Theorem 1 below).

Example I: The first higher-order identity appears for Z, and reads as

YllY33Y55=Y13Y15Y35*

Equation (5.1) is trivially violated by the following Gamma matrix y with zeros:

(5.1)

yjk = 0, otherwise.

[This is indeed a solution of [G] since all Gamma equations are of the form O=O due to the G incompatibility of all nonzero elements (see Example 3.1).]

One way to prove Eq. (5.1) for Gamma matrices without zeros is to observe that the equation,

follows from the Gamma equations,

Y23Yll=Y13Y14* Y14Y55= Yl5YOO9

and

[Since any two elements in Eq. (5.1) are G incompatible, the additional element y23 acts as a “zipper.“]

An easier proof of Eq. (5.1) is provided by the a form, which yields the obvious identity

alal a3a3 a5a5 ala3 ala5 a3a5 ---=--- a2 ~0 a4 a4 a0 Qz.

In Appendix A we will therefore use the a form to construct higher-order identities. Remark 1: The first condition for continuity (see Theorem 2.1),

is of a’similar type. It results from the Gamma equation,

y;k = 'YOO YOk 9

with yak playing the role of a zipper element. Dejinition I: Every equation of the type

YY”’ y= YY”‘Y7



Evelyn W&mar-Woods: Contractions of Lie algebras 4531

with the same number n of Gamma elements on both sides, which holds for all Gamma matrices without zeros, but which is violated by some Gamma matrix with zeros, is called a higher-order identity.

In Appendix A, we prove that n 3 3, and we give the general form of all higher-order identities with three and four factors. We give a complete list of higher-order identities for Z, , Ns 8.

Theorem 1: If y E [G] is continuous, then all higher-order identities hold for y. Proof: A Gamma matrix with zeroes that violates a higher-order identity can never be written

as a limit of Gamma matrices without zeros. cl Definition 2: We denote by [GZ] the set of all equations [G] (see Dejinition 3.1) plus all

higher-order identities. At the same time we denote by [GZ] the subset of Gamma matrices for G that satisfy all higher-order identities.

Remark 2: Note that any equation derived for Gamma matrices without zeros is, after cross- multiplying to eliminate denominators, of the form

YY”‘Y=Yy”‘Y, (5.2)

with the same number of elements on both sides, and hence holds for any y E [ GZ] . The easiest way to prove such a relation is via the a form. But, as we will show, it is always

possible to deduce Eq. (5.2) directly from [G] . There are two cases. In the first case, one can use the G compatibility of the elements in Eq.

(5.2) (i.e., [G]) to rewrite one side until it looks identical to the other side (see, e.g., the proof of Theorem 3.1).

In the other case there is no such path. However, one can multiply both sides by the same “zipper” elements so that such a path now exists (see Example 5.1). To see this, note that for each y we have a basis expansion for some natural power of y (which is a direct consequence of [G] ).

Hence, there exists a natural power of Eq. (5.2) so that we can use the basis expansion for all y’s involved. After multiplying by appropriate powers of all basis elements to eliminate denominators, one gets a dependence relation of the form of Eq. (5.2), but between basis elements only, and this must of course be a trivial identity (see Corollary 4.3).

Remark 3: The construction of a basis for Gamma matrices without zeros (Sec. IV) remains unaltered if, in addition to the Gamma equations, the higher-order identities are included.

6. Minimal sets of zeros

For Gamma matrices with zeros, the concept of a basis breaks down (Example 2 below). As a replacement we introduce the concept of minimal sets of zeros (see Definition 3 below).

In Theorem 7.2 we relate these concepts by proving that a Gamma matrix with a minimal set of zeros has null dimension one (see Definition 7.1).

Example 2: Consider the natural basis (Example 4.1) for &,

~Y11~Y~2~...~Y1.N-1~3/00~.

(i) No Gamma matrix exists with (N>3)

Yll’O, Y12=Y13=l,

because of the Gamma equation,

YllY22=Y12Y13*

Conclusion: If we allow basis elements to be zero, then a Gamma matrix does not have to exist for every assignment of values for the basis elements.

(ii) Different Gamma matrices exist with




y11=y,2=...=y,,N-1=yOO=0, N>2,

e.g., for N=4 the remaining elements yz2, yzs, y33 only have to satisfy

y22y33=O.

Conclusion: If we allow basis elements to be zero and a Gamma matrix taking on these values does exist, it need not be unique.

The concept of minimal sets of zeros stems from the question how many zeros of a Gamma matrix y are already forced by one single given zero yS (S E G X G) to be zero.

Definition 3: Given a Gamma matrix ~E[GZ] with zeros and ones where y,=O for one specljic SEGXG. Consider all Gamma matrices y’ E[GI] with zeros and ones, such that

r;=o,

Yjyilk’l, if yjk”l*

The set of zeros belonging to such a y’ for which the number of zeros is minimal, is called a minimal set of zeros (with respect to yJ.

Theorem 2: Given a Gamma matrix y E [ GZ] with zeros and ones, y can be written as a finite elementwise product of Gamma matrices, yi E [ GZ] , i = 1,2,. . . ,M, with zeros and ones, each of which has a minimal set of zeros.

Proofi We pick an arbitrary zero, yS1 (s , E G X G) of y, we choose a minimal set of zeros with respect to ySi from the total set of zeros of y, and we define yi as having this minimal set of zeros. To define yi (i> 1 ), we pick one of those zeros of y that have been converted into ones for all yl, 1 s IS i - 1, we call it ySi ( si E G X G), we choose a minimal set of zeros with respect to ySi, and we define yi as having this minimal set of zeros. We continue until-after A42 1 steps-every zero of y is a zero of at least one yi . Then

M

Yjk’P (Yi)jk*

VI. MAIN THEOREMS FOR GAMMA MATRICES WITH ZEROS

Let YE [GZ] be a Gamma matrix with zeros. We prove (Theorem 1) that all positive elements of y can be turned into ones by an appropriate change of basis of the underlying vector space. Then we prove that y is continuous and can be realized by a generalized InKi-Wigner contraction (Theorem 2).

Theorem 1: Given a Gamma matrix YE: [GZ] with zeros, a change of basis of the underlying vector space exists, which transforms all positive elements of y into ones.

Proof: By Theorem 7.1 there exists a Gamma matrix r without zeros, such that

rjk= yjk, if yjk>o.

The a form of I‘, namely (Corollary 3.1)

provides the desired change of basis,




1 Vj-- Vj.

aj

Theorem 2: Given a Gamma matrix y E [ GZ] with zeros and ones. Then y is continuous and y can be realized by a generalized In&ii-Wigner contraction.

Proof By Theorem 5.2, there exist Gamma matrices yi E [GZ], i= 1,2,...,M, with zeros and ones that have minimal sets of zeros, such that

M

Yjk= PI ( yi) jk . (6.1)

By Theorem 7.3 there exists, for every yi , a family ri( E), e>O, of Gamma matrices without zeros, with

(ri)jk( E) = dPi)jk, (Pi)jk E Q, (6.2a)

where

bi)jk=O, if (Yi)jk= 1 (6.2b)

and

(Pi)jk’o, if (yi)jk=o. (6.2~)

Clearly,

yi= lim ri( E). S-+0

(6.3)

By Remark 2.1,

is a Gamma matrix. Clearly,

y= lim r(+ S+O

so that y is continuous. Furthermore, we have, because of Eqs. (6.2a) and (6.4),

rjk( 6) = epjk,

where

M

Pjkzizl (Pi)jkE Q

and

pjk=o, if Yjk= 1

(6.4)

(6.5)

(6.6a)

(6.6b)




and

Pjk’O, if yjk=o,

Therefore the a form of I’(e) is [see Eq. (3.2)]

N-l

a:(e)= lJo rj,kfrj(E)=pj,

(6.6c)

(6.7a)

where

N-l

Nnj= rzo Pj,k+rj E Q

and

and

nj=O, if Yjk= 1, Vk (6.7b)

nj>O, if Yjk=o, for some k. (6.7~)

By combining subspaces Vj and vk with the same exponents ?Zj=nk, and by reordering the resulting subspaces so that the exponents are increasing, it is clear that y is a generalized In&C- Wigner contraction. Cl

Remark I: Since we have just proven [see Bqs. (6.5) and (6.7)]

Yjk= lim rjk( E) = lim aj(~)ak(~)

= fim ej’“k-“j+k,

C-+0 e+~ aj+k(E) S-+0 (6.8)

the exponents nj obey the mixed system of equalities and inequalities,

?lj+k=?ljf?lk, if yjk=l

and

nj+k<nj+nk, if Yjk=oa (6.9)

Since the generalized a form of y is (see Theorem 3.1)

with

Yjkaj+k=ajak 3 (6.10)

Uj= lim Uj( E)= lim Eni, C-+0 S-+0

the exponents nj obey in addition to Eq. (6.9), the conditions [cf. Bq. (6.7)]

(6.11)

nj'0, if Uj= 1, i.e., if Yjk= 1, Vk,

?lj>O, if Uj=O, i.e., if yjk'o, for some k. (6.12)

The system of Eqs. (6.9) and (6.12) is therefore equivalent to the question if YE [ GZ] is a generalized Iniinii-Wigner contraction.




In concrete examples (see Appendix C) the direct solution of Eqs. (6.9) and (6.12) seems to be the easiest way to obtain the generalized Iniinii-Wigner contraction for a given y E [ GZ] .

VII. CONSTRUCTION OF GAMMA MATRICES WITHOUT ZEROS THAT ARE RELATED TO GAMMA MATRICES WITH ZEROS

In this section we construct all Gamma matrices without zeros needed in Sec. VI. The idea is to use a basis that is linked as closely as possible to the given Gamma matrix with zeros, and to assign appropriate values to this basis (see Theorem 1 and Remark 1).

The size of the set of zeros of y is measured by its null dimension Mc3) (see Definition 1). Only the case Mc3)= 1 has to be studied in detail (see Theorem 3), since y can be written as an elementwise product of Gamma matrices with a minimal set of zeros (cf. Theorem 5.2), and a minimal set of zeros corresponds to Mc3)= 1 (Theorem 2).

Theorem 1: Given a Gamma matrix YE [GZ] with zeros, a Gamma matrix I? without zeros exists with

rjk= Yjk, if yjk>oa (7.1)

Proof: We extract from the subset of positive matrix elements of i, a set of independent matrix elements for r (Steps 1 and 2), which we then complete into a basis for y (Step 3). Finally, we assign appropriate values to this basis (Step 4).

Step I: We start the procedure of constructing a basis for r (cf. Sec. IV) by going through all those equations of [ GZ] that “survive” for y, i.e., which only link positive elements of y together. [We will see in Step 3 why higher-order identities have to be included here (see Remark 5.3).] This produces

M"b0 (7.2)

tentatively independent elements for r. The integer MC’) is well defined for y due to the replacement principle mentioned in the proof of Corollary 4.1.

Step 2: There will be M (2)>0 additional positive elements of y that have not yet been considered in Step 1, since they are multiplied in [ GZ] by vanishing elements of y only, so that they only occur in “nonsurviving” equations for y, i.e., in equations of the form O=O. We call these

Mc2b0 (7.3)

elements tentatively independent for r. The integer Mc2) is clearly well defined for y. Note that at this stage all rjk corresponding to yjk>o are either already dependent, or they belong to the set of M(‘)+M (‘)sO tentatively independent elements.

Step 3: The M (l) + Mc2) tentatively independent elements from Steps 1 and 2 are, in fact, independent, since any dependence relation between them, including higher-order identities, (cf. Corollary 4.3 and Remark 5.2) has already been included in Step 1. Hence we have the freedom, which we use, not to change their status.

But this set cannot be maximal for r, since the equations for the remaining dependent elements for r would have to survive for y too (cf. Corollary 4.2 and Remark 5.2), and hence y would have no zeros.

Therefore we can enlarge the set of M tl)+ Mc2’ independent elements by

Mc3)>0 (7.4)




additional elements into a basis for l?. Clearly, these M (3) additional elements vanish for 7, and any one of the zeros of y can be chosen as the first additional basis element. Since N= M(t) -t- MC21 + Mc3), Mc3) is also well defined.

Step 4: Finally, we define I’ by assigning to the M (*) + M(2) independent elements of the basis the same values they have for y. The remaining M (3) elements get arbitrary (positive) values. Then lY has the desired properties. 0

Corollary 1: Given a Gamma matrix y E [ GZ] with zeros, not all elements of a basis can be different from zero for -y.

Proo$ Assume all N elements of a basis are different from zero for y. It is clear from the procedure of constructing a basis for r, that we can take these N elements as independent elements in Steps 1 and 2 of the proof of Theorem 1. This gives

and hence

which contradicts Eq. (7.4)., cl Dejnition I: We call the integer d3)>0frOm Step 3 (above) the null dimension of y. Remark I: If we want to control the values of Ijk where yjk=O [e.g., if we look for a family

rye) with lim,c I .k( E) = yjk, as in Theorem 6.21, it is advantageous to work with basis expan- sions where all M d, basis elements appear on the rhs of EQ. (4.3) with non-negative exponents only. Theorem 3 shows that this is automatically true for Mc3)= 1.

Theorem 2: A Gamma matrix YE [GZ] with zeros and ones, which has a minimal set of zeros, has null dimension one.

Proof: Let {ys;3/riji = 1,2,..., K;s,ti E Z, X Z,} be a minimal set of zeros of y with respect to ys. We first sketch our argument.

We construct the most general Gamma matrix r without zeros, with

rjk=l, if yjyjk’l,

and

r-,=00.

The null dimension of y is M (3)a 1 This means we need Mc3) independent elements from the set of zeros of y, together with M”‘+&‘2’=N-M(3) ’ d m ependent elements from the set of ones of y to form a basis for l?. Since we have only fixed one of these Mc3’ elements, namely I-‘$, r exists.

We show that r is unique, which implies

To do this we select appropriate equations from [GZ] [see Eq. (7.5) below] and show that they already determine lJti, i = 1,2,. . . , K, and therefore r uniquely in terms of rs=c. We now present the proof in detail.

For each rfi, i= 1,2 ,..., K, there must exist at least one equation in [GZ] that prevents us from turning yti (perhaps together with some other yt,) into a “1” for y. Otherwise, the set of zeros of y would not be minimal. These equations look, in general, like



Evelyn Weimar-Woods: Contractions of Lie algebras

i=1,2 ,..., K,

4537

(7.5)

with

ni>O, nOi tnkizOT nojf c n&>o. k#i

In the following, we will refer to any equation of this form as an equation of type 1. At the start, the equations of type 1 that stem from [G] simply look like

I 147, or

I 1 *rtk or

* ‘rti= rsrtk or (7.6)

with i, k, 1=1,2 ,. . . , K all different. (We have written out explicitly every “1” to exhibit the structure of the Gamma equation at hand.)

We now inductively solve for Tti, i = 1,2,. .., K. We first pick one equation of type 1 for rtl. This equation determines Tl, in terms of rS and lY1,, k> 1. We replace Tt, by this expression in all other equations of [ GZ] . Assume that Tti, 16jS i < K have been determined in terms of TS and rtl, l>j, and that each lYtj has been replaced by this expression in all equations of [GZ].

Note that the replacement procedure may turn some equations into trivial identities, and may turn non-type 1 equations into type 1 equations, and vice versa. Hence there may not be a type 1 equation for rri+,.

[Example: It can happen that

i.r,,=i-rtz

is the only original type 1 equation for I’, and r,*. The replacement of lYl, by Itz in all equations of [GZ] may or may not produce another type 1 equation for rtz (e.g., an equation for rrlrtz would become a type 1 equation for lYfZ).]

If there is a type 1 equation for some rtk, k>i, we take it and renumber the remaining elements to make rft into rti+, (this simplifies the notation).

If there are no type 1 equations for r ti+, , . . . , rfK, then we can turn, e.g., yfK into a “1” without violating [ GZ], since rfK gets multiplied in all remaining equations by at least one other element TS or Tll, i+ 1 <l-c K. [Together with ytK, all y:!, ;= 1,2 ,..., i, where the rtj have been expressed in terms of rfK only, have to be turned into a 1 too.] This contradicts the assumption that we have a minimal set of zeros. It follows that the induction procedure can be completed, and hence all rti, i= 1,2 ,..., K, are expressed in terms of TS, and hence I is unique. 0

Theorem 3: Given a Gamma matrix y E [ GZ] with zeros and ones, which has a minimal set of zeros, a family I(c), c>O, of Gamma matrices without zeros exists with

rjk(c)=cPjR, PjkE Q,

where

Pjk'o, if yjk=l




and

Pjk'ot if yjk=oa

Proofi Let y have a minimal set of zeros with respect to ys, s E&,X&,. Since the null dimension of y is one (see Theorem 2), we can choose as a basis for r, (N - 1) basis elements from the set of ones of y, together with Ts.

We construct r-cc) from the values

l,l,..., 1, r,=00 for this basis. Then Eq. (4.3) yields

rjk(c)=ll if yjk=l

and

rlik(c) = rsmjk, Ik if Yjk=O,

where njk EN, and m jk E z. If mjkc 0, then the equation

r:jyC)r;mjk= 1 Jk

would have to hold for y too (see Remark 5.2). But for y it would read “O= 1.” Hence mjk>O, and we get

rjk(c)=ry=cPjk, if yjk'o,

where

ocPjk=~ E Q. I

An alternative but somewhat messy proof is to elaborate the construction of Theorem 2. Remark 2: If

Cl

M

Yjk=z~l (Yiyi)jk 9

where all yi have minimal sets of zeros and therefore null dimension one, then one might expect that the null dimension MC3) of y should satisfy MC3)sM. But we have an example where M(3)>M.

APPENDIX A: HIGHER-ORDER IDENTITIES

According to Definition 5.1, a higher-order identity is an identity of the form

yy”‘y= YY”‘Y, (Al)

with the same number n of Gamma elements on both sides, which is violated by some Gamma matrix with zeros.




To analyze the higher-order identities and to construct all of them for &, , N< 8, we use the a form. Since the a’s are independent, Eq. (Al) must be a trivial identity when expressed in terms of the a’s.

So we start by replacing the y’s in Eq. (Al) by their a form. After making as many cancellations as possible between the numerator and the denominator on both sides, then both sides must have the same u’s in the numerator and denominator, resp. We have the following possibilities for the final form.

(1) No cancellations possible,

aa aa aa aa -.-. -...- a a a a ’

(2) One cancellation possible,

aa aa aa (a) ao- 7. a*--;, due to F=ao;

aaa aa aa (b) a. a’.aa,

due to the Gamma equation

CLiajQk UiCZj Ui+jUk

-=-. -=...

ai+j+k ai+j ai+j+k

(3) Two or more cancellations possible. In Case (a> of (2) the factor a0 has to appear on both sides of Eq. (Al), and can therefore be

dropped to yield a Case (1) identity with n - 1 factors. By an elementary counting argument, one can show that (b) of Case (2) does not occur for

n < 5, and for Z, , Ns 8. The reason is, in effect, that such a dependence relation produces too many surviving Gamma equations for small N (cf. Remark 5.2 and the proof of Theorem 7.1).

For the same reason we can ignore Case (3). Therefore we restrict ourselves to Case (1).

1. Case (1) higher-order identities

Since cancellations do not occur and since the same a’s appear on both sides, all y’s in Eq. (Al) have to be pairwise G incompatible. Therefore such an identity (unless trivial) can always be violated by a Gamma matrix with zeros. We choose for such a Gamma-matrix arbitrary values for the elements appearing in Eq. (Al), while all other elements vanish (cf. Example 5.1).

The general form of a Case (1) higher-order identity is

with

j; ,k; ,jl,klEZjvr I= 1,2 ,..., n,

and (since only a rearrangement of the u’s takes place)

{j; ,k; ,...,jL ,kA}={jl ,kl,...,j, A),

{s;=j; +k;ll= I,2 ,..., n}={sl=jlfk,lZ= 1,2 ,..., n},




and (since a factor appearing on both sides can be dropped)

{j; &;}+{j,,k,), l,r= lA...,n,

and (because of G incompatibility; the exception (see Remark 3.2) does not play any role here)

and (since no cancellations occur)

For n = 1 and n = 2 every ansatz leads to a trivial identity so that higher-order identities do not occur. (The only one for n = 1, namely y ,,u= yOk, k E G, has been treated differently. It got forced onto all Gamma matrices by definition.)

The only possibility for n = 3 results from a cyclic regrouping of the indices involved, which can always be brought into the form

(A3)

with

sl=jl+kl=j3+k2=si, s2=j2+k2=jl+k3=s;, s3=j3+k3=j2fkl=s;

[A zipper element (cf. Example 5.1) for Eq. (A3) is, e.g., yk3s,, since

because of

For n = 4 we have again the cyclic type like in Eq. (A3), but it needs more than eight different digits. The only other possibility is a pairwise regrouping, which can always be brought into the form

(A4)

with

sl=jl+k,=j4+k3=s,:, s2=j2+k2=j3+k4=s;,

sg=j3+k3=j2fkl=si, s4=j4+k4=jl+k2=si,

which implies

j,+j3=h+j4, kZ+k3=k,+k4.

[Here, two zipper elements work, e.g., yk.S,, ‘yk&, since

because of




and

and

Yk,k,Yj,,k2+k3= YjZklYkqs3v Yk,k,Yj,,k,+k,= ~j4k;7/k2sl~l

For n L 5 we need more than eight different digits.

2. Complete list of higher-order identities for Z,,,, fVc8

N<5 a-.

N=6’ YllY33Y55=Y13Y15Y35

N=7 Yll Y34Y66= Yl4Y16Y36

Y16Y22Y55=Y12Y25Y56

YZSY33Y44=Y24Y34Y35 (valid for ZN9Na7)

N= 8 22 different ones with three factors

Remark I: The number of higher-order identities for G is finite (not counting trivial combi- nations of them like products). This can be seen in the following way.

Consider a Gamma matrix y$ [GZ] with a given set of zeros. If we select for such a y (as in the proof of Theorem 7.1) M(l)+ Mc2) tentatively independent elements, these elements are not necessarily independent. All higher-order identities that exist for them represent additional dependence relations. Since each such relation reduces the number of independent elements by one, clearly there can only be a finite number of them for such a y, i.e., for its given set of zeros.

Clearly there is only a finite number of different sets of zeros. Example I: For the Gamma matrix y $ [ GZ] , G=ZN , N even, defined by

Yjk>O, j,k odd,

yjk=o, otherwise,

the N(N+ 2)/8 pairwise G-incompatible elements {yjklj,k odd} (see Example 3.1) are tentatively independent (see the proof to Theorem 7. l), but for N = 6,8,. . . , clearly not independent. Therefore it is not surprising that these elements give rise to several higher-order identities.

Remark 2: If a higher-order identity is not invariant under the replacement of all indices by their negative values (see Remark 3.1), this replacement produces a new identity.

Example 2: The identity

is invariant for Z-,, but for Zs the replacement




j-+-j

leads to the new identity,

APPENDIX B: COMPLEX G-GRADED CONTRACTIONS OF COMPLEX LIE ALGEBRAS

In Sets. III-VII, we have only considered non-negative graded contractions. We now extend our results to complex graded contractions. The analog of Corollary 3.1 is the following theorem.

Theorem 1: Given a complex ZN-graded contraction y without zeros that satisfies yoo= yOk, k eZN . Then y is equivalent to the identity contraction.

Proof We show that y has the a form (see Remark 2.2). We do not follow the proof of Theorem 3.1, since the question arises how to choose the Nth root for a;. We use instead the notion of a basis in Sec. IV, the construction of which can be literally taken over.

Corollary 4.1, which states that every basis for ZN has N elements, holds for complex graded contractions too, since we can use the second proof given there.

Then we define the a’s for the “natural” basis for z,v (see Example 4.1), i.e., by

k a1

ak= ~11~12*~*~l.k-l’

k=2,3 ,..., N- 1,

with al being any solution of

4= y00ylly12-*- Yl,N-1. (Bl)

Then

ajak 'Yjk'-

ajik’ j,kEZ,.

[To extend the concept of a “natural” basis to a general finite Abelian group G, we note that G has, in general, more than one maximal cyclic subgroup. Each of them defines one element like a 1 in Eq. (Bl).] 0

The analog of Theorem 6.1 is the following. Theorem 2: Given a complex ;E,-graded contraction y with zeros that satisfies yoo= ysk,

k EZ, , and all higher-order identities. A change of basis of the underlying vector space exists that transforms all nonvanishing elements of y into ones.

Proof We construct a complex graded contraction r without zeros with

rjk= yjk, if yjk*o,

by following the proof of Theorem 7.1 (read nonvanishing elements instead of positive elements). The a form of r provides the desired change of basis (cf. the proof of Theorem 6.1). cl

Remark I: Therefore a complex y with zeros that satisfies you = yuk, k E &, , and all higher- order identities is continuous and can be realized by a generalized Inijnii-Wigner contraction. If y does not satisfy yoo= yOk and all higher-order identities, y is discrete (see Theorem 2.1 and Theorem 5.1, the proof of which holds verbatim for complex graded contractions).




APPENDIX C: REAL G-GRADED CONTRACTIONS OF REAL LIE ALGEBRAS

The natural G-graded contractions for real Lie algebras are the non-negative ones, i.e., the Gamma matrices. All theorems we have proven for them in C also hold in R, since all changes of basis we constructed for them are real.

However, negative entries produce difficulties. A real graded contraction that is discrete for 6= is clearly also discrete for W. But the same is not true for continuous graded contractions. First of all, no real graded contraction with one or more negative entries is continuous in the sense of Definition 11.3; not even the graded contraction yjR= - 1 (Vj,k), which is trivially equivalent to the identity contraction.

Therefore it seems natural to change Definition II.3 slightly and to call a real G-graded contraction real continuous if it is equivalent to a continuous Gamma matrix (see Remark 11.3).

Nevertheless, Theorems B.l and B.2 do not hold in the real case. Instead we have the following.

Theorem 1: Given a real ZN-graded contraction y without zeros that satisfies yoyoo= yOk, keZN.

Case (i): Either N is odd or ~ooy11~,2~~~y1,N-I >O. Then y is equivalent to the identity contraction.

Case (ii): N is even and yooy11y12..*y1,N-I <O. Then y is not equivalent to any (positive) Gamma matrix.

Proofi In order to prove equivalence to the identity contraction, we only have to check if the a form constructed in the proof of Theorem B.l can be made real for yjk ER Cj,k EZ~). This is exactly the case if a real solution exists for Eq. (B. 1). This is true for Case (i) and not true for Case (ii). In Case (ii), y cannot be equivalent to any other (positive) graded contraction either, since then it would have to be equivalent to the identity contraction (see Corollary 111.1). cl

Example I: The real graded contraction with

N=2, Yoo=Yol=l~ y11=-1,

relating two inequivalent real forms of a complex Lie algebra [like SO(3) and SO(2,1)] cannot be given by a real change of basis. The corresponding complex change of basis with a0 = 1 and u 1 = i is Weyl’s unitary trick.

Theorem 2: Given a real ZN-graded contraction y with zeros that satisfies yoo = yOk, k E Z,,, , and all higher-order identities. If N= 2 or N is odd, then a (real) change of basis of the underlying (real) vector space exists that transforms all nonvanishing elements of y into ones. Hence, y is real continuous and can be realized by a generalized InGnii-Wigner contraction.

Proof: We follow the proof of Theorem B.2. For N odd the a form of r can always be made real (see the proof of Theorem 1). For N even this is not always possible (see Example 2 below), although at least one of the factors Too, rll, l? 12 ---rl,N-,, in Eq. (Bl) is not yet fixed by y(see Corollary 7.1). The only exception is if N= 2. Here, ym or y1 1, resp., has to vanish so that Too or rll, resp., can be chosen to enforce

u2=roor,,>0. 1 q

Example 2: A real graded contraction y with zeros for ZN with N= 2M (M = 2,3,4,. . .), which satisfies yoo= yoyak (k= 1,2,..., N- 1) and all higher-order identities and for which

YooYMM<ot

is not real continuous. This can be easily seen in the following two different ways. (i) If Theorem 2 would hold for y, we would have




4 o’YooYMM=ao a,=&

which is not possible for uM E R. (ii) Using the Gamma equation

~lk~2M-k-l,k+l=??,2M-k-1~k.2M-kr

fork= 1,2 ,. . . ,M - 1 we can rewrite Eq. (B 1) for r, namely,

~:~=~oo~11~12~~~~1M~l,M+l~~~~1,2M-1

as

Therefore

so that no a 1 E R exists.

APPENDIX D: EXAMPLE

Given a Z6-graded Lie algebra [cf. Eq. (2.8)],

V= @ Vj, j=O

~u(vj,vk)cvj+k* j&E&,

and a graded contraction y thereof [cf. Eq. (2.9)],

with

Yll’l, Y12= Yl3=% x2=49 Y55’3,

Yjk=Ov otherwise,

which is indeed a solution of [GZ] [cf. Sec. III, Definition 5.2, Appendix A). The contracted Lie algebra looks like

P'(V5?VS)CV4, cL’( vj ,vk)=o, otherwise.

1. The positive elements of y can be turned into ones (cf. Theorem 6.1)

We construct the Gamma matrix r without zeros according to Theorem 7.1. A basis of r consists of

MC”=3 elements, we choose I’1,,r12,r22,




plus Mc2)= 1 elements, namely rss, plus M (3)= 2 elements, we choose r33 ,l? 14.

r is defined by the following values for this basis:

rll=yll=i, r12=y12=2, r22=Y22=4r rss=~ss=3, and we choose

r33= 1, r14= 1. A calculation of the u form for r (see the proof of Corollary 4.1) yields

4ao=a;= 12, u,=a:, &~~=a~, 4q=u;t, 4a:=3af

Then

1 Vi+- Vj

ui

turns all positive elements of y into ones. Hence, we consider in the following the Gamma matrix y E [ GZ] , G =2$ with zeros and ones, where

Yll=Y12=Y13=Y22=Y55=1r

yjk’o, otherwise.

2. y can be written as an elementwise product of two Gamma matrices y, and y2 with minimal sets of zeros (cf. Theorem 5.2)

We pick the zero

3/s,= Y33 *

One of the two minimal sets of zeros with respect to yS1 is

([G] yields the equation

O=YooY15,

which we solve by choosing yls as a zero; all other zeros follow unambiguously). y, is defined with this set of zeros. Only one zero of y is missing, namely yoo. Therefore we

define y2 with a minimal set of zeros with respect to

which is

(This is the only minimal set of zeros with respect to yoo. It is also the other minimal set of zeros with respect to y33.)




Then

3. y is continuous and can be realized by a generalized In6nikWigner contraction

(CY) For yl , there is a family rl(e), E>O, of Gamma matrices without zeros (cf. Theorem 7.3) belonging to the following values of its basis:

ml=(w12=(w22=rl)55=(rl)oo= 1, r1)33=e.

Since Eq. (7.5) reads for the minimal set of zeros of y,,

this family is

r14=r15=r23=r25=r35=r45,

r24= r34= r44= r23r35 = r33,

so that

yl= lim rl(+ E-+0 (p) For y2, the corresponding family r2(E), EBO, is defined by

(~2)ll=(~2)12=(r2)22=(r2)55=(r2)15=1, (r2)00=e.

Here, Eq. (7.5) reads as

r14=r23=r24=r25=r33=r35=r45=roo, r34=r44=roor24, which yields

so that

( y) Therefore

with

y2= lim r2cE). r-0

y= lim IyE), S-+0




where

r11=r12=r13=r22=r55=i, r15=C2, roo=E,

r14=r23=r25=r35=r45=E3/2, r24=r33=c2, r34=r44=P.

The u form of r is

a, = P2, ao=u2=a5=e, u3 = E3i2, a4 = e2,

which gives the exponents

n,=+, no=n2=n5=1, n3=5, n4=2, (Dl)

for the generalized Inonii-Wigner contraction. Therefore y can be realized as a generalized Inonii-Wigner contraction of a Lie algebra with

the structure [cf. Eqs. (2.4), (2.5), and (2.6)]

v= V(O)@ v(l)@ V(2)@ v(3),

where

v(O) = Vl , v(‘)=vocBv2cBv5, v(2)=v3, v(3’=v4,

and where

with

no= 4, n”,=l, - n2= ;, ii3=2.

The contracted Lie algebra is

p’($j),V(k))=O otherwise.

4. Direct solution of the exponents for a generalized In&&Wigner contraction for y (cf. Remark 6.1)

In our concrete example it is very easy to determine the exponents nj , j = 0, 1, . . . ,5, directly. We have to solve the system [cf. Eq. (6.5)]




n3+4+n5, n5< i

nl+n4 n2+n3’

under the condition [cf. Eq. (6.6)]

ni>O.

The general solution is

n2=n5=2n1, n3=3n1, n4=4nl,

O<n0<3nl, n,>O.

Equation (Dl) gives a special solution.

‘I. E. &gal, Duke Math. J. 18, 221 (1951). *E. Inijnii and E. P. Wigner, Proc. Natl. Acad. Sci. USA 39, 510 (1953). 3E. J. Saletan, J. Math. Phys. 2, 1 (1961). 4E. Weimar-Woods, J. Math. Phys. 32, 2028 (1991). 5E. Weimar-Woods, J. Math. Phys. 32, 2660 (1991); R. J. B. Fawcett and A. J. Bracken, J. Phys. A 24, 2743 (1991). 6M. de Montigny and J. Patera, J. Phys. A 24, 525 (1991). 7R. V. Moody and J. Patera, J. Phys. A 24, 2227 (1991). 8A. L. Onishchik and E. B. Vi&erg, in Lie Gmups and Lie Algebras III (Springer, Berlin, 1994). Chap. 7, Sec. 2. 9J. J. Rotman, The Theory of Groups. An Introduction (Allyn and Bacon, Boston, 1973).



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Contractions of Lie algebras: Generalized Inönü–Wigner contractions versus graded contractions

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