Let U, V, W be finite dimensional vector spaces over a field k and let : U x V → W be a bilinear mapping. The (multiplieative) complexity L(~) of ~ is defined as the least r ∈~ such that there are linear forms Ul,...,u r , Vl,...,v r ∈ (U xV) and elements Wl,...,w r ∈ W satisfying r ~(x,y) = ~ Up(X,y) Vp(X,y) Wp for all (x~y) ∈ U× V .Departamento de Matemátic

Groote, Hans F. de

Heintz, Joos

Möhler, Stefan

Schmidt, Heinz

Servicio de Difusión de la Creación Intelectual

ON THE COMPLEXITY OF LIE ALGEBRAS Hans F. de Groote I Joos Heintz ! ,2 t Stefan MOhler I , and Heinz Schmidt I J.W. Goethe- Universit~t Fachbereich Mathematik Robert-Mayer - Str. 6-10 D-6000 Frankfurt/Main I F RG 2 Consejo Nacional de Znvestigaciones Cientificas y T~cnicas (CONICET) Departamento de Matem~ticas de la Universidad Nacional de La Plata La Plata, Pcia. de Buenos Aires Argentina I. Introduction Some general notions from complexity theory. Complexity of Lie algebras Let U,V,W be finite dimensional vector spaces over a field k and let : U x V ~ W be a bilinear mapping. The (multiplieative) complexity L(~) of ~ is defined as the least r 6~ such that there are linear forms Ul,...,u r , Vl,...,v r 6 (U xV) and elements Wl,...,w r 6 W satisfying r ~(x,y) = ~ Up(X,y) Vp(X,y) Wp for all (x~y) £ U× V . p=1 The r-tuple ((Up,Vp,Wp)6(U×V)×(U×V) × W, IS p5r) is then called an o p t i m a l  quadratic algorithm f o r  9 .  L(~) is the number of non linear arithmetic operations that are neces- sary and sufficient to compute ~(x,y) from x,y by a straight line program (cf. [Strassen 1973] and [deGroote 1987] for further details). We shall use a somewhat coarser but more feasible computational model: A bil inear algorithm for ~ is an r-tuple B = ((Up,Vp,Wp) 6 U × V × W , 1~p~r) with the property that r @(x,y) = Z Up(X) Vp(y)Wp for all (x,y) 6 U x V . p=1 Writing L(B) := r for the length of B , R(@) := min { L(~) ; ~ bilinear algorithm for ~ } is called the 6ilinear complexity or rank of ~ [Strassen 1973]. In case L(B)= R(@) we call B an optimal bilinear algorithm for @ . 173 Obviously we have L(~) ~ R(#) , and it is easy to see that R(~) S 2 L(¢). Hence the complexity measures L(~) and R(~) have the same size. The main theme of the complexity theory of bilinear mappings, e.g. the multiplication in algebras, is the determination of lower bounds for L(~) and R(#) . Another problem, closely related to the rank problem for bilinear map- pings, is the determination of the isotropy group of a given bilinear mapping. Isotropy groups belong to the decisive tools in the investigation of varieties of all optimal algorithms for bilinear mappings (cf. [de Groo- te 1978]). Let GL(U) , GL(V) , GL(W) be the groups of k-linear automorphisms of U , V , W respectively. Then the group of all ~ ® ~® X £ GL(U~®V~®W) with ¢ 6 GL(U) , @ 6 GL(V) , X 6 GL(W) such that ¢(x,y) = X(~(¢(x),~(y))) for all (x,y) 6 U × V is called the proper isotropy group of ¢ and denoted by F(¢) (cf. [de Groote 1978]). (Here we write ~,~* for the dual mappings of ¢ , 4.) The elements of F(@) transform bilinear algorithms for ¢ of length r into bilinear algorithms of the same length : Let B = ((Up,Vp,W@)£ U~xV*xW, ISQSr) be a bilinear algorithm for @ of length r . Then for (x,y) 6 U × V we have r @(x,y) = p=II Up(X) vp(y) w@ = X(¢(¢(x) , ~(y))) r r = P=IZ Up(¢(x))Vp(~(y)) X(W@) = p~1 ¢*(Uo)(X) ~*(v )(y)w X So, ((@*(Up) ,~(vp) , X(Wp))6 U~xv~×w , 1SpSr) is a new bilinear algorithm for ~ of length r . In particular, F(@) can be considered as a group operating on the va- riety A@ of optimal algorithms for # . Let r := R(~) and denote by S r the symmetric group of permutations of r elements. For any optimal algorithm B = ((Up,Vp,Wp) 6 U~×V~×W , 1~psr) for ¢ and any ~ 6 S r , 6z-z:= ((U~(p),Vz[D),wz(pl)6 U*×V~×W , Isp~r) is also an optimal algorithm for # . Hence S r operates on A@ . Looking at F(#) as a group acting on A¢ , we see that S r and F(~) commute elementwise. The compositum of S r and F(#) is called the extended isotropy group of ~ (of. [deGroote 1978]). 174 In this paper we extend the results of [deGroote - Heintz 1986] on the complexity of certain classes of Lie algebras. A Lie algebra over a field k is a (finite dimensional) k-vector space g , together with a bilinear mapping to g such that (i) [X,X] = 0 (~) [X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] (X,Y) ~ [X,Y] from g × g = 0 for all X 6 g and for all X,Y,Z 6 g • The rank of the bilinear mapping (X,Y) ~ IX,Y] , which we consider as a bilinear mapping [,] : g × g ~ [g,g] , is called the rank of g , and we denote it by R(g) . The notions "bilinear algorithm for g " and "isotropy group of g " always refer to the bilinear map (X,Y) ~ [X,Y] . The proper isotropy group of g is denoted by F(g) . From now on let k be algebraically closed and of characteristic 0 , for simplicity : k = C . Lie algebras we are going to consider mostly are semisimple  or Borel subalgebras of semisimple Lie algebras over C. Specific attention will be paid to the case of simple Lie algebras. (We will freely use defini- tions and results from Lie algebra theory and recommend [Humphreys 1980] and [Goto - Grosshans 1978] as references.) 2. Complexity and rank of classical simple Lie algebras and their Borel subalgebras Let g be a simple Lie algebra over C with n := dim C g , Cartan sub- algebra h and £ := dim C h (Usually, £ is called the rank of g . How- ever we will use "rank of g " only in the sense of complexity theory.) Let g be a c l a s s i c a l  s imple  Lie algebra,  i.e. g := si(£+I,C) g := o(2~+I,C) g := sp(2Z, C) or g := o( 2£ , C) Concerning the complexity L(g) with Dynkin diagram A£ ( £ ~ I ) , with Dynkin diagram B£ ( £ ~ 2 ) , with Dynkin diagram C£ ( £ Z 3 ) , with Dynkin diagram D£ ( £ Z 4 ) we have the following lower bounds : Theorem I (i) L(si(£+I,C) Z 2n - 2£ , (~) L(o(2£+I,C) Z 2n - 4£ + 2 , (i i i) L(sp( 2Z ,C) a 2n - 4£ + 2 , (iv) L( o ( 2£ ,C) Z 2n - 4£ + 4 175 Unfortunately, these lower bounds are unlikely to be reached. For example, we have L(sI(2,C))= 5 , whereas our lower bound is 4 in this case. For the rank R(g) our lower bounds are more realistic. However, we haven't yet results for g = sp(2~,C) . Theorem 2 (~) R(g) Z 2 dim C g - dim C h = 2n - £ for g = sl(~+1,C) , g = o(2~+I,C), and g = o{2£,C) . Moreover, we have g = sl(2,C) iff equality holds in (*). In [de Groote - Heintz 1986] the problem to find lower bounds for the rank of a semisimple Lie algebra g is reduced to a purely algebraical counterpart, namely to find upper bounds for the dimension of so-called generic subalgebras of g . A subalgebra a of g is called generic iff there exists an element A0 £ g such that Cg(A0) = Cg(a) (the centralizer Cg(m) of m c g is the set of all X6 g that commute with m ). A 0 is called a generic element of a . Generic subalgebras are abelian and a generic subalgebra with generic element A0 is contained in the double centralizer C2(A0) := Cg(Cg(A0)) . C2(A0) , in turn, is a generic subalgebra with g g generic element A0 • Thus we are left with the problem of estimating ~(A) , A £ g . the dimension of double centralizers Cg In case si(£+I,C) we can use the classical double centralizer theorem C 2 (A0) = { F(A0) ; F 6 C[T] } gl(~+1,e) T an indeterminate over C (cf. [Greub 1981], p. 422). Then it easily can be shown that for A0£ sl(Z+1, C) C2 (A0) = C 2 (A0) N sl(£+1,e) sI(Z+I,C) gI(Z+I,C) Z+I = { F(A0) ; F6 C[T] , i~I F(li)= 0 } , where 11,...,I£+ I £ C are the (not necessarily distinct) eigenvalues of A0 • Hence dim C C 2 sI(£+I,c)(A0 ) ~ From this we infer (~) in case @=sl(Z+ItC) (cf. [de Groote- Heintz 1986]). In ease g = o(2Z+I ,C) or g = o(2Z, C) the proof that double centra- lizers C ~(A0) have dimension S £ is much more involved and relies essentially on the theory of elementary divisors for skew symmetric matrices over C . 176 Theorem 3 Let B be a Borel subalgebra of a simple Lie algebra g with Caftan subalgebra h . Then L(B) ~ 2 dim C [B,B] - dim C h The smallest Borel subalgebra among those mentioned in Theorem 3 is the non abelian two dimensional Lie algebra, the Borel subalgebra of sl(2,C). Its structure can be generalized also in the following way : Let g be a finite dimensional vector space and ~ £ g~ a non trivial linear form. Then [X,Y] := ~(X) Y - ~(Y) X for all X,Y £ g defines a Lie structure on g , to which we refer as Lie null algebra g~. The name arose from the similarity of g~ with the associative null al- gebra, that was treated in [deGroote 1987]. In both cases the rank is known. Theorem 4 R(g~) = 2 dim C g~ - 2 Proof : Certainly R(g~) S 2 dim C g~- 2. Now consider an optimal bill- near algorithm for g~ ((u~,v~,wp) ~ ~j~gj~ [~,~], 1~p~r) and let n := dim C g~ . After a suitable renumbering we may assume that the restrictions of Ul,...,Un_ I to [g~,g~] form a basis of [ge,ge]~ with dual basis X1,...,Xn_ I of [g~,g~] . We denote by a the orthogonal complement of Vn,...,v r a is not contained in [g~,ge] = ker e , for the converse would imply a ~ 1~p~rN ker Vp = O and therefore O = dim C a _> 2n-I - r , which con- tradicts the choice of r . Thus for some element A0 6 a ~ker ~ : [Xi,A 0] = - ~(A0)X i = vi(A 0) W i ~ O for i ~ n-1 , showing that {WI,...,Wn_ I} is a basis of [g~,g~] . In this case, how- ever, a is even a generic subalgebra of g~ and forthwhith abelian. But abelian subalgebras of g~ containing A0 are one dimensional, i.e. I = dim~ a ~ 2n- I - r . [] 177 3. The isotropy group Let k be any field and g be an arbitrary, finite dimensional Lie al- gebra over k . Let ~,~,X 6 GL(g) such that ~®~® X 6 F(g) . As one easily sees ~o~ -I is an endomorphism of the k-vector space g, symmetric in the sense that [ ~ o~-1(X) , Y ] = [X , ~o~-1(y)] holds for all X,Y 6 g • So, in a first attempt to characterize F(g) , one has to determine the symmetric mappings of g . These are special cases of the transposable mappings of g: We call a k - v e c t o r  space endomorphism o: g ~ g t r a n s p o s a b l e ,  if there exists a k-vector space endomorphism T: g ~g such that [ a(X) , Y ] = [ X , T(Y) ] holds for all X,Y 6 g . If g has trivial centre, then T is uniquely determined by o , and we write o8 := T The transposable mappings form a finite dimensional associative k-algebra T(g) with involution ~ A k- linear endomorphism a : g ~ g is symmetric iff ~* = ~ . Now let us assume that g is a semisimple Lie algebra over C . Then m g -- • ( gie ... e gi ) i=I n. summands i where the g i  a r e  s i m p l e  L i e  a l g e b r a s  s u c h  t h a t  g i  ~ g j  f o r  i ¢ j a n d  g l  - s l ( 2 , C )  . m Let B be a Borel subalgebra of g . Then B ~ • (Bie...eB i) (n i sum- i=I mands), where B i i s  a n  i d e a l  o f  B b e i n g  a B o r e l  s u b a l g e b r a  o f  g i  " With this notation we have Theorem 5 (i) (A) T ( g )  - C s T ( B )  _ ]M2(C) n l  x ( C[T]/(T 2) ) s-nl m where s = iZ__1 n i , ~2(e) is the associative C-algebra of 2 x 2-matrices over C , and T is an indeterminate over C. 0 6 T ( g )  symmetric iff 0 6 C idg I • ... • C idg s and o 6 T(B) symmetric iff o £ C idB1 • ... • C idBs 178 For the isotropy group of g we have Theorem 6 (i) m m F(g) ~ ~I Sni K (c0sx GLCgl) x ... ×GL(gl)x i=2X(Aut(gi) ..... Aut(gi)) ) n 1 f a c t o r s  n i f a c t o r s  for m > I s+1 (A) F(g) ~ Snl K (~0 x (GL(gl) ×... ×GL(gl))/C0) for m = I, v n I factors m where s := n I - 2 + 2 ~ n, and C O := C TM {0} i=2 1 In particular, for g simple we obtain F(g) ~ Aut(g) if g ~ sl(2, C) and F(g) ~ GL(g)/c 0 if g ~ sl(2, C) (compare [deGroote - Heintz 1986], [Mirwald 1986]). The same results hold, if we replace in the above formulae each Lie al- gebra by its Borel subalgebra. Finally let us mention that the extended isotropy group operates tran- sitively on the algorithm variety of sl(2,C) . This means that there exists essentially only one optimal bilinear algorithm which computes the bilinear mapping [,] : sl(2,C) x sl(2,C) --~ sl(2,C) . This is by now the only case of a Lie algebra of which the algorithm variety is known ([Mirwald 1986]). References Goto, M° & Grosshans, F.D. Greub, W. deGroote, H.F. : Semisimple Lie Algebras. M. Dekker, New York & Basel 1978. : Linear Algebra. 4th edition. GTM 23, Springer, New York 1981. : On varieties of optimal algorithms for the computation of bilinear mappings. I. The isotropy group of a bilinear map- ping. Theoret. Comput. Sci. 7 (1978) 1-24. 179 de Groote, H.F. : Lectures on the Complexity of Bilinear Problems. LN Comput. Sci. 245, Springer, Berlin 1987. de Groote, H.F. & Heintz, J. : A lower bound for the bilinear complexity of some semisimple Lie algebras. in: Algebraic Algorithms and Error Correct- ing Codes. Proc. AAECC-3, Grenoble 1985. LN Comput. Sci. 229 (1986) 211-222. Humphreys, J.E. : Introduction to Lie Algebras and Represen- tation Theory. 3rd printing revised. GTM 9, Springer, New York 1980. Mirwald, R. : The algorithmic structure of sl(2,k). in: Algebraic Algorithms and Error Correc~ ing Codes. Proc. AAECC-3, Grenoble 1985. LN Comput. Sci. 229 (1986) 274-287. Strassen, V. : Vermeidung yon Divisionen. J. Reine Angew. Math. 264 (1973) 184-202. 

On the Complexity of Lie Algebras

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On the Complexity of Lie Algebras

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