RENDICONTI DI-L CIRCOLO MATEMNrI(70 DI PALERMO

Scrie II, Tomo XLVIII (1999). pp. 153-162

IDENTITIES OF REPRESENTATIONS OF LIE A L G E B R A S

AND *-POLYNOMIAL IDENTITIES

TSETSKA RASHKOVA - VESSELIN DRENSKY

In this paper we have found a necessary and sufficient condi~m for the existence of *-identities of special kind and in skew-symmetric variables in matrix algebras with symplectic involution *. The same result holds for the identities of the natural representation of the symplectic simple Lie algebra.

Let R = ( R , , ) be an associated algebra with involution �9 over a field K of characteristic 0. The notion of , -polynomial identities for R is similar to that of ordinary polynomial identities. We start with the free associative algebra with involution K ( X U X*). An e l emen t f ( x l . . . . . Xm) from K ( X U X * ) is a ,-polynomial identity for ( R , , ) if f ( r t . . . . . rm) = 0 for all ri . . . . . rm 6 R. Since (R, , ) = R + | - , where R • = {r ~ RIr* = +r}, we can replace K ( X U X * ) with K ( Y U Z), where Y and Z are, respectively, sets of symmetric and skew-symmetric variables.

The main purpose of the present paper is to study the , -polyno- mial identities of the 2n • 2n matrix algebra M2n(K, *) with symplectic involution �9 defined by

(A B)*= (D t -B t) C D - C t A t '

Partially supported by Grant MM605/96 of the Bulgarian Foundation for Scientific Research.

154 T S E T S K A R A S t t K O V A - V E S S F L I N D R F N S K Y

where A, B, C, D are n • n matrices and t is the usual transpose. For a background on the . -polynomial identities for Mzn(K, *) see e.g. [61. We are interested in the ,-identities in skew-symmetric variables of the following type considered in the ordinary case by Formanek [51 and Bergman [I] and by the authors [4] in the case of weak polynomial identities for Lie algebras. To a polynomial in commuting variables

(1) g(tl , .. t n + l ) - ~ ' ~ O t l , t~ 'l �9 �9 , �9 . . ~ + j ~ Kltl . . . . ,tn+l] tPn-~ l

we relate a polynomial v(g) from the free associative algebra K ( x , Yl, . . . . y,,)

v(g) = v (g) (x , Yl . . . . . Yn)

(2) _ , ~ x P l y l X I ' 2 y 2 X p ' ' ' X l'''+l - - ~ p . . . . r

Any multilinear in yj . . . . . Yn polynomial f ( x , yj . . . . . y~) written as

f ( x , yj . . . . . Yn) =- Z v(gi ) (x , Yij . . . . . Yi,,), (3)

gi E K l q . . . . . t,,+ll.

can be

In the paper we prove the following main result.

THEOREM 1. The polynomial v(g)(x , Yl . . . . . Yn) from (2) is a , - polynomial identi~ in skew-symmetric variables for Mzn(K, *) i f and only if

l<_p<q<_n+l

divides the polynomial g(tl . . . . . t,,+l from (1).

The vector space of the skew-symmetric elements in Men(K, *) equipped with multiplication [u, v] = uv - vu, u, v ~ M2n(K, *), is iso- morphic to the symplectic Lie algebra (i.e. the simple Lie algebra from the series C,,) with its canonical representation in the 2n-dimensional vector space. Recall that the polynomial f ( x l . . . . . Xm) from the free as- sociative algebra K ( X ) is an identity for the representation v of the Lie algebra G if f ( v ( g l ) . . . . . V(gm)) = 0 for all gj . . . . . gm ~ G. Theorem 1 can be restated in the following way.

THEOREM 2. The polynomial v (g) (x , Yl . . . . . y,,) from (2) is an iden- tity f o r the natural 2n-dimensional representation o f the simplectic Lie

ID|'NTITIES O[: REPRESENTATIONS OF LIE AI.GEBRAS AND *-POLYNOMIAL IDENTITIES 155

algebra c. if and only r 1-I ( ( , - l<_p<q.<.n+l

g ( t l , . . . , tn+l) from (1).

divides the polynomial

Our investigation is based on the following two propositions. The first is well known and the second is due to Bergman [1].

PROPOSITION I. Every .-identity in skew-symmetric variables for M2n(K, *) is an ordinary polynomial identity for Mn(K).

PROPOSITION 2. [l, Section 6, (27)] (i) The polynomial v(g)(x, Yl , . . . , Yn) from (2) is an identity for M, (K)

if and only if

H (tp -- tq) l<_p<q<_n+l

divides g(tl . . . . . tn+l) from (1).

(ii) The po(vnomial f ( x , Yl . . . . . Yn) from (3) is an identity for Mn(K) if and only if every summand v(gi) is also an identity for Mn(K).

Since [1] is in a preprint form, we refer to [4, Proposition 1.2] for the proof of Proposition 2. As in [1] and [5], it is very important to treat x in (2) as a diagonal matrix of special type. For this purpose we apply some classical results on simple Lie algebras.

We emphasize on the fact that in Theorems 1 and 2 the y ' s are in a special ordering in the polynomial (2). In our case, there exist examples showing that for small n a polynomial of type (3) can be a . - ident i ty in skew-symmetric variables for M2n(K, *) while the corresponding v ( g i ) ' S

are not. We are not able to describe the polynomials (3) which vanish on M2n(K,*)-. Newertheless, if in (3) we replace x by a skew-symmetric matrix and allow the y 's to be any 2n x 2n matrices which is useful for constructing central polynomials by the method of Razmyslov [8] (see also Halpin [7]) we obtain an analogue of the result of Bergman.

THEOREM 3. Considered in M2n(K, *), the polynomial f from (3) satisfies f (a , rl . . . . . rn) = 0 for any skew-symmetric matrix a and all matrices rl . . . . . r,, if and only if

H I<p<q<n+l

divides the polynomials gi(tl . . . . . tn-~ l) for all i = (il . . . . . in).

156 TSETSKA RASHKOVA VESSEl.IN DRENSKY

Finally, we want to mention a recent result of Domokos [3] giving relations between Gr6bner bases and determinantal ideals in polynomial algebras and polynomial identities of type (2).

1. Lie groups and algebras.

In this section we give some background on simple Lie groups and algebras. For details see e.g. 12J. We fix and algebraically closed field K of characteristic 0. Let 0 be a simple Lie algebra of rank r and of dimension m. The Zariski topology on 0 identifies 0 with the m- dimensional affine space and considers the sets of common zeros of sets of polynomials as closed subsets of 0. For h 6 0 we denote by 0 ~ the kernel of the vector space endomorphism adh and consider the characteristic polynomial of ad h

m

det(t - ad h) = ~ ai (h)t, i =o

ai 6 K .

The element h is regular, if ar(h)TkO.

PROPOSITION 3. Let 0 be a finite dimensional simple Lie algebra.

(i) The set of the regular elements of g is open and dense in 0 with respect to the Zariski topology.

(ii) If h is a regular element, then 0 ~ is a Cartan subalgebra of 0 and all Cartan subalgebras are of the type 0 ~ where h is regular.

Let V be a 2n-dimensional vector space with nondegenerated skew- symmetric bilinear form qJ. The symplectic group Sp(qJ) of V consists of all g ~ GL(V) such that

tP(g(vj), g(v2 ) ) = KI/(vl, V2), Vl, V2 E V.

Similarly the symplectic Lie algebra sp(qJ) is the subalgebra of the Lie algebra 0I(V) consisting of all g with

~P(g(vl), v2) + kO(vl, g(v2)) = 0, vl, v2 E V.

It is well known that ~p(~) is a simple Lie algebra isomorphic to the algebra of the series Cn.

IDENTITIES OF REPRESENTATIONS OF LIE ALGEBRAS AND *-POLYNOMIAL IDENTITIES 157

that A Witt basis el . . . . . en, e-n . . . . . e - l of V is a basis of V such

~iei, Z rIiei = ~-~-(~irl-i i=-n i=-n i = 1

Clearly, a matrix g ~ G L ( V ) is from Sp(qQ if and only if g transfers a Witt basis to a Witt basis.

Choosing a Witt basis of V and identifying the elements of End(V) with the algebra M2n(K) we define a mapping �9 : M2n(K) -+ M2n(K) by (A (O' . ' )

C = - C t A t '

where A, B, C, D are n • n matrices and t is the usual transpose.

Then

Sp(qQ = {g ~ G L z , ( K ) I g * g = 1},

and the algebra

5p(qJ) = {g E 9[2n(K)lg* + g = 0}

11 = p i ( e i i - en+i,n+i)lPi E K i = 1

is a Cartan subalgebra of sp(qJ). For every two Cartan subalgebras bl and t12 of sp(q~) there exists a matrix g ~ Sp(qJ) such that g-1111g = b2. Since the set of all regular elements in sp(qJ) is dense in the Zariski to- pology and every regular element is contained in some Cartan subalgebra of ~p(qJ) we obtain immediately the following.

PROPOSITION 4. The set o f all elements h r sp(qJ) with the proper O, that there exists a matrix g ~ Sp(qQ such that g - l h g belongs to the diagonal Cartan subalgebra 11 is dense in ~p(qJ) with respect to the Zariski topology.

2. The main results.

In this section we shall establish the main results of our paper. Since Theorems 1 and 2 are equivalent, we shall prove Theorem 2. We

1 5 8 T S E T S K A R A S H K O V A - VESSEL1N D R E N S K Y

use arguments similar to those from [1] combined with arguments from Section 1.

Proof o f Theorem 2. We consider the natural isomorphism of the symplectic Lie algebra Cn and the algebra sp(qJ), where qJ is a non- degenerated skew-symmetric bilinear form of the 2n-dimensional vector space.

(i) Let v(g) vanish on sp(qJ). We consider a monomial m,~ in the expression of v(g) from (2), namely

m a : Ola xa l y l Xa2 �9 �9 �9 y n X an+t �9

We make the following substitution with elements from sp(qJ)"

n

X = Z Pi (e i i - - en+i ,n+i ) , i : 1

Yl = e l 2 - - en+2,n+l ,

Y2 = e 2 3 - - en+3,n+2,

Y n - I = e n - l , n - - e 2 n , 2 n - l ,

Yn = e n , n + l q- e l , 2 n ;

follow the summand containing el,n+1 and denote by arrow the result

m, ---> OtaPl' . . . Pa"(-Pl)a"+' al,,+l.

Hence the coefficient of el,n+l in the substitution in v(g) is equal to g(Pl . . . . . P n , - P l ) . Since the base field is infinite and v(g) vanishes on sp(qQ, we obtain that g(q . . . . . t n , - t l ) = O, i.e. tl +tn+l divides the polynomial g(tl . . . . . tn+l).

We make n - 1 more substitutions, changing only Yn, namely Yn = en,n+k q-ek,2n for k = 2 . . . . . n - 1 and y~ = en,2n and follow the terms containing el,n+k and el,2n, respectively. In this way we obtain that tk + t~+l and tn + tn+l also divide g.

Now we fix i and k (1 < i < k < n ) and substitute

IDENTITIES OF REPRESENTATIONS OF LIE ALGEBRAS AND *-POLYNOMIAL IDENTITIES 159

x = p l e l l + �9 " + P k - l e k - l , k - I + Pk+lekk

-+" "'" + Pn+lenn

-- P l en+l ,n+ l . . . . . P k - l e n + k - l , n + k - I

- - Pk+ 1C~ +k . . . . . Pn+ 1 e2n,2n,

Yl ~ e 1 2 - - en+2,n+l,

Y2 ~ e 2 3 - - en+3,n+2

Y k - 2 ~ e k - 2 , k - I -- e n + k - l , n + k - 2 ,

Y k - I ~ e k - l , n + i ~ e i ,n+k-1 ,

Yk =en+i ,k q- en+k,i ,

Yk+l ~ e k , k + l -- en+k+l,n+k,

Y n - I ~ e n - 2 , n - 1 -- e 2 n - l , 2 n - 2 ,

Yn ~ e n - l , n -- e2n,2n-1.

Following the term, containing eln we come to

ma --+ o[op~ 1 -ak-I (--t'~'~ak t"~ak+l an+l "" " P k - I " t"t; ~ k + l "" "Pn+l e ln .

Again we obtain that

g ( P l , �9 - - , P k - l , - - P i , Pk+l . . . . , P n + l ) = 0

meaning that ti q-tk divides g for i = 1 ,2 . . . . . k - 1. Allowing k to be consequently 2 . . . . , n we come to the final result that 1-I (tp+tq)

l<_p<q<n+l divides g.

By Proposition 1 v(g) is an ordinary polynomial identity for M.(K) and we apply Proposition 2 to see that VI ( tp - tq) divides

l<p<q<n+l

g ( q . . . . . tn+l ) and this completes the "if" part of the theorem.

(ii) Let k be the algebraic closure of the field K. Since M z n ( K ) c M2.( /~) and a similar inclusion holds for the symplectic Lie algebras, we many replace K by /('. We shall prove a statement whi- ch is stronger than the "only if" part of the theorem. If K is al- gebraically closed and I - I (t 2 - t q z) divides g( t l . . . . ,t~+,), then

l<p<q<n+l

160 T S E T S K A R A S H K O V A - V E S S E L I N D R E N S K Y

v ( g ) ( a , rl . . . . , rn) = 0 for any a 6 ~p(qJ) and all r l , . . . , rn E M z n ( K ) .

Let rl . . . . . rn be arbitrary matrices from Mzn(K). Since v ( g ) ( x , Yl . . . . . Yn) is multilinear in Yl . . . . . yn, without loss of generality we may assu- me that r~ . . . . . rn are matrix units.

We think of v ( g ) ( x , rl . . . . . rn) from (2) as a polynomial function in the coordinates of x with respect to any basis of sp(qJ). Every poly- nomial function which vanishes on a dense (with respect to the Zariski topology) subset of sp(kO) vanishes everywhere. Hence, by Proposition 4, it is sufficient to consider the case when x is a diagonal matrix from sp(kO), i.e.

(4) x = n

Z Pi(eii -- en+i,n+i. i = 1

Then v ( g ) ( x , eft j, . . . . . ei,,j,,) = 0 if ( j l . . . . . j n - l ) ~ : ( i 2 . . . . . in) and

v (g ) (x , eili2, ei2i3 . . . . . eini.+t ) = g(s lPh . . . . . 8nPi,,, 6n+lPjn)eilin+j,

where Sk = 1 if ik < n and ek = --1 if ik > n. Since we have n + l variables Pil . . . . . P i . + l , there exist two different indices ik, il such that ik = il (modn), i.e. Pik = - 4 - P i l " Since tk q-t l divides g ( q . . . . . tn+l), this implies that g(e lPi l . . . . . enPi,,, sn+lPj , ) = 0 and v ( g ) ( x , Yl . . . . , Yn) = O.

P r o o f o f Theorem 3.

(i) Let f from (3) be such that f ( a , rl . . . . . rn) = 0 for any a M 2 n ( K , *)- and all rl . . . . . rn ~ M2n(K). As in the part (i) in the proof of Theorem 2, we obtain that for k = 1 . . . . . n and i0 = (1,2 . . . . . n)

u(gio) ( ~-~pi(eii-en+i'n+i)'e12,e23i=l . . . . . en-l'n'en'n+k) =

= gio(Pl, ' ' ' , Pn, --pk)el,n+k = 0

and tk + tn+l divides gio. Similarly, fixing i and k (1 < i < k < n) and

IDENTITIES OF REPRESENTATIONS OF LIE ALGEBRAS AND *-POLYNOMIAL IDENTITIES 161

substituting

x = p l e l l + " '" + P k - l e g - l , k - I + Pk+lekk

+ �9 . . + Pn+lenn

- - plen+l ,n+l . . . . . P k - l e n + k - l , n + k - 1

- - Pk+ I en+k,n+k . . . . . Pn + 1 e2n,2n,

Y.I ~e12

Y2 --~-e23

Yk-2 =ek-2,k-1,

Yk-I ~-ek-l,n+i,

Yk =en+i,k,

Yk+l ~ek,k+l,

Yn- I =en-2,nl ,

Yn ~en-l ,n,

we see that ti + tk divides gio. Similarly we obtain that ti + t~ divides the other g's in the expression of f . By Proposition 1 f is an ordinary polynomial identity for M n ( K ) and we apply Proposition 2 to see that

H - t q ) divides all This the "if" of the (tp g's. completes part l < p < q < _ n + l

theorem.

(ii) The proof of the "only if" part of the theorem follows imme- diately from the considerations in the corresponding part of the proof of Theorem 2, where we have seen that the division by ti-4-tj of each g in the expression of f implies that each v ( g ) ( a , rl . . . . , rn) = 0, as desired.

REFERENCES

[1] G. M. Bergman, Wild automorphisms of free P.L algebras and some new identities, preprint.

[2] N. Bourbaki, Groupes et Algkbres de Lie, Chs. VII, VIII Hermann, Paris, 1975.

[3] M. Domokos, GrObner bases, determinantal ideals and identities of matrices, in preparation.

162 "I'St-~'SKA RASHKOVA - VESSELIN DRENSKY

[4] V. Drensky, Ts. Rashkova, Weak polynomial identities for the matrix algebra, Commun. in Algebra 21 (1993), 3779-3795.

[5] E. Formanek, Central polynomials for matrix rings, J. Algebra 23 (1972), 129- 132.

[6] A. Giambruno, A. Valenti, On minimal *-identities of matrices, Linear Multi- lin. Algebra 39 (1995), 309-323.

[7] E Halpin, Central and weak identities for matrices, Commun. in Algebra 11 (1983), 2237-2248.

[8] Yu. E Razmyslov, On a problem of Kaplansky (Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 37 (1973), 483-501. Translation: Math. USSR Izv. 7 (1973), 479-496.

Pervenuto il 9 setlembre 1997.

Centre of Mathematics, University of Rousse "A. Kanchev" Studentska Str. 8, 7017, Rousse, Bulgaria

e-mail: tcetcka@ami.ru.acad.bg

Institute of Mathematics and lnformatics, Bulgarian Academy of Sciences

Acad. G. Bonchev Str., block 8, 1113, Sofia, Bulgaria e-mail: drensky@ banmatpc.math.acad.bg