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On Invariants of F4 and the Center of the Albert Algebra

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On Invariants of F4 and the Center of the Albert Algebra JOURNAL OF ALGEBRA 179, 838]851Ž. 1996 ARTICLE NO. 0039 On Invariants of F4 and the Center of the Albert Algebra A. V. Iltyakov and I. P. Shestakov Institute of Mathematics, No¨osibirsk, 630090, Russia* Communicated by Georgia Benkart Received October 26, 1994 1. INTRODUCTION It is known that some simple algebraic groups can be represented as groups of automorphisms of simple algebras. For example, the group PSLk is isomorphic to AutŽ.Mkk, where M is the algebra of k = k-matrices; the exceptional groups G24and F can be represented as AutŽ.O and Aut Ž.A , respectively, where O is a split Cayley algebra and A is a split Albert algebra, i.e., the algebra of Hermitian 3 = 3-matrices over O, with the symmetric multiplication a( b s 1r2Ž.ab q ba . Let G s AutŽ.A for some algebra A over a field F. Consider the diagonal action of G on the direct sum nA of n copies of A, which can be nm considered as an affine space F , m s dimŽ.A . It induces naturally an action of AutŽ.A on the field FnA Ž .of rational functions on nA, i.e., the field of rational functions on A in n vector variables. One of the main problems concerning the structure of the invariant field FnAŽ.Gis whether this field is purely transcendental over F wx1.If F is an algebraically closed field of characteristic zero and A is a central simple algebra, then the field G FnAŽ.coincides with the quotient field KAnŽ.of the centroid of the algebra FnŽ.A generated over F by n generic elements of A wx4Ž a ``free affine algebra of type A,'' in notations ofwx 8. , so, the problem of rational- G ity of FnAŽ.reduces to that of KAnŽ.. Ž. Ž. It is known that the field KAnkis rational over F if A s M s MFk, kF42,3,orwxAsO wx8 ; so the fields of rational invariants of corre- sponding representations of PSLk , k F 4, and G2 are rational. The main result of this paper is that K nŽ.A is also rational over F in the case of the Albert algebra A for any field F of characteristic / 2, 3. * During 1993r94 Becarios de FICYT at the University of Oviedo, Asturias, Spain. 838 0021-8693r96 $12.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. INVARIANTS OF F4 839 In particular, if F is algebraically closed field of characteristic zero, the Ž.G Ž. field of invariants FnA is rational over F for G s Aut A of type F4. Moreover, the subspace A 0 of zero trace elements from A is known to be an irreducible G-module of minimal dimensionwx 6, p. 382 , and we obtain as a corollary of the above result that the field of rational invariants of a minimal representation of a simple algebraic group of type F4 in several variables is rational over F. Ž. Ž Ž .. The field K113A , KMF is generated by traces of a generic ele- ment and, therefore, it is rational; the rationality of K2Ž.A is proved inwx 9 . Moreover, there it is also proved that K nŽ.A is purely transcendental over K33Ž.A, so it suffices to prove the rationality of K Ž.A . We do it by giving an explicit list of free generators for this field. 2. PRELIMINARY RESULTS Let us recall some properties of Cayley]Dickson and Albert algebras Žseewx 6, 11. The elements of a split Cayley]Dickson algebra O can be written in the form a ¨ , ž/¨*b 3 where a, b g F and ¨, ¨* g F s F = F = F. The multiplication is defined by the rule a11¨ a 22¨ UU? ž/ž/¨11b ¨ 22b U U U aa12qŽ.¨ 1,¨ 2 a12¨ qb 21¨ y¨ 1=¨ 2 s UU U , ž/a21¨qb 12¨q¨ 1=¨¨ 2Ž. 1,¨ 2qbb 1 2 3 where Ž.¨, w , ¨ = w are the scalar and vector product in F , respectively. The algebra O is equipped with the involution a ¨ b y¨ s . ž/ž/¨*b y¨*a Observe that the trace tŽ.x of a matrix x g O equals x q x g 1 ? F , F. Let O O F be the algebra of 3 3-matrices over O.Itis 3s mF 3 = equipped with the involution a m b ª a m bii, where b ª b is the matrix transposition. The subspace HŽ.O3 of symmetric elements with the 840 ILTYAKOV AND SHESTAKOV multiplication a( b s 1r2Ž.ab q ba is a central simple Jordan algebra A of dimension 27, which is called aŽ. split Albert algebra. Ž Ä4.Ž Let eiiiijs e , ei,jg1, 2, 3 , i / j be matrix units of O3we identify . 1mF33with F . Then an arbitrary element of A can be written in the form 3 csÝÝeiiaq wxc ijk , is1 Ž.ijk Ž. where aig F, c i g O, wxc jk means c ? ejkq c ? e k j, and ijk runs over the subgroup A3 of even permutations. Denote by trŽ.c the trace of c, that is a123q a q a . We frequently use the following multiplication rules Ž.i, j, k are distinct : 11 wxaij( wxb jk s22 wab xik; wxa ij( wxb ij s tŽ.Žab eijq e .. It is easy to check that the trace trŽ.Rc of the linear transformation Ž. Rc :xªx(cof the vector space A equals to 9 ? tr c for any c g A. A permutation of the idempotents e123, e , e induces an automorphism of A. The set of such automorphisms is a subgroup of G s AutŽ.A , which Ž. is isomorphic to S3. Further, an automorphism f g Aut O defines an Ž. automorphism of O3 by a ? eij ª f a ? eij. Its restriction on A is an automorphism of A. So, we can assume that AutŽ.O is embedded in AutŽ.A under this map. Let s : Ž.Ža ¨ b y¨ * .. It is easy to see that s AutŽ.O . We set ¨ * b ª y¨ a g Ä4 Hsgr S33, s . Since sf s fs for each f g S , this group is isomorphic to S32= Z . In particular, <<H s 12. Ž. Recall that a free Albert algebra Bnns F A of rank n G 3 is generated over F by generic elements Ž.j s 1,...,n 27 X a x A FYŽ.Ž n. ,1Ž. jiis Ým jFg m is1 Žn.Ä4 where a127,...,a is a basis of A and Y s xij N i s 1, . , 27; j s 1,...,n is a set of variables. The algebra A is generated by three elementswx 11, p. 58, Ex. 12 . Hence, by Proposition 2.1 inwx 8Ž see also Lemma 2 inwx 4. the center Cn of the Žn. algebra Bnnnlies in FYŽ .. Besides, if K is the quotient field of C , then BKnnis a central simple algebra over K nof dimension 27. Note that the algebra BKnnhas unit and, therefore, K ncoincides with the quotient field of the centroid of Bn. Let b B . Then, since it is an element of the Albert algebra A g n m F Ž Žn..ŽŽn..Ž.ŽŽn..Ž. FY over FY , the trace tr b g FY . It turns out that tr b g K n INVARIANTS OF F4 841 wx9, Section 1.2 . Moreover, in the same way one can check that for any central simple algebra B A FYŽ Žn..Žover a subfield K FYŽn..of : m F : dimension 27 the trace trŽ.b of an element b g B lies in K. We will show that X12, X can be transformed to a special form. Let us take a set of variables U X s Ä4xiiiijijikik, y , z ; y , z ; ¨ , ¨ N i, j, k s 1,2,3; i / j Ž.UŽUUU. and put ¨iis ¨ 1, ¨i2, ¨i3, ¨iis ¨ 1, ¨i2, ¨i3. Consider the elements from A A FXŽ., smF 3 xsÝexii, is1 3 ysÝÝeyiiq wxy ijk , is1 Ž.ijk 3 zsÝÝeziiq wxzijk , is1 Ž.ijk yjk 0 zjk¨ i where y Ž., z Ž.U are from O FXŽ.. These elements iis 0 ykj s ¨ ikjz m F turn to be free generators of B3 Žcf.wx 2. LEMMA 1. The algebra alg F Ä4x, y, z is isomorphic to B3. Proof. Let K FYŽ Ž2.., consider the subalgebra B ? K A A s 2 : KFs m 3Ž.2 Ž. Ž. Ž. K. Since X111121131q tXXqtXXqtXs0, where tXi1gKand tX31Ž./0wx 6, p. 232 , the unit of A lies in this subalgebra. Hence, by w 6, p. 363, Ex. 2xwBK2 is a central simple algebra of dimension 9. From 8, Lemma 1.3x it follows that BK22is a central simple Jordan algebra over K22of dimension 9 and K : K. ŽŽ3.. Let P be the algebraic closure of the field FY . Then B3 : A P s A P and K P. The algebra BK Pis a central simple one mmF22: 2K2 Ž.Žq. over P wx11, p. 137 ; therefore, it is isomorphic to BP23,MP s MFŽ.Žq. P A 6, p. 204 . 3 m FP: wx Ž. Ž . Ž.Žq. Bywx 6, p. 389, Ex.3 there is f g Aut A P such that f BP23sMP and fŽ.X1 has a diagonal form. Hence, there is an epimorphism from Ä4Ž. Ž.Ž.Ž. alg F x, y, z to f B3, which sends x ª f X 123, y ª f X , z ª f X .
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