On Involutions which Preserve Natural Filtration

Transcription

1 Proceedigs of Istitute of Mathematics of NAS of Ukraie 00, Vol. 43, Part, O Ivolutios which Preserve Natural Filtratio Alexader V. STRELETS Istitute of Mathematics of the NAS of Ukraie, 3 Tereshchekivska Str., Kyiv-4, Ukraie sav@imath.kiev.ua I this work we study ivolutios i fiitely preseted -algebras which preserve the atural filtratio. 1 Itroductio Itroducig additioal structures is ofte useful i a study of algebraic objects, i particular fiitely preseted algebras ad their represetatios, itroducig topology i algebras gives a comprehesive theory of Baach algebras or, more geerally, a theory of locally covex algebras; itroducig a ivolutio, which we ca cosider as some ier symmetry, calls ito beig the theory of -algebras; cosiderig a ivolutio together with the correspodig orm gives the theory of C -algebras. Moreover, o the oe had, studyig ot all represetatios but oly those which coserve this additioal structure (for example, -represetatios) is simple (for example, -represetatios are idecomposable if ad oly if they is irreducible, see [1]) o the other had, this is ofte sufficiet for applicatios. I [1] the theory of -represetatios of fiitely preseted -algebras is studied, ad the ivolutio i the cosidered -algebras ofte preserves filtratio (see Defiitio 1). I this article we cosider the followig questio. Let F be a free algebra with geerators x 1,...,x ad a idetity e, ad lets us also have a uital fiitely preseted algebra A = C x 1,...,x q 1 =0,...,q m =0, where q k F, k =1,...,m. We ca assume, without loss of geerality, that all relatios q k are oliear, for otherwise, the algebra A is isomorphic to a algebra with a smaller umber of geerators (roughly speakig, we ca exclude geerators that are liear combiatios of the others). We will deote by V (A) the liear subspace of A geerated by the elemets x 0 = e, x 1,...,x. The the questio is how may ivolutios which map V (A) ito itself exist i the algebra A such that the correspodig -algebras are ot -isomorphic. The aswer is that such a ivolutio is uique ad so we ca always suppose that the geerators are self-adjoit (see Theorem 1 ad Propositio 1). Moreover, i some cases there is a -isomorphism betwee the correspodig -algebras such as it coserves the relatios (see Theorem 1 ad examples). Mai result We will deote the free -algebra with self-adjoit geerators z k by F. Some other ivolutio will be deoted by. It is give by defiig its values o geerators. We will deote the free -algebra with such a ivolutio by F = C x 1,...,x x k = p k,k =1,...,, where p k F.

2 O Ivolutios which Preserve Filtratio 491 Defiitio 1. We say that a ivolutio of a -algebra A preserves the atural filtratio iff the ivolutio maps V(A ) ito itself. Theorem 1. Let a ivolutio of the -algebra F preserve the atural filtratio. The there is a -isomorphism ϕ : F F. Moreover, ϕ(v (F )) = V (F ). Proof. We ca assume that the first l geerators are self-adjoit ad the others are ot, such otherwise, we ca reumber the geerators. We will prove the theorem by iductio o the umber l of the geerators that are ot self-adjoit. If l = 0 the there is othig to prove, sice all the geerators are self-adjoit. Let 1 l. Put y k = x k + x k, k =0,...,. It is evidet that y k = y k. Because the ivolutio preserves the filtratio, x k V (F )adso y k V (F ). If y 0 = e, y 1,...,y are liearly idepedet the we defie ϕ : F F o the geerators by ϕ(z k )=y k,k=0,...,, z 0 = e. Sice dimv (F )= + 1 ad y 0,y 1,...,y are liearly idepedet ad lie i V (F ), y k, k =0,...,, is a liear basis of V (F )adso x k = α j k y j, α j k C. j=0 The the homomorphism iverse to ϕ is defied o the geerators by ϕ 1 (x k )= α j k z j. j=0 So ϕ is a isomorphism of the algebras F ad F. It is evidet that ϕ is also a -homomorphism ad ϕ(v (F )) = V (F ). Let ow y 0 = e, y 1,...,y be liearly depedet. The, sice the first l geerators are self-adjoit, y j = x j for j =0,..., l ad, cosequetly, y j are liearly idepedet. The there exists k ( l<k ) such that λ j y j, λ j C. Ad sice y j are self-adjoit, λ j y j, λ j C. If we put a j =(λ j + λ j )/ the we get a j y j, a j R. Reumberig the geerators we ca suppose that k = l +1. Put F 1 = C v 1,...,v v 1 j = v j,j =1,...,k,v 1 j = q j,j >k,

3 49 A.V. Strelets where q j = p j v 1,...,v k 1, iv k + a j v j,v k+1,...,v, j > k. Defie ψ : F 1 F o geerators by the formula ψ(v j )=x j, if j k, ad ψ(v k )= i x k a j x j. It is evidet that ψ is a isomorphism of the algebras F 1 ad F. Let us show that ψ is a -homomorphism. If j<k, the ψ(v j ) = x j = x j = ψ(v j )=ψ(v 1 j ). If j>k, the ψ(v j ) = x j = p j ad agai ψ(v 1 j )=ψ(q j)=ψ p j v 1,...,v k 1, iv k + a j v j,v k+1,...,v Fially, = p j x 1,...,x k 1,x k a j x j + a j x j,x k+1,...,x = p j = ψ(v 1 j ). ψ(v k ) = i x k a j x j ad ψ(v 1 k )=ψ(v k)= i x k a j x j. So ψ(v 1 k ) ψ(v k) = i y k a j y j =0, i.e., ψ(vk )=ψ(v k). We have proved that F ad F 1 are -isomorphic. Further, by the defiitio of ψ we agai that are ot self-adjoit ad is -isomorphic to F ad, cosequetly, F is -isomorphic have ψ(v (F 1 )) = V (F ). Ad ow we have l 1 geerators i F 1 so, by the iductive assumptio, F 1 to F. 3 Corollary ad examples I this sectio we will obtai a corollary of Theorem 1 ad cosider some examples. Cosider the -algebra A = C x 1,...,x x k = p k,k =1,...,, r 1 =0,...,r m =0, where r k F, k =0,...,m. LetIbea -ideal geerated by r 1,...,r m, i.e., A is a -isomorphic to the factor F /I. By icreasig the umber of geerators (ot more tha two times) ad addig ew relatios we always ca costruct a -algebra which is -isomorphic to A such that its geerators are self-adjoit. The corollary of Theorem 1 claims that if the ivolutio is good the we ca leave the umber of the geerators ad relatios the same as i A ad the legth of words i the relatios does ot grow.

4 O Ivolutios which Preserve Filtratio 493 Propositio 1. Let the ivolutio preserves the filtratio. The the -algebra A is - isomorphic to the -algebra B = C z 1,...,z z k = z k,k =1,...,, s 1 =0,...,s m =0, where s k have the same degrees as r k, k =1,...,m. Proof. Sice the ivolutio preserves the filtratio the, there exists a -isomorphism ϕ : F F. DeotebyJ=ϕ(I) the -ideal geerated by the relatios s 1 = ϕ(r 1 ),...,s m = ϕ(r m ). It is evidet that so defied s k have the same degrees as r k. The we ca put B = F /J. Let i be a ijectio of I ito F ad π a projectio of the latter ito A. Similarly, let i 0 be a ijectio of J ito F ad π 0 a projectio ito B. The restrictio of ϕ to I will be deoted by ϕ 0. The we get a commutative diagram of -homomorphisms, i 0 I F A 0 ψ 0 J ϕ 0 ϕ π i 0 F π 0 B 0 where ψ is defied by the formula ψ(π(a)) = π 0 (ϕ(a)), for ay a F. Now we show that ψ is well-defied. Ideed, sice π is surjective, ψ is defied for all elemets of A. Ifπ(a) = 0 the a Iadsoϕ(a) J, cosequetly, ψ(π(a)) = π 0 (ϕ(a)) = 0. It is evidet that ψ is surjective. Now we show that it is ijective.ideed, if ψ(π(a)) = 0, the it meas that π 0 (ϕ(a)) = 0 ad so ϕ(a) J, cosequetly, a I, from where we get π(a) =0. It is also evidet that ψ is a -homomorphism. So we have costructed a -isomorphism of the -algebras A ad B. Actually we have chaged the geerators i A so that the ew geerators are self-adjoit. But the ext example shows that, geerally speakig, the relatios are chaged too. Example 1. Cosider the -algebra Q = C q 1,q q 1 = q,q = q 1,q 1 = q 1,q = q. A -isomorphism ϕ : F F is defied by the formulas The similarly ϕ(q 1 )=z 1 + iz, ϕ(q )=z 1 iz. ϕ(q 1 q 1 )=(z 1 + iz ) z 1 iz = z 1 z + i{z 1,z } z 1 iz, ϕ(q q )=z 1 z i{z 1,z } z 1 + iz, where {, } is the aticommutator. It is evidet that the ideal geerated by these relatios is also geerated by the relatios z 1 z = z 1 ad {z 1,z } = z. So Q is -isomorphic to the -algebra C z 1,z z 1 = z 1,z = z,z 1 z = z 1, {z 1,z } = z 1. O the other had, it is ot difficult to show that there is o -isomorphisms betwee Q ad the -algebra C x 1,x x 1 = x 1,x = x,x 1 = x 1,x = x.

5 494 A.V. Strelets The ext two examples show that there are algebras that are ot free for which a aalogue of Theorem 1 is also true. Example. Cosider the -algebra of polyomials i variables, P. It is a factor of the free algebra by the ideal I geerated by the relatios [x j,x k ]=0, j,k =1,..., where [, ] is the commutator. All elemets of the ideal I ca be writte as [p 1,p ], where p 1,p F. The, for ay ivolutio i F,[p 1,p ] =[p,p 1 ] IsoIisa -ideal. Let preserves the filtratio. The the -ideal ϕ(i) cosists of all elemets which ca be writte as [ϕ(p 1 ),ϕ(p )]. So it is geerated by the relatios [z j,z k ]=0, j,k =1,..., Ad we have the -isomorphism of P ad P. Example 3. Cosider oe more algebra for which a theorem aalogous to Theorem 1 holds. Let A = C p, q [[p, q],p]=0, [[p, q],q]=0. Let I be a ideal geerated by the correspodig relatios. The it is evidet that for ay a, b, c V (F )wehave[[a, b],c] I. Now, let us itroduce i A a ivolutio which preserves the filtratio. Let us show that the ideal I is a -ideal, [[p, q],p] =[p, [p, q]] =[[p, q],p ]=[[q,p ],p ], but p,q V (F )so[[p, q],p] I. Similarly, [[p, q],q] I. Sice preserves the filtratio, by Theorem 1 there is a -isomorphism ϕ : F F ad there exist elemets a 1,a V (F ) such that ϕ(a 1)=z 1 ad ϕ(a )=z, where z 1 ad z are geerators of F. The the -ideal ϕ(i) is geerated by the relatios [[z 1,z ],z 1 ]=0, [[z 1,z ],z ]=0. So we have a -isomorphism of A ad the -algebra C z 1,z z 1 = z 1,z = z, [[z 1,z ],z 1 ]=0, [[z 1,z ],z ]=0. [1] Ostrovskyĭ V.L. ad Samoĭleko Yu.S., Itroductio to the theory of represetatios of fiitely preseted -algebras. I. Represetatios by bouded operators, Rev. Math. ad Math. Phys., 1999, V.11, 1 61.