《BCN graded Lie algebras arising from fermionic representations.doc》由会员分享，可在线阅读，更多相关《BCN graded Lie algebras arising from fermionic representations.doc（30页珍藏版）》请在三一办公上搜索。
1、精品论文大全BCN -graded Lie algebras arising from fermionic representationsHongjia Chen and Yun GaoDepartment of Mathematics, USTC (230026)9AbstractNWe use fermionic representations to obtain a class of BC-gradedLie algebras coordinatized by quantum tori with nontrivial central extensions.Keywords: Lie al
2、gebra graded by finite root systems of BC type; Quantum tori;Fermionic and bosonic representations; Central extensions0 IntroductionLie algebras graded by the reduced finite root systems were first intro- duced by Berman-Moody BM in order to understand the generalized in- tersection matrix algebras
3、of Slodowy. BM classified Lie algebras graded by the root systems of type Al, l 2, Dl, l 4 and E6 , E7, E8 up to central extensions. Benkart-Zelmanov BZ classified Lie algebras graded by the root systems of type A1 , Bl, l 2, Cl, l 3, F4 and G2 up to central extensions.Neher N gave a different appro
4、ach for all the (reduced) root systems exceptE8, F4 and G2. The main idea of root graded Lie algebras can be traced back to Tits T and Seligman S.ABG1 completed the classification of the above root graded Lie algebras by figuring out explicitly the centers of the universal coverings of those root gr
5、aded Lie algebras. It turns out that the classification of those root graded Lie algebras played a crucial role in classifying the newly developed extended2laffine Lie algebras. All affine Kac-Moody Lie algebras except A(2)are im-portant examples of Lie algebras graded by reduced finite root systems
6、. To2linclude the twisted affine Lie algebra A(2)and for the purpose of the clas-sification of the extended affine Lie algebras of non-reduced types, ABG2Research was partially supported by NSERC of Canada and Chinese Academy ofScience.精品论文大全2lintroduced Lie algebras graded by the non-reduced root s
7、ystem BCN . BCN - graded Lie algebras do appear not only in the extended affine Lie algebras (see AABGP) including the twisted affine Lie algebra A(2) but also in the finite-dimensional isotropic simple Lie algebras studied by Seligman S. Theother important examples include the “odd symplectic” Lie
8、algebras studied by Gelfand-Zelevinsky GeZ, Maliakas Ma and Proctor P.The Cliffold(or Weyl) algebras have natural representations on the ex-terior(or symmetric) algebras of polynomials over half of generators. Those representations are important in quantum and statistical mechanics where the generat
9、ors are interpreted as operators which create or annihilate par- ticles and satisfy Fermi(or Bose) statistics.Fermionic representations for the affine Kac-Moody Lie algebras were first developed by Frenkel F1 and Kac-Peterson KP independently. Feingold-Frenkel FF systematically con- structed represe
10、ntations for all classical affine Lie algebras by using Clifford or Weyl algebras with infinitely many generators. G constructed bosonicand fermionic representations for the extended affine Lie algebragC ),lN ( qwhere Cq is the quantum torus in two variables. Thereafter Lau L gave a more general con
11、struction.In this paper, we will construct fermions depending on the parameter q which will lead to representations for some BCN -graded Lie algebras coordi- natized by quantum tori with nontrivial central extensions. Since CN -graded Lie algebras are also BCN -graded Lie algebras we will treat boso
12、ns as well in a unified way.The organization of the paper is as follows. In Section 1, we review the definition of BCN -graded Lie algebras and give examples of BCN -graded Liealgebras which are subalgebras of g C ) or gl C ). In Section 2, wel2N ( q2N +1( quse fermions or bosons to construct repres
13、entations for those examples of BCN -graded Lie algebras by using Clifford or Weyl algebras with infinitely many generators. Although we get BCN -graded Lie algebras with the grading subalgebras of type BN , CN and DN , there is only one which is a genuine BCN -graded Lie algebra arising from the fe
14、rmionic construction.Throughout this paper, we denote the field of complex numbers and thering of integers by C and Z respectively.1 BCN -graded Lie AlgebrasIn this section, we first recall the definition of quantum tori and BCN - graded Lie algebras. We then go on to provide some examples of BCN -g
15、raded Lie algebras. For more information on BCN -graded Lie algebras, see ABG2.Let q be a non-zero complex number. A quantum torus associated to q(see M) is the unital associative C-algebra Cq x, y (or simply Cq ) with generators x,y and relations(1.1) xx1 = x1 x = yy1 = y1y = 1andyx = qxy.Then(1.2)
16、 xmynxpys = qnpxm+pyn+sand(1.3) Cq = Xm,nZCxm yn.Set (q) = n Z|qn = 1. From BGK we see that Cq , Cq has a basis consisting of monomials xmyn for m / (q) or n / (q).Let be the anti-involution on Cq given by(1.4)x = x,y = y1.We have Cq = C+ C, where C = s Cq |s = s, thenqqqC+m n m nZ, n 0,(1.5)q = spa
17、nx y+ x y|m C m nq = spanx y xmyn|m Z, n 0.Now we form a central extension of glr(Cq ) (cf. G),glr(1.6) Cq )= glr(Cq ) Xn(q)Cc(n) Ccywith Lie bracketeij (xmyn), ekl(xpys) = jkqnpeil (xm+pyn+s) il qmsekj (xm+pyn+s)(1.7)+mqnpjk ilm+p,0n+s,0 c(n + s)+nqnpjk ilm+p,0n+s,0cyfor m, p, n, s Z, where c(u), f
18、or u (q) and cy are central elements ofglr(Cq ), t means t Z/(q), for t Z.Next we recall the definition of BCN -graded Lie algebra and constructthree types of BCN -graded Lie algebras. Let(1.8)B = i j |1 i = j N i|i = 1, , N C = i j |1 i = j N 2i|i = 1, , N D = i j |1 i = j N .be root systems of typ
19、e B,C and D respectively, and(1.9) = i j |1 i = j N i, 2i|i = 1, , N be a root system of type BCN in the sense of Bourbaki Bo, Chapitre VI.Definition 1.1 (BCN -graded Lie Algebras) A Lie algebra L over a fieldF of characteristic 0 is graded by the root system BCN or is BCN -graded if(i) L contained
20、as a subalgebra a finite-dimentional split “simple” Lie al-Xgebra g = h Lg whose root system relative to a split Cartansubalgebra h = g0 is X , X=B,C, or D;(ii) L =L0L, where L = x L|h, x = (h)x, for all h h for 0, and is the root system BCN as in (1.9); and(iii) L0 = PL, L.In Definition 1.1 the wor
21、d simple is in quotes, because in every case but two the Lie algebra g associated with X is simple; the sole exceptions beingwhen X = D2 or D1. The D2 root system is the same as A1 A1 , and gis the sum g = g(1) g(2) of two copies of sl2 in this case. In the D1 case,g = Fh, a one-dimensional subalgeb
22、ra.We refer to g as the grading subalgebra of L, and we say L is BCN -gradedwith grading subalgebra g of type XN (where X = B, C, or D) to mean thatthe root system of g is of type XN .Any Lie algebra which is graded by a finite root system of type BN , CN ,or DN is also BCN -graded with grading suba
23、lgebra of type BN , CN , or DNrespectively. For such a Lie algebra L, the space L = (0) for all not inB , C , or D respectively.1.1 Type C and DFor BCN -graded Lie algebras with grading subalgebra of type CN ( =1) and DN ( = 1), we putG = 0ININ0 M2N(Cq ).Then, G is an invertible 2N 2N matrix and G t
24、 = G. Using the matrixG, we define a map : M2N (Cq ) M2N (Cq ) by A = G1AtG.Since G t = G, is an involution of the associative algebra M2N (Cq ). As inAABGP, we defineS(M2N (Cq ), ) = A M2N (Cq ) : A = Ain which case S(M2N (Cq ), ) is a Lie subalgebra of gl2N (Cq ) over C. The general form of a matr
25、ix in S(M2N (Cq ), ) is(1.10) A STAtwith St= S andTt= Twhere A, S, T MN (Cq ). Then the Lie algebraG = S(M2N (Cq ), ), S(M2N (Cq ), )is a BCN -graded Lie algebra with grading subalgebra of type CN ( = 1)and DN ( = 1). Using the method in AABGP, we easily know thatG = Y S(M2N (Cq ), )|tr(Y ) 0 mod Cq
26、 , Cq .We putN,(1.11) H = nX ai(eii eN +i,N +i)|ai Coi=1then H is a N -dimensional abelian subalgebra of G. Defining i H, i =1, , N , by(1.12) i NX aj (ejj eN +j,N +j )j=1= aifor i = 1, , N. Putting G = x G|h, x = (h)x, for all h H as usual, we have(1.13) G = G0 X G X(G + G ) X(G2 G2 )whereiji=jijij
27、ij iiimGi j = spanCfij (m, n) = xyneij xmyneN +j,N +i|m, n Z,mmGi +j = spanCgij (m, n) = xynei,N +j xmynej,N +i|m, n Z,(1.14)Gi j = spanC h ij (m, n) = xyneN +i,j xmyneN +j,i|m, n Z,mnG2i = spanC gii(m, n) = (x y xmyn)ei,N +i|m, n Z,mnG2i = spanC h ii(m, n) = (x y xmyn)eN +i,i|m, n Z,andG0 = spanCfi
28、i(m, n)f11 (m, n), f11(p, s)|2 i N, m, n Z, p / (q) or s / (q).Note that gij (m, n) = qmngji(m, n), h ij (m, n) = qmnh ji(m, n).Now we form a central extension of G(1.15)Gb = G Xn(q)Cc(n) Ccywith Lie brackets as (1.7).We haveProposition 1.1(1.16) gij (m, n), gkl(p, s) = 0 (1.17)gij (m, n), fkl(p, s)
29、 = il qmsgkj (m + p, n + s) + jlq(sn)mgki(m + p, s n)gij (m, n), h kl(p, s)= ik qn(m+p)fjl(m + p, s n) + jkqnpfil(m + p, n + s)(1.18)+ilq(mn+np+ps)fjk(m + p, (n + s) jlq(ns)pfik (m + p, n s)+mqnpjk ilm+p,0n+s,0(c(n + s) + c(n s)mik jlm+p,0ns,0(c(n s) + c(s n)fij (m, n), fkl(p, s) = jkqnpfil(m + p, n
30、 + s) ilqsmfkj (m + p, n + s)(1.19)+2mqnpjkil m+p,0n+s,0c(n + s)(1.20)fij (m, n), hkl(p, s) = ik qn(m+p) h jl(m + p, s n) ilqmsh kj (m + p, n + s)(1.21) h ij (m, n), h kl(p, s) = 0for all m, p, n, s Z and 1 i, j, k, l N .Proof. We only check (1.18).gij (m, n), h kl(p, s)= xmynei,N +j xmynej,N +i, xp
31、yseN +k,l xpyseN +l,k = xmynei,N +j , xpyseN +k,l xm ynei,N +j , xpyseN +l,k xmynej,N +i, xpyseN +k,l+xmynej,N +i, xpyseN +l,k = jkxmynxpyseil ilxpysxmyneN +k,N +j + mqnpjkil m+p,0n+s,0c(n + s) jlxmynxpyseik kixpysxmyneN +l,N +j + mjlik m+p,0ns,0c(n s) ik xmynxpysejl lj xpysxmyneN +k,N +i + mjlik m+
32、p,0ns,0c(s n) + ilxpysxmynej k kj xmynxpyseN +l,N +i + mqnpjlik m+p,0n+s,0c(n s) +njkil m+p,0n+s,0cy njlik m+p,0ns,0cy + njlik m+p,0ns,0cynjk ilm+p,0n+s,0cy= ik qn(m+p)fjl(m + p, s n) + jk qnpfil(m + p, n + s)+ilq(mn+np+ps)fjk (m + p, (n + s) jlq(ns)pfik (m + p, n s)+mqnpjkilm+p,0n+s,0(c(n + s) + c(
33、n s)mik jlm+p,0ns,0(c(n s) + c(s n).The proof of the others is similar. 1.2 Type BFor type B, we put 100 G = 00IN0 IN0 M2N +1 (Cq ).Then, G is an invertible (2N + 1) (2N + 1)-matrix and Gt = G. Using the matrix G, we define a map : M2N +1 (Cq ) M2N +1 (Cq ) by A = G1AtG.Since G t = G, is an involuti
34、on of the associative algebra M2N +1 (Cq ). As in AABGP, we defineS(M2N +1 (Cq ), ) = A M2N +1 (Cq ) : A = Ain which case S(M2N +1(Cq ), ) is a Lie subalgebra of gl2N +1(Cq ) over C. The general form of a matrix in S(M2N +1 (Cq ), ) isab1b2t(1.22) b2A S with a = aSt = S andTt = T1b t TAtwhere A, S,
35、T MN (Cq ). Then the Lie algebraG = S(M2N +1 (Cq ), ), S(M2N +1(Cq ), )is a BCN -graded Lie algebra with grading subalgebra of type BN . Following from AABGP, we easily know thatG = Y S(M2N +1 (Cq ), )|tr(Y ) 0 mod Cq , Cq As in Section 1.1, we setN(1.23) H = nX ai(eii eN +i,N +i)|ai Co,i=1then H is
36、 a N -dimensional abelian subalgebra of G. Defining i H, i =1, , N , by(1.24) i lX aj (ejj eN +j,N +j )j=1= aifor i = 1, , N. Putting G= x G|h, x = (h)x, for all h H asusual, we have(1.25) G = G X GX(GG)X(G GGG )0i=ji jiji +ji jiii2i2iwherei j = spanCfij (m, n) = xy eij xy eN +j,N +i|m, n Z,G m n m
37、ni +j = spanCgij (m, n) = xy ei,N +j xy ej,N +i|m, n Z,G m n m ni j = spanCh ij (m, n) = xy eN +i,j xy eN +j,i|m, n Z,G m n m n(1.26)G2i= spanCgii(m, n) = (xmyn xmyn)ei,N +i|m, n Z,2i = spanC h ii(m, n) = (x y x y)eN +i,i|m, n Z,G m n m ni = spanCei(m, n) = xy ei,0 xy e0,N +i|m, n Z,G m n m ni = spa
38、nCei (m, n) = xy eN +i,0 xy e0,i|m, n Z,G m n m nandG0 = spanCfii(m, n)e0 (m, n), e0(p, s)|1 i N, m, n Z, p / (q) or s / (q),where e0(m, n) = (xmyn xmyn)e0,0.Next we form a central extension of G(1.27)Gb = G Xn(q)Cc(n) Ccywith Lie brackets as (1.7).Remark 1.1 Note that the index of the matrices in M
39、2N +1 (Cq ) ranges from0 to 2N .Now we haveProposition 1.2(1.28) gij (m, n), gkl(p, s) = 0 (1.29)gij (m, n), fkl(p, s) = ilqmsgkj (m + p, n + s) + jlq(sn)mgki(m + p, s n)(1.30)gij (m, n), h kl(p, s)= ik qn(m+p)fjl(m + p, s n) + jkqnpfil(m + p, n + s)+ilq(mn+np+ps)fjk(m + p, (n + s) jlq(ns)pfik (m + p, n s)+mqnpjkil m+p,0n+s,0(c(n + s) + c(n s)mik jlm+p,0ns,0(c(n s) + c(s n)(1.31) gij (m, n), ek (p, s) = 0(1.32)kgij (m, n), e (p, s) = ik qn(m+p)ej (m + p, s n) + jkqnpei(m + p, n + s)(1.33) gij (m, n), e0 (p, s) = 0fij (m, n), fkl(p, s) = jkqnpfil(m + p, n + s) ilqsmfkj
