《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）》请在三一文库上搜索。
1、精品论文大全http:/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:
2、Lie algebra 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 alg
3、ebras 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
4、 approach 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 r
5、oot graded 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 s
6、ystems. 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
7、root system 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
8、” Lie 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 g
9、enerators 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 r
10、epresentations 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 gener
11、al construction.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 trea
12、t bosons 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
13、representations 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
14、the fermionic 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
15、BCN -graded 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.The
16、n(1.2) 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
17、 = spanx 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
18、(u), for 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
19、of type 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 cont
20、ained 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 t
21、he word 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 su
22、balgebra.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 gradin
23、g subalgebra 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 a
24、nd G t = 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
25、a matrix 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
26、mod Cq , 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
27、=jijijij 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 = s
28、panCfii(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
29、(p, s) = 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
30、+ p, n + 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
31、+i, xpyseN +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 + mj
32、lik m+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
33、) + c(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 in
34、volution 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
35、A, S, 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=1the
36、n H is 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
37、m n m 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
38、 = spanCei (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 matrice
39、s in M2N +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