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Spin And. Wedge Representations of Infinite-Dimensional Lie. Algebras

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Spin And. Wedge Representations of Infinite-Dimensional Lie. Algebras Proc. NatL Acad. Sci. USA Vol. 78, No. 6, pp. 3308-3312, June 1981 Mathematics Spin and. wedge representations of infinite-dimensional Lie. algebras. and groups (Clifford algebra/spin representation/affine Kae-Moody Lie algebra/highest weight representation/line bundle) VICTOR G. KAC* AND DALE H. PETERSONt *Massachusetts Institute ofTechnology, Cambridge, Massachusetts 02139; and tUniversity of Michigan, Ann Arbor, Michigan 48109 Communicated by Bertram Kostant, November 24, 1980 ABSTRACT We suggest a purely algebraic construction ofthe o(V, U;4)) on the space s(V, U) by spin representation of. an infinite-dimensional orthogonal Lie al- gebra (sections 1 and 2) and a corresponding group (section 4). ovvu(a)[x + (CfV)U] = *na)x + xd + (COV)U. From this we deduce a construction ofall level-one highest-weight representations of orthogonal affmne Lie algebras in terms ofcre- Note that if a E Ofin(V) = CC2V, then *(a)x = ax - xa, and ation and annihilation operators on an infinite-dimensional Grass- we may take d = a, so that we obtain the usual definition (ref. mann algebra (section 3). We also give a similar construction of 2): ovvu(a)[x + (CtV)U] = ax + (CfV)U. Hence, on Ofin(V), OV U the level-one representations of the general linear affine Lie al- is the usual spin representation. gebra in an infinite-dimensional "wedge space." Along these lines Let a be the involutive automorphism ofCIV defined by a(x) we construct the corresponding representations of the universal - -x,x E V and CCV = Ce+v ff Cf V be the corresponding central extension of the group SL.(ktt,t-1]) in spaces of sections eigenspace decomposition. This induces the decomposition ofline bundles over infinite-dimensional homogeneous spaces (sec- s(V,U) = s+(V, U) s-(V,U) into adirect sum ofirreducible half- tion 5). spin representations, Ocvwu and -v As soon as the choice a a-+ d is made, we obtain a k-valued 1. Let V be a vector space over a field k with 2 # 0 (we do not two-cocycle y of the Lie algebra o(V, U;O0): assume that dim V < 00) and 4 be a nondegenerate k-valued symmetric bilinear form on V. Define Lie algebras (with the -Xa,b)Is(vmO = [oVu(a), ovu(b)] - avu([a,b]). usual bracket): To make this choice, suppose that U' is a subspace of U with o(V;4) = {a E Endk V 4(ax,y) + O(x,ay) = 0, x,y E V}, dim U/U' < 0 and that U" is a subspace ofV such that we have (nonorthogonal) direct sum decompositions V = U' (D U", V Ofin(V;4)) = {a E o(V;c) dim a(V) <oo}. = U" DU"'. Letp': V-i' U', p" : V- U"', p": V-> U", p"': V-k U"' be the associated projections. Then for a E o(V,U;O), Suppose U is an isotropic subspace of V satisfying we choose U"L = U; dim U'/U s 1. [1] a=pap [3] Introduce a Lie subalgebra of o(V;O): Then we have o(V,U;O) = {a E o(V;O) dim [U + a(U)]/U < x0}. = - trace p'(ap"b - bp"a)p". Y(a,b) 2 [4] We shall construct the projective spin representation a u of the Lie algebra o(V, U;O). Now suppose in addition that U"' is isotropic and that there Recall (ref. 2) that the Clifford algebra CeV associated to exists a (indexed) subset {u,: i E I} C U' and a basis {ui: i E I} (V, 4)) is an associative k-algebra with unit 1, defined as the quo- of U"' such that O(ui,,u) = SY. Let po be a projection of V along tient of the tensor algebra over V by the two-sided ideal gen- U' + U"'E onto the finite-dimensional subspace U" n U" of V. erated by elements of the form x 0& y + y 09 x - O(xy)1, xy Then for any a E o(V, U;O)), there exists a set {aj:j E J} C CfV E V. We identify V with a subspace of Cfv. Ifa FE o(V;O), the such that for any x E CfV, there exists a finite setJ(x) C J such action of a on V extends uniquely to a derivation *(a) of CfV, that ajx E (CfV)U forj V J(x), and o(a)[x + (CfV)U] = IjeJ(x) and ir is a representation of o(V;)) on Cfv. ajx + (CfV)U. In this case, we write ota) = Yjj aj. Then for Let Cf2V be the linear span in CeV of elements of the form a E o(V,U;4)), we have [x,y]: = xy - yx for x,y CE V. Then Cf2V is a Lie subalgebra of CeV. If a E Cf2V, x E V, then [a,x] E V, and x '-+ [a,x] lies in o(v u(a) = (apo + poa) + E (au')ui - u'(au,). [5] ofi(V;4)), allowing us to identify Cf2V and ofi.(V). For a E Cf2V, 2 2 ~~~~ir~ x E cV, one has *(a)x = [a,x]. Let (CfV)U be the left ideal of CfV generated by U, and set s(V, U) = CeV/(CfV)U. For a 2. Let k[t,t-'] be the ring offinite Laurent series over k. For- E o(V,U;O), there exists EaCf2V such that P = YjEZcjti E k[t,t-1], set Res P = c 1. Let V be a finite di- mensional vector space over k, and let V = k[tt-] %kV = *(a-&)U C U. [2] ESez(t'0 V) be the associated loop space, regarded as a vector Furthermore, a is defined modulo (CfV)U + k by Eq. 2. This space over k. allows us to define the projective spin representation ovu of Call A E Endk V homogeneous of degree m ifA(t' 09 V) C ts+m 0 V for s E Z. In this case, we assign to A the sequence The publication costs ofthis article were defrayed in part by page charge A(') E Endk V, i E Z. defined by A(t' 09 x) = ti+m 0 Al" (x), x payment. This article must therefore be hereby marked "advertise- E V. Let gl,(V) be the Lie subalgebra ofEndk V spanned by the ment" in accordance with 18 U. S. C. §1734 solely to indicate this fact. homogeneous A E Endk V. 3308 Downloaded by guest on September 27, 2021 Mathematics: Kac and Peterson Proc. Natl. Acad. Sci. USA 78 (1981) 3309 Fix r E Z, and set r' = [r/2]. Let 4 be a nondegenerate k- i }, called the set ofdual simple roots. Define reflections r, valued symmetric bilinear form on V. 'Define a nondegenerate 8 GL()*), 0 . i < e, by ri(A) = A - A(h2)a, and let W C k-valued symmetric bilinear form 4)r on V by GL(b*) be the group generated by the ri. We regard W as the = tr subgroup of W generated by rl...re. Set nf+ = n+ ED (tk[t] ),r(P 0 x, Q 0 y) Res PQ O<x,y), P,Q E k[t,t-F], x,y E V. 0) g). Finally, define derivations do, s 8 Z, by [dsc] = 0, [d8,A] Set o8(V;r) = gl(V) n o(V;4r). For s E Z, define a decomposition = ts+idA/dt for A 8 g. Define fundamental weights Ai fi6* by Ai(h.) = 8.,, and set V= Vs (e3V;,v8=ed3(ti C) V),V=E (t ) V). [6] = iS p 4=o Ai. Let P+ be the semigroup generated by the Ai. For KS<8 A E P+, there exists a unique irreducible '-module L(A), called If r is even, set ,r = V_r; if r is odd, suppose U is an isotropic the highest weight module (ref. 3), admitting a nonzero v E L(A) subspace ofV satisfying Eq. 1 and set UrV-= r e(tr'i0 U). such that. n+(v) = 0 and h(v) = A(h)v for h EE 6. A(c) is called Then Tr is a 4)-isotropic subspace of V satisfying Eq. 1, and the level of the module L(A); it is a nonnegative integer and is o*(V;r) C o(Vj,Ur; ,r). Therefore, we may define the projective 0 if and only if L(A) is the trivial one-dimensional module. > spin representation ar ofo.(V;r) on s(V,Ur) to be the restriction Let V be a vector space over k with basis ul,,.. IUn_ n 3, of oVOr to o*(V;r). and let 4 be the symmetric bilinear form on V defined by Set U V__ U' = V' r and define dby Eq. 3 for a E 0(uiun-J+1) = 8,,. Then g = o(V;4)) is identified with the Lie o(V,Ur;4r). Then the cocycle y is given on o*(V;r) by Eq. 4 as algebra of n X n matrices skew symmetric with respect to the follows. If A and B are homogeneous of degrees m and s, re- side diagonal. Let b be the set of diagonal matrices in g and spectively, .then y(A,B) = 0 unless m + s = 0, and ifm 2 0, n+ be the set of strictly upper triangular matrices in g. Set e rm-1 = [n/2], and let U be the ne-stable maximal isotropic subspace ,<A,B) = 2 8m-s tracev AU-r'-m) B(j-r') [7] kul ...E ED ku, of V.
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