Lecture 19 - Clifford and Spin Representations

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1 Lecture 19 - lifford and Spin Representations April 5, 2013 References: Lawson and Michelsohn, Spin Geometry. J. Baez, The Octonions. 1 The Lie Algebras so(n) and spin(n) We know the Lie algebra so(n) consists of antisymmetric matrices, which can be identified with 2 R n. The standard action of so(n) on R n can also be expressed this way: (v w) (x) i x (v w) = (v, x) w (w, x) v (1) which leads to a bilinear product on 2 R n (equivalent to matrix multiplication) and a lie bracket. Now consider the Lie group Spin(n). We have seen that the paths γ ij (t) = (e i sin(t) + e j cos(t)) ( e i sin(t) + e j cos(t)) = cos(2t) + e i e j sin(2t) (2) are in Spin(n) where γ ij (0) = 1. This the Lie algebra spin(n) contains {e i e j i j}, so by dimension counting, we know that these are equal. Theorem 1.1 The vector space span R { e i e j i j } l n is closed under the lifford bracket, and is the Lie algebra of spin(n). Theorem 1.2 The twisted adjoint representation Ãd : Spin(n) SO(n) provides the following Lie algebra isomorphism between spin(n) and so(n) 2 R n e i e j 1 2 e i e j. (3) 1

2 Pf. omputing the twisted conjugation by the path γ ij (t), where α(γ ij (t)) 1 = sin(t) e i e j cos(t), we obtain Ãd γij(t)v = (sin(2t) + cos(2t)e i e j ) v (sin(2t) e i e j cos(2t)) = ( v sin 2 (2t) e i e j ve i e j cos 2 (2t) ) + cos(2t) sin(2t) (e i e j v ve i e j ) Taking a derivative at t = 0 we obtain the action of e i e j spin(n) on v R n, which is (e i e j )(v) = 2 (e i e j v ve i e j ) = 2 (v, e i ) e j 2 (v, e j ) e i = 2 (e i e j ) (v) (4) (5) 2 The lifford representation on the Exterior Algebra onsider a vector space V with an inner product, not necessarily definite, and let = V denote its exterior algebra. Theorem 2.1 ( Poincare-Birkhoff-Witt for lifford algebras) The lifford algebra l(v ) V/I obtains a filtration from V. The associated graded algebra is the exterior algebra. onsequently, l(v ) V as vector spaces. Pf. The proof is so much simpler than the analogous result for Lie algebras that it is almost an insult to Poincare, Birkhoff, and Witt to give it this name. Let l (k) denote the k th filtered subset, meaning l (k) is the image of k i=0 i V under the canonical projection. hoose an orthonormal basis e 1,..., e n for V (if the metric is indefinite, some of these can have length 0). Then because each e i commutes with all e j except when i = j, any monomial e i1... e ik l (k) \ l (k 1) can be rearranged to obtain e i1... e ik = ±e l1 j 1 e l2 j 2... e ln j n, j 1 < j 2... j n, l i N (6) If any l m > 1, then the monomial on the right is in l (k 1). Therefore l (k) /l (k 1) consists of sums of monomials with k factors and no repeats. Each monomial in l (k) /l (k 1) can be rearranged so the indices are in increasing order. This gives a vector space isomorphism l (k) /l (k 1) k The product operation l (k) /l (k 1) l (p) /l (p 1) l (k+p) /l (k+p 1) proceeds by lifford multiplication, then re-arranging the factors until their indices are increasing. If any repeats exist, the product is zero. If no repeats exist, then a sign is involved in every transposition, so we get the same product as k V p V k+p V. A natural lifford representation exists on V. Given v V and η V, define v.η = v η i v η. (7) 2

3 One checks v.v.η = v i v η i v (v η) = v 2 η (8) so by the universal property this extends to a lifford representation. Notice that this passes to a representation of l 0 (V ) on V. This preserves parity, so splits into two l 0 (V ) representations, one on even (V ) and one on odd (V ). These are equivalent representations; more will be said about them below. A second representation, this time of l 0 on k V, exists. Given an element η k V and a generator vw of l 0, we define More will be said about this below as well. vw.η = π p (v.w.η). (9) 3 Maximal Tori and Spin Representations 3.1 Maximal Tori In SO(n) for n = 2m, resp. n = 2m + 1, we have the maximal torus cos(x 1 ) sin(x 1 ) sin(x 1 ) cos(x 1 ) resp. cos(x 1 ) sin(x 1 ) sin(x 1 ) cos(x 1 ) cos(xm) sin(xm) sin(x m ) cos(x m ) cos(xm) sin(xm) sin(x m ) cos(x m ), (10) as x 1,..., x m each range over R. Both tori come from the maximal toral subalgebra 1 (11) span R { e 1 e 2, e 3 e 4,..., e 2m 1 e 2m } (12) Let R n have orthonormal basis e 1,..., e n. A maximal torus in Spin(n), where n = 2m or n = 2m + 1 has dimension m. We can take it to be m (sin( 12 x i) + cos( 12 ) m x i)e 2i 1 e 2i = Exp( x i 2 e 2i 1e 2i ) (13) i=1 3 i=1

4 as x 1,..., x m each range over R. Note that the factor of 1 2 was chosen so the natural 2-1 map T Spin(n) T SO(n) does not dilate the torus. The corresponding maximal toral subalgebras are t 0 = span R { e 1 e 2, e 3 e 4,..., e 2m 1 e 2m }. (14) 3.2 Weights and spin(n) representations Now the x i can be considered a basis of t 0 so the 1x i are a basis of weights. To proceed, we recall Humphrey s concrete realization of the B n and D n root systems. Let the y i n be dual to the 1x i n. For B n we have the root system consisting of vectors ± (y i ± y j ) (where possibly i = j), with simple roots y 1 y 2, y 2 y 3,..., y n 1 y n, 2y n (15) For D n we have the root system ±(y i ± y j ) with i j, and simple roots y 1 y 2, y 2 y 3,..., y n 1 y n, y n 1 + y n. (16) As an example, consider the spin action on 1 R n. To speak of weights, we must be able to solve polynomial equation, so we must complexify: 1 n. Diagonalizing the vectors in 1 via the action of the maximal torus, we obtain a weight-vector decomposition e 1 ± 1e 2, e 3 ± 1e 4,..., e 2m 1 ± 1e 2m (17) whose weights are ± 1x 1, ± 1x 2,..., ± 1x n. Now, using the descriptions of the root systems B n and D n, we see that + 1x 1 is the only dominant integral weight; it s Dynkin coefficients are (1, 0,..., 0) Likewise, we have an action of the Spin groups on 2, whose weight vectors are (e i e j ± e i+1 e j+1 ) + 1 (e i e j+1 e i+1 e j ) (e i e j ± e i+1 e j+1 ) 1 (e i e j+1 e i+1 e j ) (18) with weights ± 1x i ± 1x j. Here we notice the only dominant integral weight is 1x1 + 1x 2, which has Dynkin coefficients (0, 1, 0,..., 0). Proceeding similarly, we obtain an irreducible spin(n) so(2, ) representations on each i for i {1,..., m 1} in the case that n = 2m + 1, and for i {1,..., m 2} in the case n = 2m. Above these levels, we obtain representations that are no longer fundamental (their maximal weights have more than one non-zero Dynkin coefficient, and/or have a Dynkin coefficient of 2). 4

5 3.3 onstruction of the spin representations To find the missing representations, we expand our purview. Given a vector v R n, consider the action on the tensor algebra given by v.η = v η i v η. (19) We compute v.v.η = v 2 η so this passes to a lifford representation l n Hom(, ). Now this is not sufficient to find our missing representations; we must first split the vectors in two. 3.4 Dynkin Diagrams In the case of B n we have the following spin(2n + 1) representations: 1 2 n 1 Adj. In the case of D n we have the following spin(2n) representations: n 3 n 2 Adj. (20) 5