Centralizers of spin subalgebras
aa r X i v : . [ m a t h . DG ] J un Centralizers of spin subalgebras
Gerardo Arizmendi ∗† and Rafael Herrera ‡§ June 14, 2018
Abstract
We determine the centralizers of certain isomorphic copies of spin subalgebras spin ( r ) in so ( d r m ), where d r is the dimension of a real irreducible representation of Cl r , the even Clif-ford algebra determined by the positive definite inner product on R r , where r, m ∈ N . In this paper, we determine the centralizer subalgebras of (the isomorphic images under certainmonomorphisms of) subalgebras spin ( r ) in so ( d r m ), where d r is the dimension of the irreduciblerepresentations of Cl r , the even Clifford algebra determined by R r endowed with the standard positivedefinite inner product, and r, m ∈ N . The need to determine such centralizers has arisen in variousgeometrical settings such as the following: • The holonomy algebra of Riemannian manifolds endowed with a parallel even Clifford structure[6]. • The automorphism group of manifolds with (almost) even Clifford (hermitian) structures [1].The centralizers determined in this paper help generalize the results on automorphisms groups ofRiemannian manifolds [10, 11], almost hermitian manifolds [9], and almost quaternion-hermitianmanifolds [7]. • The structure group of Riemannian manifolds admitting twisted spin structures carrying purespinors [4]. More precisely, if M is a smooth oriented Riemannian manifold, F is an auxiliaryRiemannian vector bundle of rank r , S ( T M ) and S ( F ) are the locally defined spinor vectorbundles of M and F respectively, ( f , · · · , f r ) is a local orthonormal frame of F , and m ∈ N is suchthat the bundle S ( T M ) ⊗ S ( F ) ⊗ m is globally defined, a pure spinor field φ ∈ Γ( S ( T M ) ⊗ S ( F ) ⊗ m )is a spinor such that its local 2-forms η φkl ( X, Y ) = h X ∧ Y · κ mr ∗ ( f k f l ) · φ, φ i induce at each point x ∈ M a representation of Cl r on T x M without trivial summands. The centralizers determinedin this paper are the orthogonal complements of spin ( r ) in the annihilator algebra of such aspinor. Should the spinor be parallel, such annihilator will contain the holonomy algebra of themanifold and thus be related to the special holonomies of the Berger-Simons holonomy list [2, 8].The paper is organized as follows. In Section 2 we recall some background material and prove threeresults which will be required later in the main theorems. More precisely, in Subection 2.1, we recallstandard material about Clifford algebras, Spin groups, Spin algebras, and their representations. In ∗ Centro de Investigaci´on en Matem´aticas, A. P. 402, Guanajuato, Gto., C.P. 36000, M´exico. E-mail: [email protected] † Partially supported by a CONACYT scholarship ‡ Centro de Investigaci´on en Matem´aticas, A. P. 402, Guanajuato, Gto., C.P. 36000, M´exico. E-mail: [email protected] § Partially supported by grants from CONACyT and LAISLA (CONACyT-CNRS) spin ( r ) representations ˜∆ r , decompositions intoirreducible summands of ˜∆ r ⊗ ˜∆ r , and calculate various basic centralizers. In Section 3, we prove themain results of the paper, Theorems 3.1 and 3.2. Namely, in Subsection 3.1, we find the centralizers of spin ( r ) in so ( d r m ) for r spin ( r ) in so ( d r m + d r m ) for r ≡ Cl r for r r ≡ Acknowledgements . The second named author would like to thank the International Centre forTheoretical Physics and the Institut des Hautes ´Etudes Scientifiques for their hospitality and support.
In this section we recall material that can also be consulted in [3, 5]. Let Cl n denote the Cliffordalgebra generated by all the products of the orthonormal vectors e , e , . . . , e n ∈ R n subject to therelations e j e k + e k e j = − δ jk , for 1 ≤ j, k ≤ n .We will often write e ...s := e e · · · e s . Let C l n = Cl n ⊗ R C , the complexification of Cl n . It is well known that C l n ∼ = ( End( C k ) , if n = 2 k End( C k ) ⊗ End( C k ) , if n = 2 k + 1 , where C k = C ⊗ . . . ⊗ C the tensor product of k = [ n ] copies of C . Let us denote∆ n = C k , and consider the map κ : C l n −→ End( C k )which is an isomorphism for n even and the projection onto the first summand for n odd. In order tomake κ n explicit consider the following matrices with complex entries Id = (cid:18) (cid:19) , g = (cid:18) i − i (cid:19) , g = (cid:18) ii (cid:19) , T = (cid:18) − ii (cid:19) . Now, consider the generators of the Clifford algebra e , . . . , e n so that κ n can be described as follows e Id ⊗ Id ⊗ . . . ⊗ Id ⊗ Id ⊗ g e Id ⊗ Id ⊗ . . . ⊗ Id ⊗ Id ⊗ g e Id ⊗ Id ⊗ . . . ⊗ Id ⊗ g ⊗ T Id ⊗ Id ⊗ . . . ⊗ Id ⊗ g ⊗ T ... . . .e k − g ⊗ T ⊗ . . . ⊗ T ⊗ T ⊗ Te k g ⊗ T ⊗ . . . ⊗ T ⊗ T ⊗ T, and the last generator e k +1 i T ⊗ T ⊗ . . . ⊗ T ⊗ T ⊗ T if n = 2 k + 1.Let u +1 = 1 √ , − i ) , u − = 1 √ , i )which forms an orthonormal basis of C with respect to the standard Hermitian product. Note that g ( u ± ) = iu ∓ , g ( u ± ) = ± u ∓ , T ( u ± ) = ∓ u ± . Thus, we get a unitary basis of ∆ n = C k B = { u ε ,...,ε k = u ε ⊗ . . . ⊗ u ε k | ε j = ± , j = 1 , . . . , k } , with respect to the induced Hermitian product on C k .The Clifford multiplication of a vector e and a spinor ψ is defined by e · ψ = κ n ( e )( ψ ) . Thus, if1 ≤ j ≤ k e j − · u ε ,...,ε k = i ( − j − k Y α = k − j +2 ε α u ε ,..., ( − ε k − j +1 ) ,...,ε k e j · u ε ,...,ε k = ( − j − k Y α = k − j +1 ε α u ε ,..., ( − ε k − j +1 ) ,...,ε k and e k +1 · u ε ,...,ε k = i ( − k k Y α =1 ε α ! u ε ,...,ε k if n = 2 k + 1 is odd.The Spin group Spin ( n ) ⊂ Cl n is the subset Spin ( n ) = { x x · · · x l − x l | x j ∈ R n , | x j | = 1 , l ∈ N } , endowed with the product of the Clifford algebra. The Lie algebra of Spin ( n ) is spin ( n ) = span { e i e j | ≤ i < j ≤ n } . The restriction of κ to Spin ( n ) defines the Lie group representation κ n := κ | Spin ( n ) : Spin ( n ) −→ GL (∆ n ) , which is, in fact, special unitary [3].There exist either real or quaternionic structures on the spin representations. A quaternionicstructure α on C is given by α (cid:18) z z (cid:19) = (cid:18) − z z (cid:19) , β on C is given by β (cid:18) z z (cid:19) = (cid:18) z z (cid:19) . Note that these structures satisfy h α ( v ) , w i = h v, α ( w ) i , h α ( v ) , α ( w ) i = h v, w i , h β ( v ) , w i = h v, β ( w ) i , h β ( v ) , β ( w ) i = h v, w i , with respect to the standard hermitian product in C , where v, w ∈ C . The real and quaternionicstructures γ n on ∆ n = ( C ) ⊗ [ n/ are built as follows γ n = ( α ⊗ β ) ⊗ k if n = 8 k, k + 1 (real), γ n = α ⊗ ( β ⊗ α ) ⊗ k if n = 8 k + 2 , k + 3 (quaternionic), γ n = ( α ⊗ β ) ⊗ k +1 if n = 8 k + 4 , k + 5 (quaternionic), γ n = α ⊗ ( β ⊗ α ) ⊗ k +1 if n = 8 k + 6 , k + 7 (real).which also satisfy h γ n ( v ) , w i = h v, γ n ( w ) i , h γ n ( v ) , γ n ( w ) i = h v, w i , where v, w ∈ ∆ n . This means h v + γ n ( v ) , w + γ n ( w ) i ∈ R . (1) Lemma 2.1
Let m ≥ r and let e i . . . e i r =: e i ...i r = e I ∈ Cl m . Then e I commutes with spin ( r ) = span { e i e j | ≤ i < j ≤ r } if and only if I ⊂ { r + 1 , . . . , m } or { , . . . , r } ⊂ I .Proof . Suppose that neither I ⊂ { r + 1 , . . . , m } nor { , . . . , r } ⊂ I then there exist j, k ∈ { , . . . , r } such that j ∈ I and k I . Rearranging the other of the i l ’s if necessary we can suppose that j = i , sothat e I e j e k = e i . . . e i r e i e k = ( − r e i . . . e i r .e k and e j e k e I = e i e k e i . . . e i r = ( − r +1 e i . . . e i r e k .Conversely, the volume form on Cl r commutes with spin ( r ) in every dimension and if k
6∈ { , . . . , r } then for all i, j ∈ { , . . . , r } we have that e i e j e k = e k e i e j . ✷ Now, we summarize some results about real representations of Cl r in the next table (cf. [5]).Here d r denotes the dimension of an irreducible representation of Cl r and v r the number of distinctirreducible representations. r (mod 8) Cl r d r v r R ( d r ) 2 ⌊ r ⌋ C ( d r /
2) 2 r H ( d r /
4) 2 ⌊ r ⌋ +1 H ( d r / ⊕ H ( d r /
4) 2 r H ( d r /
4) 2 ⌊ r ⌋ +1 C ( d r /
2) 2 r R ( d r ) 2 ⌊ r ⌋ R ( d r ) ⊕ R ( d r ) 2 r − r denote the irreducible representation of Cl r for r ± r denote the irre-ducible representations for r ≡ r ≡ , r ≡ , , r ≡ , , , d r = d r − and if r ≡ , , , d r = 2 d r − .By restricting to a standard subagebra Cl r − ⊂ Cl r , the representations decompose as follows: r (mod 8) ˜∆ r | Cl r − r ∼ = ˜∆ + r − + ˜∆ − r − r ∼ = ˜∆ r − + ˜∆ r − r ∼ = ˜∆ r − + ˜∆ r − ± r ∼ = ˜∆ r − r ∼ = ˜∆ + r − + ˜∆ − r − r ∼ = ˜∆ r − r ∼ = ˜∆ r − ± r ∼ = ˜∆ r − Table 2
In this section we prove results which are essential in Theorems 3.1 and 3.2. Let V V and S V denotethe second exterior and symmetric power of a finite dimensional vector space respectively. In addition,if the vector space is endowed with an inner product, let S V denote the orthogonal complement ofthe identity endomorphism within the symmetric endomorphisms of V . Proposition 2.1
The centralizers of the spin subalgebras under consideration are: r (mod 8) C so ( d r ) ( spin ( r )) C so ( d r ) ⊕ so ( d r ) ( spin ( r ))0 { }± { }± u (1) ± sp (1)4 sp (1) ⊕ sp (1) Furthermore, the representations V ˜∆ r , V ˜∆ ± r , S ˜∆ r , S ˜∆ ± r and ˜∆ + r ⊗ ˜∆ − r have the following trivial Spin ( r ) subrepresentations: r (mod 8) V ˜∆ r , V ˜∆ ± r S ˜∆ r , S ˜∆ ± r ˜∆ + r ⊗ ˜∆ − r { } { } { }± { } { }± u (1) { }± sp (1) { } sp (1) { } { } Proof of the PropositionCase r ≡ ± γ = γ r on ∆ r . By using these real structures, we can describethe underlying real space ˜∆ r ⊂ ∆ r as follows. Recall the unitary basis B of ∆ r and let B = { u ε ,...,ε [ r/ + γ ( u ε ,...,ε [ r/ ) , iu ε ,...,ε [ r/ + γ ( iu ε ,...,ε [ r/ ) | ε j = ± , j = 1 , . . . , [ r/ } , r = span( B ) = { v + γ r ( v ) | v ∈ ∆ r } . since the hermitian product of ∆ r restricts to a real inner product on ˜∆ r (cf. (1)). Consider the spin ( r ) equivariant morphism Φ : ˜∆ r ⊗ ˜∆ r → M k V k R r . defined byΦ[( v + γ ( v )) ⊗ ( w + γ ( w ))] = [ r/ X k =0 X j < ··· 6s representations of spin ( r ), and ˜∆ r = span( B ) . ˜∆ r ⊗ ˜∆ r ∼ = ˜∆ r +1 ⊗ ˜∆ r +1 . Since r + 1 ≡ − r +1 ⊗ ˜∆ r +1 ∼ = V ev R r +1 with respect to spin ( r + 1), as proved in the previous subsection. Furthermore, R r +1 = R r ⊕ V R r +1 = 1 , V R r +1 = V R r + R r , V R r +1 = V R r + V R r , ... V r R r +1 = V r R r + V r − R r . and ˜∆ r ⊗ ˜∆ r = V ∗ R r . On the other hand, ˜∆ r ⊗ ˜∆ r = V ˜∆ r + S ˜∆ r + 1 . and V ˜∆ r ∼ = V ˜∆ r +1 ∼ = M V l +2 R r +1 , ∼ = M l ≥ V l +1 R r M l ≥ V l +2 R r ,S ˜∆ r ∼ = S ˜∆ r +1 ∼ = M l> V l R r +1 ∼ = M l> V l R r M l ≥ V l +3 R r . We see that V ˜∆ r contains a 1-dimensional trivial spin ( r ) representation.Recall that we wish to find the centralizer of spin ( r ) in so ( d r ) ⊂ End( ˜∆ r ) ∼ = Cl r . Note that anyelement of Cl r which commutes with spin ( r ) must commute with the volume element e e · · · e r ∈ Cl r ,and such elements are precisely Cl r . Thus, by Lemma 2.1 C Cl r ( spin ( r )) ⊆ C Cl r ( spin ( r )) = span(1) ⊕ span( e · · · e r ) , where e · · · e r acts as an orthogonal complex structure J on ˜∆ r which generates the afore mentioned1-dimensional trivial summand in V ˜∆ r . Hence, C so ( d r ) ( spin ( r )) = span( J ) ∼ = u (1) ⊂ so ( d r ) ∼ = V ˜∆ r . ase r ≡ γ r on ∆ r which commutes with Clifford multipli-cation. We can describe the real space ˜∆ r ⊂ ∆ r as follows. Recall the unitary basis B of ∆ r andlet B = { u ε ,...,ε r/ + γ ( u ε ,...,ε r/ ) , iu ε ,...,ε r/ + γ ( iu ε ,...,ε r/ ) | ε j = ± , j = 1 , . . . , r/ } . Note that the space generated by the orthogonal basis B is preserved by the action of spin ( r ) and Cl r , the hermitian product in ∆ r restricts to an inner product to ˜∆ r (cf. (1)), and its dimension is d r . Therefore ˜∆ r = span( B ) . Now consider the spin ( r ) equivariant morphismΦ : ˜∆ r ⊗ ˜∆ r → M k V k R r defined byΦ[( v + γ ( v )) ⊗ ( w + γ ( w ))] = r X k =0 X j < ··· 12 (1 + e . . . e r ) · e . . . e r = 12 (1 + e . . . e r ) , 12 (1 + e . . . e r ) · e r +1 = 12 ( e r +1 + e ,...,r +1 ) , 12 (1 + e . . . e r ) · e . . . e r +1 = 12 ( e r +1 + e ,...,r +1 ) , and, for v ∈ ∆ r +3 ,12 (1 + e . . . e r ) · 12 (1 + e . . . e r ) · ( v + γ r +3 ( v )) = 12 (1 + e . . . e r ) · ( v + γ r +3 ( v ))12 (( e r +1 + e ,...,r +1 )) · 12 (1 + e . . . e r ) · ( v + γ r +3 ( v )) = 0 , so (1 + e . . . e r ) acts as the identity element on this copy of ˜∆ r and ( e r +1 + e ,...,r +1 ) acts as thenull endmorphism on this copy of ˜∆ r . It is not hard to check that the only projections that inducenonzero endomorphisms are (1 + e . . . e r ), (1 + e . . . e r ) · e r +1 ,r +2 , (1 + e . . . e r ) · e r +1 ,r +3 and (1+ e . . . e r ) · e r +2 ,r +3 . Note that the Hermitian product of ∆ r +3 restricts to a positive definite innerproduct on ˜∆ r +3 (cf (1)). Now we will check whether the endomorphisms induced by (1 + e . . . e r ), (1 + e . . . e r ) · e r +1 ,r +2 , (1 + e . . . e r ) · e r +1 ,r +3 and (1 + e . . . e r ) · e r +2 ,r +3 are symmetric orantisymmetric: • The element (1 + e . . . e r ) ∈ Cl r +3 acts as the identity on this copy of ˜∆ r so is a symmetricautomorphism. • For v, w ∈ ˜∆ r +3 , the element (1 + e . . . e r ) · e r +1 ,r +2 ∈ spin ( r + 3) is such that (cid:28) 12 (1 + e . . . e r ) · e r +1 ,r +2 · 12 (1 + e . . . e r ) · ( v + γ r +3 ( v )) , 12 (1 + e . . . e r ) · ( w + γ r +3 ( w )) (cid:29) = − (cid:28) 12 (1 + e . . . e r ) · ( v + γ r +3 ( v )) , 12 (1 + e . . . e r ) · e r +1 ,r +2 · 12 (1 + e . . . e r ) · ( w + γ r +3 ( w )) (cid:29) , so that (1 + e . . . e r ) · e r +1 ,r +2 induces a complex structure I on ˜∆ r . Indeed, it is a complexstructure. 10 Similarly, (1 + e . . . e r ) · e r +1 ,r +3 and (1 + e . . . e r ) · e r +2 ,r +3 induce complex structures J and K on ˜∆ r .Thus, C so ( d r ) ( spin ( r )) = sp (1) = span ( I, J, K ) ⊂ so ( d r ) ∼ = V ˜∆ r . Case r ≡ − r ∼ = ˜∆ r +2 as spin ( r ) representations, and ˜∆ r ⊗ ˜∆ r ∼ = ˜∆ r +2 ⊗ ˜∆ r +2 . Since r + 2 ≡ − r +2 ⊗ ˜∆ r +2 ∼ = V ev R r +2 as a spin ( r + 2) representation and R r +2 = R r ⊕ V R r +2 = 1 , V R r +2 = V R r + 2 R r + 1 , V R r +2 = V R r + 2 V R r + V R r , ... V r +1 R r +2 = V r +1 R r + 2 V r R r + V r − R r , so that ˜∆ r ⊗ ˜∆ r = 2 V ∗ R r . Recall that we wish to compute the centralizer of spin ( r ) in so ( d r ) ⊂ End( ˜∆ r ) = End( ˜∆ r +2 ) ∼ = Cl r +2 . By Lemma 2.1, C Cl r +2 ( spin ( r )) = span(1 , e r +1 e r +2 , e . . . e r +1 , e . . . e r e r +2 ) , where the last three elements form a copy of sp (1). By means of Clifford multiplication, these threeelements act as orthogonal complex structures I , J , K on ˜∆ r and behave as quaternions, i.e.span( I, J, K ) = sp (1) ⊂ so ( d r ) ∼ = V ˜∆ r . Case r ≡ spin ( r ) has two irreducible representations given by˜∆ ± r = 12 (1 ± e . . . e r ) · ˜∆ r +1 , so that ˜∆ r +1 ∼ = ˜∆ + r ⊕ ˜∆ − r , and ( ˜∆ + r ⊕ ˜∆ − r ) ⊗ ( ˜∆ + r ⊕ ˜∆ − r ) ∼ = ˜∆ r +1 ⊗ ˜∆ r +1 as spin ( r ) representations. Since r + 1 ≡ r +1 ⊗ ˜∆ r +1 ∼ = V ev R r +1 11s a spin ( r + 1) representation, and R r +1 = R r ⊕ V R r +1 = 1 , V R r +1 = V R r + R r , V R r +1 = V R r + V R r , ... V r R r +1 = V r R r + V r − R r , i.e. ( ˜∆ + r ⊕ ˜∆ − r ) ⊗ ( ˜∆ + r ⊕ ˜∆ − r ) = V ∗ R r , which has only 2 trivial summands with respect to spin ( r ). On the other hand,( ˜∆ + r ⊕ ˜∆ − r ) ⊗ ( ˜∆ + r ⊕ ˜∆ − r ) = V ˜∆ + r ⊕ S ˜∆ + r ⊕ ⊕ V ˜∆ − r ⊕ S ˜∆ − r ⊕ ⊕ ˜∆ + r ⊗ ˜∆ − r ⊕ ˜∆ − r ⊗ ˜∆ + r , i.e. no other summand contains a trivial spin ( r ) representation.Recall that we wish to find the centralizer of spin ( r ) in so ( d r ) ⊕ so ( d r ) ⊂ End( ˜∆ + r ) ⊕ End( ˜∆ − r ) ∼ = Cl r . By Lemma 2.1, C Cl r ( spin ( r )) = span(1) ⊕ span( e . . . e r ) . Since both 1 and e . . . e r induce symmetric endomorphisms on ˜∆ r , C so ( d r ) ⊕ so ( d r ) ( spin ( r )) = { } ⊂ so ( d r ) ∼ = V ˜∆ r . Case r ≡ spin ( r ) has two irreducible representations and˜∆ + r ⊕ ˜∆ − r ∼ = ˜∆ r +1 ∼ = ˜∆ r +2 ∼ = ˜∆ r +3 as representations of spin ( r ). Since r + 3 ≡ − γ r +3 is a real structure and˜∆ r +3 = { v + γ r +3 ( v ) | v ∈ ∆ r +3 } . Moreover ˜∆ ± r = 12 (1 ± e . . . e r ) · ˜∆ r +3 and ˜∆ r +3 ⊗ ˜∆ r +3 ∼ = ( ˜∆ + r ⊕ ˜∆ − r ) ⊗ ( ˜∆ + r ⊕ ˜∆ − r )with respect to spin ( r ). With respect to spin ( r + 3),˜∆ r +3 ⊗ ˜∆ r +3 ∼ = V ev R r +3 . Now, R r +3 = R r ⊕ V R r +3 = 1 , V R r +3 = V R r + 3 R r + 3 , V R r +3 = V R r + 3 V R r + 3 V R r + R r , ... V r +2 R r +3 = V r +2 R r + 3 V r +1 R r + 3 V r R r + V r − R r = 3 + V r − R r , + r ⊕ ˜∆ − r ) ⊗ ( ˜∆ + r ⊕ ˜∆ − r ) = 4 V ∗ R r . Recall that we wish to compute the centralizer of spin ( r ) in so ( d r ) ⊕ so ( d r ) ⊂ End( ˜∆ + r ⊕ ˜∆ − r ) =End( ˜∆ r +3 ) ∼ = Cl r +3 . First we will compute C Cl r +3 ( spin ( r )). If η = X | I |≡ η I e I ∈ C Cl r +3 ( spin ( r ))then it must commute in Clifford product with every e i e j ∈ spin ( r ), 1 ≤ i < j ≤ r . By Lemma 2.1, theonly free coefficients are η ∅ , η r +1 ,r +2 , η r +1 ,r +3 , η r +2 ,r +3 , η ,...,r , η ,...,r +2 , η ,...,r +1 ,r +3 , η ,...,r,r +2 ,r +3 ,i.e. C Cl r +3 ( spin ( r )) = span(1 , e r +1 ,r +2 , e r +1 ,r +3 , e r +2 ,r +3 , e ,...,r , e ,...,r +2 , e ,...,r +1 ,r +3 , e ,...,r,r +2 ,r +3 )= span (cid:18) 12 (1 ± e ...r ) , 12 (1 ± e ...r ) e r +1 e r +2 , 12 (1 ± e ...r ) e r +1 e r +3 , 12 (1 ± e ...r ) e r +2 e r +3 (cid:19) . Now we need to check which of these elements induce antisymmetric endomorphisms on ˜∆ ± r . respec-tively. • The element (1 ± e ...r ) ∈ Cl r +3 induces the identity endomorphism on ˜∆ ± r and the nullendomorphism on ˜∆ ∓ r , both of which are symmetric. • The elements (1 ± e . . . e r ) e r +1 ,r +2 , (1 ± e . . . e r ) e r +1 ,r +3 and (1 ± e . . . e r ) e r +2 ,r +3 inducealmost complex structures I ± , J ± , K ± on ˜∆ ± r respectively, and the null endomorphism on ˜∆ ∓ .Such elements also commute with the elements of spin ( r ). In other words, sp (1) ± = span( I ± , J ± , K ± ) ⊂ V ˜∆ ± r are trivial spin ( r ) representations.Hence, C so ( d r ) ⊕ so ( d r ) ( spin ( r )) ∼ = sp (1) + ⊕ sp (1) − . ✷ Due to geometric considerations in [6, 4], we will consider spin ( r ) embedded in so ( N ) in the followingway. Suppose that Cl r is represented on R N , for some N ∈ N , in such a way that each bivector e i e j is mapped to an antisymmetric endomorphism J ij satisfying J ij = − Id R N . (2) spin ( r ) in so ( d r m ) , r , r > Let us assume r r > 1. In this case, R N decomposes into a sum of irreducible repre-sentations of Cl r . Since this algebra is simple, such irreducible representations can only be trivial orcopies of the standard representation ˜∆ r of Cl r (cf. [5]). Due to (2), there are no trivial summandsin such a decomposition so that R N = ˜∆ r ⊕ · · · ⊕ ˜∆ r | {z } m times . 13y restricting to spin ( r ) ⊂ Cl r , R N = ˜∆ r ⊗ R R m we see that spin ( r ) has an isomorphic image \ spin ( r ) = spin ( r ) ⊗ { Id m × m } ⊂ so ( d r m ) , which is the subalgebra of so ( d r m ) whose centralizer C so ( d r m ) ( \ spin ( r )) we wish to find. Theorem 3.1 Let r and let \ spin ( r ) ⊂ so ( d r m ) as described before. The centralizer of \ spin ( r ) in so ( d r m ) is isomorphic to r (mod 8) C so ( d r m ) ( \ spin ( r ))1 so ( m )2 u ( m )3 sp ( m )5 sp ( m )6 u ( m )7 so ( m ) Proof . Consider the real ( d r m )-dimensional real Grassmannian G = SO ( d r + m ) SO ( d r ) × SO ( m ) . The tangent space factors as follows T [Id ( dr + m ) × ( dr + m ) ] G ∼ = R d r ⊗ R m ∼ = R d r m . so that the differential of the isotropy representation is so ( d r ) ⊕ so ( m ) −→ [ so ( d r ) ⊗ { Id m × m } ] ⊕ [ { Id d r × d r } ⊗ so ( m )] ⊂ so ( d r m )( A, B ) A ⊗ Id m × m ⊕ Id d r × d r ⊗ B. Let \ so ( m ) = { Id d r × d r } ⊗ so ( m ) and \ so ( d r ) = so ( d r ) ⊗ { Id m × m } . Thus, we see that \ so ( m ) centralizes \ so ( d r ) in so ( d r m ), and \ so ( m ) ⊆ C so ( d r m ) ( \ spin ( r )) . Let us consider the following orthogonal decomposition so ( d r m ) = [ \ so ( m ) ⊕ \ so ( d r )] ⊕ m , and set g = so ( d r m ) , h = \ so ( m ) ⊕ \ so ( d r ) . Since the homogeneous space F = SO ( d r m ) SO ( d r ) ⊗ SO ( m )14s Riemannian homogeneous, it is reductive, i.e.[ h , m ] ⊂ m . Let X = X + X + X ∈ g where X ∈ \ so ( m ) ,X ∈ \ so ( d r ) ,X ∈ m , and assume that X ∈ C so ( d r m ) ( \ spin ( r )), i.e. [ X, Y ] = 0for all Y ∈ \ spin ( r ). Thus, 0 = [ X , Y ] + [ X , Y ] + [ X , Y ] . Note that [ X , Y ] ∈ h , [ X , Y ] ∈ h , [ X , Y ] ∈ m , so that [ X + X , Y ] = 0 , [ X , Y ] = 0 . Since X ∈ \ so ( m ) and Y ∈ \ spin ( r ) ⊂ \ so ( d r ), [ X , Y ] = 0 , which implies [ X , Y ] = 0 . On the other hand, since [ X , Y ] = 0for all Y ∈ \ spin ( r ), the subalgebra \ spin ( r ) ⊂ h acts trivially on the 1-dimensional subspace of thetangent space m of F at [Id ( d r m ) × ( d r m ) ] generated by X . Now, as a representation of h = \ so ( d r ) ⊕ \ so ( m ) ∼ = so ( d r ) ⊕ so ( m ), m ∼ = hV R d r ⊗ S R m i ⊕ h S R d r ⊗ V R m i . By restricting to \ so ( d r ) m ∼ = (cid:20)V R d r ⊗ (cid:18)(cid:18) m + 12 (cid:19) − (cid:19)(cid:21) ⊕ (cid:20) S R d r ⊗ (cid:18) m (cid:19)(cid:21) , i.e. m decomposes as the sum of multiple copies of the irreducible so ( d r ) representations V R d r and S R d r . By restricting further to \ spin ( r ) ⊂ \ so ( d r ), m decomposes as m ∼ = (cid:20)V ˜∆ r ⊗ (cid:18)(cid:18) m + 12 (cid:19) − (cid:19)(cid:21) ⊕ (cid:20) S ˜∆ r ⊗ (cid:18) m (cid:19)(cid:21) . (3)Both spin ( r ) representations V ˜∆ r and S ˜∆ r decompose further into irreducible summands. Now weneed to work out three cases separately. 15 ase r ≡ ± : By Proposition 2.1, the centralizer of \ spin ( r ) in \ so ( d r ) is trivial, i.e. X = 0 . Recall that \ spin ( r ) preserves each summand in (3) and annihilates X . By Proposition 2.1, there areno trivial summands in either V ˜∆ r nor S ˜∆ r , i.e. X = 0 . Hence X = X ∈ \ so ( m ) . Case r ≡ ± : By Proposition 2.1, the centralizer of \ spin ( r ) in \ so ( d r ) is a copy of u (1), i.e. X = λJ ⊗ Id m × m , where J is an orthogonal complex structure that generates u (1) and λ ∈ R . Recall that \ spin ( r )preserves each summand in (3) and annihilates X . There are no trivial summands in S ˜∆ r , but thereis a trivial summand in V ˜∆ r generated precisely by J , since it is an antisymmetric endomorphism.We see that m contains span( J ) ⊗ S R m as a trivial \ spin ( r ) representation. Hence X ∈ \ so ( m ) ⊕ span( J ) ⊗ (span(Id m × m ) ⊕ S R m ) ⊂ so ( d r m ) . In order to recogize which Lie algebra h \ so ( m ) ⊕ span( J ) ⊗ S R m i is, notice that if A ∈ u ( m ), byseparating real and imaginary parts A = A + iA ,A ∈ so ( m ) is antisymmetric and A is symmetric, i.e. A ∈ S R m . Here, a canonical summand u (1)is spanned by the element i Id m × m . Note that due to the existence of J , we can work instead with acomplex vector space, where J corresponds to i , J ⊗ S R m corresponds to iS R m and \ so ( m ) ⊕ span( J ) ⊗ S R m ∼ = u ( m ) . Case r ≡ ± : By Proposition 2.1, the centralizer of \ spin ( r ) in \ so ( d r ) is a copy of sp (1) = span( I, J, K ), where I, J, K are three orthogonal complex structures which behave as imaginary quaternions. Thus, X ∈ sp (1) ⊗ span(Id m × m ) . By Proposition 2.1, S ˜∆ r contains no trivial \ spin ( r ) representations, but V ˜∆ r does contain a 3-dimensional one given by sp (1) = span( I, J, K ). We have the trivial \ spin ( r ) representation in m span( I, J, K ) ⊗ S R m = sp (1) ⊗ S R m . X ∈ = so ( m ) ⊕ sp (1) ⊗ S R m . In order to recognize this Lie algebra, notice that if A ∈ sp ( m ), by separating real and imaginaryparts A = A + iA + jA + kA ,A ∈ so ( m ) is antisymmetric and A , A , A are symmetric, i.e. A , A , A ∈ S R m . The summand sp (1) is spanned by the elements i Id m × m , j Id m × m , k Id m × m . Moreover, due to the existence of I, J, K ,we can work instead with a quaternionic vector space, in which, I corresponds to i , J corresponds to j , K corresponds to k , and span( I, J, K ) ⊗ S R m corresponds to iS R m ⊕ jS R m ⊕ kS R m so that \ so ( m ) ⊕ sp (1) ⊗ S R m ∼ = sp ( m ) . ✷ spin ( r ) in so ( d r m + d r m ) , r ≡ Let us assume r ≡ r is the irreducible representation of Cl r , then byrestricting this representation to Cl r it splits as the sum of two inequivalent irreducible representationsˆ∆ r = ˜∆ + r ⊕ ˜∆ − r . Since R N is a representation of Cl r satisfying (2), there are no trivial summands in such a decompo-sition so that R N = ˜∆ + r ⊗ R m ⊕ ˜∆ − r ⊗ R m . By restricting this representation to spin ( r ) ⊂ Cl r , consider \ spin ( r ) = spin ( r ) + ⊗ (Id m × m ⊕ m × m ) ⊕ spin ( r ) − ⊗ ( m × m ⊕ Id m × m ) ⊂ so ( d r m + d r m ) , where spin ( r ) ± are the images of spin ( r ) in End( ˜∆ ± r ) respectively. We wish to find the centralizer C so ( d r m + d r m ) ( \ spin ( r )). Theorem 3.2 Let r ≡ . The centralizer of \ spin ( r ) in so ( d r m + d r m ) is isomorphic to r (mod 8) C so ( d r m + d r m ) ( \ spin ( r ))0 so ( m ) ⊕ so ( m )4 sp ( m ) ⊕ sp ( m ) Proof . Consider the homogeneous space G = SO ( m + d r ) × SO ( m + d r )( SO ( d r ) × SO ( m )) × ( SO ( d r ) × SO ( m )) . with the obvious inclusions of subgroups. The tangent space decomposes as follows T [Id (2 dr + m m × ( dr + m m ] G ∼ = R m ⊗ R d r ⊕ R m ⊗ R d r , Let \ so ( m ) = (Id d r × d r ⊕ d r × d r ) ⊗ so ( m ) , so ( m ) = ( d r × d r ⊕ Id d r × d r ) ⊗ so ( m ) , \ so ( d r ) = so ( d r ) ⊗ (Id m × m ⊕ m × m ) , \ so ( d r ) = so ( dr ) ⊗ ( m × m ⊕ Id m × m ) . We see that \ so ( m ) ⊕ \ so ( m ) centralizes \ so ( d r ) ⊕ \ so ( d r ) in so ( d r m + d r m ), \ so ( m ) ⊕ \ so ( m ) ⊆ C so ( d r m + d r m ) ( \ spin ( r )) . Let us consider the following orthogonal decomposition so ( d r m + d r m ) = [ \ so ( m ) ⊕ \ so ( d r ) ] ⊕ [ \ so ( m ) ⊕ \ so ( d r ) ] ⊕ m , and set g = so ( d r m + d r m ) , h = \ so ( m ) ⊕ \ so ( d r ) ⊕ \ so ( m ) ⊕ \ so ( d r ) . Since the homogeneous space F = SO ( d r m + d r m )( SO ( m ) ⊗ SO ( d r )) × ( SO ( m ) ⊗ SO ( d r ))is Riemannian homogeneous, it is reductive, and[ h , m ] ⊂ m . Let X = X + X + X ∈ g where X ∈ \ so ( m ) ⊕ \ so ( m ) ,X ∈ \ so ( d r ) ⊕ \ so ( d r ) ,X ∈ m , and assume that X ∈ C so ( d r m + d r m ) ( \ spin ( r )), i.e.[ X, Y ] = 0for all Y ∈ \ spin ( r ). Thus, 0 = [ X , Y ] + [ X , Y ] + [ X , Y ] . Note that [ X , Y ] ∈ h , [ X , Y ] ∈ h , [ X , Y ] ∈ m , so that [ X + X , Y ] = 0 , [ X , Y ] = 0 . X ∈ \ so ( m ) ⊕ \ so ( m ) and Y ∈ \ spin ( r ) ⊂ \ so ( d r ) ⊕ \ so ( d r ) ,[ X , Y ] = 0 , which implies [ X , Y ] = 0 . Since [ X , Y ] = 0for all Y ∈ \ spin ( r ), the subalgebra \ spin ( r ) ⊂ h acts trivially on the 1-dimensional subspace of thetangent space m of F at [Id ( d r m + d r m ) × ( d r m + d r m ) ] generated by X . Note that so ( d r m + d r m ) = V ( R d r ⊗ R m ⊕ R d r ⊗ R m )= V ( R d r ⊗ R m ) ⊕ ( R d r ⊗ R m ) ⊗ ( R d r ⊗ R m ) ⊕ V ( R d r ⊗ R m ) ∼ = \ so ( d r ) ⊕ \ so ( m ) ⊕ hV R d r ⊗ S R m ⊕ S R d r ⊗ V R m i ⊕ R d r ⊗ R d r ⊗ R m ⊗ R m ⊕ \ so ( d r ) ⊕ \ so ( m ) ⊕ hV R d r ⊗ S R m ⊕ S R d r ⊗ V R m i , so that, by restricting to \ spin ( r ), m = hV ˜∆ + r ⊗ S R m ⊕ S ˜∆ + r ⊗ V R m i ⊕ ˜∆ + r ⊗ ˜∆ − r ⊗ R m ⊗ R m ⊕ hV ˜∆ − r ⊗ S R m ⊕ S ˜∆ − r ⊗ V R m i . Now we need to check two cases separately. Case r ≡ : By Proposition 2.1, the centralizer of \ spin ( r ) in \ so ( d r ) ⊕ \ so ( d r ) is trivial, i.e. X = 0 . By Proposition 2.1, m has no trivial summands, i.e. X = 0 . Hence X = X ∈ \ so ( m ) ⊕ \ so ( m ) . Case r ≡ : By Proposition 2.1, the centralizer of \ spin ( r ) in \ so ( d r ) ⊕ \ so ( d r ) is a copy of sp (1) ⊕ sp (1), i.e. X ∈ [ sp (1) ⊗ (Id m × m ⊕ m × m )] ⊕ [ sp (1) ⊗ ( m × m ⊕ Id m × m )]By Proposition 2.1, the only \ spin ( r ) representations in m containing trivial spin ( r ) summands are V ˜∆ ± r . More precisely, V ˜∆ ± r constains a 3-dimensional trivial spin ( r ) representation sp (1) ± =19pan( I ± , J ± , K ± ), where I ± , J ± , K ± are orthogonal complex structures on ˜∆ ± r which behave asquaternions. Thus, we have the trivial spin ( r ) representation in msp (1) + ⊗ S R m ⊕ sp (1) − ⊗ S R m . Altogether, we have X ∈ [ \ so ( m ) ⊕ sp (1) + ⊗ S R m ] ⊕ [ \ so ( m ) ⊕ sp (1) − ⊗ S R m ] ∼ = sp ( m ) ⊕ sp ( m ) . ✷ References [1] Arizmendi, G.; Herrera, R.; Santana, N.: Almost even-Clifford hermitian manifolds with large automor-phism group. 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