GENERALIZED SPIN REPRESENTATIONS. PART 2: CARTAN BOTT PERIODICITY FOR THE SPLIT REAL E n SERIES

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1 GENERALIZED SPIN REPRESENTATIONS. PART : CARTAN BOTT PERIODICITY FOR THE SPLIT REAL E SERIES MAX HORN AND RALF KÖHL (NÉ GRAMLICH) Abstract. I this article we aalyze the quotiets of the maximal compact subalgebras of the split real Kac Moody algebras of the E series resultig from the geeralized spi represetatio itroduced i [HKL3]. It turs out that these quotiets satisfy a Carta Bott periodicity. Our fidigs are also meaigful i the fiite-dimesioal cases of A A, A 4, D 5, E 6, E 7, E 8, where it turs out that the geeralized spi represetatio is ijective. Cosequetly the observed Carta Bott periodicity provides a structural explaatio for the seemigly sporadic isomorphism types of the maximal compact Lie subalgebras of the split real Lie algebras of types E 6, E 7, E 8.. Itroductio I this article we cotiue the ivestigatio of the geeralized spi represetatios itroduced i the first part [HKL3]. We focus o the E series ad use the origial descriptio of the geeralized spi represetatio from [DKN06], [DBHP06], [HKL3] via Clifford algebras. The E series is traditioally oly defied for {6, 7, 8}. However, usig the Bourbaki style labelig show i Figure, it aturally exteds to arbitrary N. Usig this descriptio, oe has E = A, E = A A, E 3 = A A, E 4 = A 4, E 5 = D 5 (see Figure ). E Figure. The Dyki diagram of type E A elemetary combiatorial coutig argumet usig biomial coefficiets allows us to determie lower bouds for the R-dimesio of the images of the geeralized spi represetatio. These images have to be compact, whece reductive by [HKL3, Theorem 4.] ad eve semisimple, if the diagram be irreducible, thus providig a upper boud for the R-dimesio via the maximal compact Lie subalgebras of the Clifford algebras. As it turs out, the lower ad the upper bouds coicide, providig the followig Carta Bott periodicity. Theorem A (Carta Bott periodicity of the E series). Let N with 4, let k be the maximal compact Lie subalgebra of the split real Kac Moody Lie algebra of type E, let C = C(R, q) be the Clifford algebra with respect to the stadard positive defiite quadratic form q ad let ρ : k C be the stadard geeralized spi represetatio. The im(ρ) is isomorphic to

2 MAX HORN AND RALF KÖHL (NÉ GRAMLICH) (0) so( )) R R M(, R), if 0 (mod 8), () so( ) so( ) (R R) R M(, R), if (mod 8), () so( ) M(, R) R M(, R), if (mod 8), (3) su( ) M(, C) R M( 3, R), if 3 (mod 8), (4) sp( ) M(, H) R M( 4, R), if 4 (mod 8), (5) sp( 3 ) sp( 3 ) (M(, H) M(, H)) R M( 5, R), if 5 (mod 8), (6) sp( ) M(4, H) R M( 6, R), if 6 (mod 8), (7) su( ) M(8, C) R M( 7, R), if 7 (mod 8), i.e., im(ρ) is a semisimple maximal compact Lie subalgebra of C. Alog the way we arrive at a structural explaatio for the isomorphism types of the maximal compact Lie subalgebras of the semisimple split real Lie algebras of types E 3 = A A, E 4 = A 4, E 5 = D 5, E 6, E 7, E 8. Theorem B. The maximal compact Lie subalgebras of the semisimple split real Lie algebras of types A A, A 4, D 5, E 6, E 7, E 8 are isomorphic to u(), sp() = so(5), sp() sp() = so(5) so(5), sp(4), su(8), so(6), respectively. Ackowledgemets. We thak Klaus Metsch for poitig out to us the idetity of sums of biomial coefficiets i Propositio 4.. This research has bee partially fuded by the EPRSC grat EP/H083X. The secod author gratefully ackowledges the hospitality of the IHES at Bures-sur-Yvette ad of the Albert Eistei Istitute at Golm. E 3 = A A E 4 = A E 5 = D 5 E E E Figure. The Dyki diagrams of types E 3 to E 8.

3 GENERALIZED SPIN REPRESENTATIONS.PART : CARTAN BOTT PERIODICITY FOR THE SPLIT REAL E SERIES3. Carta Bott periodicity of Clifford algebras Let N = {,, 3,...} be the set of atural umbers, ad let R, C, resp. H deote the reals, complex umbers resp. quaterios. For N ad a divisio rig D, deote by M(, D) the D-algebra of matrices over D. Let V be a R-vector space ad q : V R a quadratic form with associated biliear form b. The the Clifford algebra C(V, q) is defied as C(V, q) := T (V )/ vw + wv b(v, w) where T (V ) is the tesor algebra of V ; cf. [KY05, Sectio 4.3], [LM89, Chapter, ]. Let V = R with stadard basis vectors v i, let q = x + + x. The i C(V, q) we have v i = ad v iv j = v j v i. Propositio. (Carta Bott periodicity). For, the Clifford algebra C(R, q) is isomorphic to the followig algebra: (0) R R M(, R), if 0 (mod 8), () (R R) R M(, R), if (mod 8), () M(, R) R M(, R), if (mod 8), (3) M(, C) R M( 3, R), if 3 (mod 8), (4) M(, H) R M( 4, R), if 4 (mod 8), (5) (M(, H) M(, H)) R M( 5, R), if 5 (mod 8), (6) M(4, H) R M( 6, R), if 6 (mod 8), (7) M(8, C) R M( 7, R), if 7 (mod 8). Proof. See e.g. [KY05, Propositio Table 4.4.]. Sice C(V, q) is a associative algebra, it becomes a Lie algebra by settig [A, B] := AB BA. With this i mid, Propositio. implies the followig: Corollary.. For, the maximal semisimple compact Lie subalgebra of the Clifford algebra C(R, q) is isomorphic to the followig Lie algebra: (0) so( ), if 0 (mod 8), () so( ) so( ), if (mod 8), () so( ), if (mod 8), (3) su( ), if 3 (mod 8), (4) sp( ), if 4 (mod 8), (5) sp( 3 ) sp( 3 ), if 5 (mod 8), (6) sp( ), if 6 (mod 8), (7) su( ), if 7 (mod 8). 3. A lower boud o the dimesio of a subalgebra Defiitio 3.. For 3 let m be the Lie subalgebra of C(R, q) geerated by v v v 3 ad by v i v i+, i <. Lemma 3.. Let 3. The m cotais all products of the form v j v j v jk for k ad k, 3 (mod 4) with pairwise distict j t {,..., }, with the possible exceptio of v v v. The exceptio ca oly happe if 3 (mod 4). Proof. It is well-kow that all products v j v j, j j, are cotaied i m: Ideed, Λ R = so() (cf., e.g., [LM89, Propositio 6.]) is geerated as a Lie algebra by the v i v i+, i < (cf., e.g., [Ber89, Theorem.3], [HKL3, Theorem.]).

4 4 MAX HORN AND RALF KÖHL (NÉ GRAMLICH) Moreover, for pairwise distict j t, t k +, oe has [v j v j, v j v j3 v k+ ] = v j v j3 v jk+. Sice re-orderig of the factors simply yields scalar multiples, this shows iductively that, as log as k +, oce a arbitrary factor of the form v j v j v jk is cotaied i the Lie subalgebra, all factors of that form are cotaied i the Lie subalgebra. This statemet is also true i the situatio k =, because i that case all factors of that form are scalar multiples of oe aother. We fially prove the claim by iductio over k. For k = ad k = 3, this is obvious. Suppose the claim holds for k 3 (mod 4), the the ext value for k to cosider is k + 3 (mod 4). By iductio hypothesis v 4 v 5 v k+3 m ad 0 [v v v 3, v 4 v 5 v k+3 ] = v v v 3 v 4 v k+3. If o the other had the claim holds for k (mod 4), the the ext value for k to cosider is k + 3 (mod 4). If k +, the by iductio hypothesis v 3 v 4 v k+ m ad 0 [v v v 3, v 3 v 4 v k+ ] = v v v 4 v k+. That is, the presece of all elemets of the form v j v j v j3 with pairwise distict j t {,..., } iductively allows us to costruct all elemets of the form v j v j v jk for k, 3 (mod 4) with pairwise distict j t {,..., } for all k, with the possible exceptio of the situatio k = 3 (mod 4), as the elemet v k+ does ot exist i that case. Remark 3.3. It will tur out later, as a cosequece of the proof of Theorem A based o dimesio argumets, that the above elemets i fact geerate m as a R-vector space ad that for 3 (mod 4) the elemet v v v ideed is ot cotaied i m, uless of course = 3. Defiitio 3.4. For k {0,,, 3}, let δ k : N N : i=0, i k (mod 4) ( ). i Remark 3.5. Let N ad let M be a set of size. The the umber of subsets of M of size k (mod 4) is precisely δ k (). Therefore δ 0 () + δ () + δ () + δ 3 () =. Cosequece 3.6. Let 3. The { δ () + δ 3 () if 3 (mod 4), dim m δ () + δ 3 () if 3 (mod 4). 4. Combiatorics of biomial coefficiets We ow tur the lower boud from Cosequece 3.6 ito a umerically explicit boud by derivig a closed formula i for the fuctios δ k. Propositio 4.. Let N ad k {0,,, 3}. (0) If 0 (mod 4), the { for k {, 3}, δ k () = + ( ) 4 + k for k {0, }. () If (mod 4), the δ k () = { + ( ) 4 3 for k {0, }, ( ) 4 3 for k {, 3}.

5 GENERALIZED SPIN REPRESENTATIONS.PART : CARTAN BOTT PERIODICITY FOR THE SPLIT REAL E SERIES5 () If (mod 4), the { for k {0, }, δ k () = + ( ) 4 + k for k {, 3}. (3) If 3 (mod 4), the δ k () = { ( ) for k {0, 3}, + ( ) for k {, }. Proof. Note first that the claimed idetities hold for {, }. The pairig S S {m}, where deotes symmetric differece, provides a bijectio betwee the set of subsets of M of eve order with the set of subsets of M of odd order. Combied with Remark 3.5 we coclude () δ 0 () + δ () = δ () + δ 3 () =. Moreover, the pairig S M \ S provides a bijectio (i) betwee the set of subsets of M of order (mod 4) ad the set of subsets of M of order 3 (mod 4), if 0 (mod 4), (ii) betwee the set of subsets of M of order 0 (mod 4) ad the set of subsets of M of order (mod 4) ad betwee the set of subsets of M of order (mod 4) ad the set of subsets of M of order 3 (mod 4), if (mod 4), (iii) betwee the set of subsets of M of order 0 (mod 4) ad the set of subsets of M of order (mod 4), if (mod 4), (iv) betwee the set of subsets of M of order 0 (mod 4) ad the set of subsets of M of order 3 (mod 4) ad betwee the set of subsets of M of order (mod 4) ad the set of subsets of M of order (mod 4), if 3 (mod 4). Hece δ () = δ 3 () for 0 (mod 4), δ 0 () = δ () ad δ () = δ 3 () for (mod 4), δ 0 () = δ () for (mod 4), δ 0 () = δ 3 () ad δ () = δ () for 3 (mod 4). Together with Equatio, this already yields the claim for (a), case k {, 3} ad for (c), case k {0, }. We will ow prove case k = 0 of (b), (d) by iductio, which by the above observatios implies all claims made i (b), (d). Let M be a set of order + ad let a, b M be distict elemets so that M = M {a, b} for a set M of cardiality. A subset S M of cardiality 0 (mod 4) satisfies exactly oe of the followig: (i) S M has cardiality 0 (mod 4), (ii) S \ {a} M has cardiality 3 (mod 4), (iii) S \ {b} M has cardiality 3 (mod 4), (iv) S \ {a, b} M has cardiality (mod 4). Hece for (mod 4) resp. + 3 (mod 4) we have δ 0 ( + ) = δ 0 () + δ () + δ 3 () = + δ 3 () ) = + ( ( ) 3 4 = ( ) 4 = (+) ( ) (+) 3 4 (+) 3,

6 6 MAX HORN AND RALF KÖHL (NÉ GRAMLICH) ad similarly for 3 (mod 4) resp. + (mod 4) we have δ 0 ( + ) = δ 0 () + δ () + δ 3 () = + δ 3 () ) = + ( ( ) = ( ) 3 4 = (+) + ( ) (+) 4 (+) 3. Next we prove case k = 0 of (a) usig (c) as a iductio hypothesis ad afterwards case k = of (c) usig (a) as a iductio hypothesis. By the above observatios this implies all claims made i (a) ad (c). I order to establish case k = 0 of (a) we use the exact same combiatorial iductio step as above ad arrive agai at δ 0 ( + ) = δ 0 () + δ () + δ 3 () = + δ 3 () = + ( + ( ) ) = + ( ) + 4 = (+) + ( ) as claimed. I order to establish case k = of (c) we use the same combiatorial iductio step as above but eed to observe that if S M is a subset of cardiality (mod 4), the S \ {a, b} may have cardiality (mod 4), 3 (mod 4) or, i two differet ways, 0 (mod 4). Therefore δ ( + ) = δ 0 () + δ () + δ 3 () = δ 0 () + ( = + ( ) ) + = + ( ) = (+) + ( ) (+) Combiig this with Cosequece 3.6 yields the followig: Cosequece 4.. Let N ad. (0) If 0 (mod 8), the dim m δ () + δ 3 () = + = ( ) () If (mod 8), the () If (mod 8), the dim m δ () + δ 3 () = = dim R (so( )). ) ( 3 = ( ) = dim R (so( ) so( )). dim m δ () + δ 3 () = + = ( ) = dim R (so( )).

7 GENERALIZED SPIN REPRESENTATIONS.PART : CARTAN BOTT PERIODICITY FOR THE SPLIT REAL E SERIES7 (3) If 3 (mod 8), the dim m + δ () + δ 3 () = = (4) If 4 (mod 8), the = dim R (su( )) +. dim m δ () + δ 3 () = + + = ( + ) (5) If 5 (mod 8), the (6) If 6 (mod 8), the dim m δ () + δ 3 () = = dim R (sp( )). ) ( + 3 = ( + ) = dim R (sp( 3 ) sp( 3 )). dim m δ () + δ 3 () = + + = ( + ) (7) If 7 (mod 8), the = dim R (sp( )). dim m + δ () + δ 3 () = = = dim R (su( )) Geeralized spi represetatios of the split real E series ad the resultig quotiets The example of a geeralized spi represetatio of the maximal compact subalgebra of the split real Kac Moody Lie algebra of type E 0 described i [DKN06], [DBHP06], [HKL3] geeralizes directly to the whole E series as follows. Let N, let g be the split real Kac Moody Lie algebra of type E, let k be its maximal compact subalgebra, ad let X i, i, be the Berma geerators of k (cf. [Ber89, Theorem.3], [HKL3, Theorem.]) eumerated i Bourbaki style as show i Figure, i.e., X, X 3, X 4,..., X belog to the A subdiagram, geeratig so(), ad X to the additioal ode. As i Sectio let q be the stadard positive defiite quadratic form o R ad let C = C(R, q) be the correspodig Clifford algebra, cosidered as a Lie algebra. Propositio 5.. Let 3. The assigmet X v v, X v v v 3, X j v j v j for 3 j defies a Lie algebra homomorphism ρ from k to the Lie subalgebra m of C geerated by v v v 3 ad by v i v i+, i <, called the stadard geeralized spi represetatio of k. Proof. The proof is based o the criterio established i [HKL3, Remark 4.5] ad is exactly the same as i the E 0 case discussed i [HKL3, Example 4.]. Proof of Theorem A. By [HKL3, Theorem 4.] ad sice E is simply laced ad coected for 4, the image m of ρ is semisimple ad compact. By Lemma 3. ad Cosequece 4., dim R (m) is at least as large as the dimesio of the semisimple maximal compact Lie subalgebra of C as give i Corollary.. The claim follows.

8 8 MAX HORN AND RALF KÖHL (NÉ GRAMLICH) Proof of Theorem B. Let g be a semisimple split real Lie algebra of type E 4 = A 4, E 5 = D 5, E 6, E 7 or E 8 ad g = k a its Iwasawa decompositio. Sice dim R (k) = dim R (), from the combiatorics of the respective root system we coclude that the maximal compact Lie subalgebra k has dimesio 0 = 4 5 = 4 ( 4 + ) = dim R (sp()) = dim R (so(5)) if = 4, 0 = 0 = dim R (sp() sp()) = dim R (so(5) so(5)) if = 5, 36 = 4 9 = 6 ( 6 + ) = dimr (sp( )) if = 6, 63 = 6 = dim R (su(8)) if = 7, = = 8 ( 8 ) = dim R (so(6)) if = 8. For 4 we may ow apply Theorem A ad deduce that the stadard geeralized spi represetatio ρ has to be ijective i these cases. This leaves the case E 3 = A A. Sice this diagram is ot irreducible, [HKL3, Theorem 4.] oly implies that im(ρ) = m is compact but ot that it is semisimple (ad ideed, it is ot). However, = 3 is also a exceptioal case for Lemma 3.. Takig that ito cosideratio, it follows that dim R (m) (, v v, v v 3, v v v 3 is a basis of m). O the other had, the Clifford algebra C is isomorphic to M(, C), hece k = u(), ad this has dimesio 4. Thus ρ is also ijective whe = 3. The claim follows. Refereces [Ber89] Stephe Berma. O geerators ad relatios for certai ivolutory subalgebras of Kac-Moody Lie algebras. Comm. Algebra 7 (989)(), pp [DBHP06] Sophie De Buyl, Marc Heeaux, Louis Paulot. Exteded E 8 ivariace of -dimesioal supergravity. J. High Eergy Phys. (006)(), pp. 056, pp. (electroic). [DKN06] Thibault Damour, Axel Kleischmidt, Herma Nicolai. Hidde symmetries ad the fermioic sector of eleve-dimesioal supergravity. Phys. Lett. B 634 (006)(-3), pp [HKL3] Gutram Haike, Ralf Köhl, Paul Levy. Geeralized spi represetatios. Part : reductive fiitedimesioal quotiets of maximal compact subalgebras of Kac Moody algebras. Preprit. 03. [KY05] Toshiyuki Kobayashi, Taro Yoshio. Compact Clifford Klei forms of symmetric spaces revisited. Pure Appl. Math. Q. (005), pp [LM89] H. Blaie Lawso, Marie-Louise Michelsoh. Spi geometry. Priceto, New Jersey JLU Giesse, Mathematisches Istitut, Ardtstrasse, 3539 Giesse, Germay address: max.hor@math.ui-giesse.de JLU Giesse, Mathematisches Istitut, Ardtstrasse, 3539 Giesse, Germay address: ralf.koehl@math.ui-giesse.de