Generalized spin representations ohl Ralf K 30 May 2013 n e - PowerPoint PPT Presentation

Generalized spin representations ohl ∗ Ralf K¨ 30 May 2013 ∗ n´ e Gramlich ralf.koehl@math.uni-giessen.de

Outline of talk Part 1: Generalized spin representations of ‘maximal com- pact’ subalgebras of simply laced Kac–Moody algebras • Berman’s presentation • Damour et al./Henneaux et al. description of E 10 GSR • GSR’s for arbitrary simply laced diagrams Part 2: ‘Maximal compact’ subgroups of simply laced Kac– Moody groups as amalgams of Lie groups • geometric group theory • buildings • integrated Berman-style/Borovoi-style presentation Part 3: Spin covers • lifting of presentation • construction of extended Weyl group

Part 1: Generalized spin representations of ‘maximal compact’ subalgebras of simply laced Kac–Moody algebras (joint with Hainke)

Simply laced real Kac–Moody algebras Let g be a simply laced real Kac–Moody algebra, presented by Gabber–Kac using Serre’s relations: The Kac–Moody algebra g is the quotient of the free Lie algebra over R generated by e i , f i , h i , i = 1 , . . . , n , subject to the relations [ h i , h j ] = 0 , [ h i , e j ] = a ij e j , [ h i , f j ] = − a ij f j , [ e i , f j ] = 0 , [ e i , f i ] = h i , (ad e i ) − a ij +1 ( e j ) = 0 , (ad f i ) − a ij +1 ( f j ) = 0 for i � = j with a ii = 2 and a ij ∈ { 0 , − 1 } for i � = j .

‘Maximal compact’ subalgebras of Kac–Moody algebras Let ω ∈ Aut( g ) be the Cartan–Chevalley involution: ω ( e i ) = − f i , ω ( f i ) = − e i , ω ( h i ) = − h i . The ‘maximal compact’ subalgebra is defined as k := { X ∈ g | ω ( X ) = X } . Theorem 1 (Berman 1989) The ‘maximal compact’ subalgebra k is isomorphic to the quo- tient of the free Lie algebra over R generated by X 1 , . . . , X n subject to the relations [ X i , [ X i , X j ]] = − X j , if the simple roots α i , α j form an edge, [ X i , X j ] = 0 , otherwise, via the map X i �→ e i − f i . The X i are called Berman generators .

Generalized spin representations of k A representation ρ : k → End( C s ) is called a generalized spin representation if the images of the Berman generators satisfy ρ ( X i ) 2 = − 1 4id s for i = 1 , . . . , n. Put A := ρ ( X i ), B := ρ ( X j ). If α i , α j do not form an edge: [ A, B ] 1 = 0 ⇐ ⇒ AB = BA. If α i , α j form an edge: = [ A, [ A, B ]] = [ A, AB − BA ] = A 2 B − 2 ABA + BA 2 = − 1 − B 1 2 B − 2 ABA Left-multiplying with − 4 A = A − 1 ( ⇐ ⇒ A 2 = − 1 4 id s ) yields: 4 AB = 2 AB − 2 BA ⇐ ⇒ AB = − BA

How to construct generalized spin representations? Conversely, suppose that there are matrices A i ∈ C s × s satisfying i = − 1 (i) A 2 4 · id s , (ii) A i A j = A j A i , if α i , α j do not form an edge, (iii) A i A j = − A j A i , if α i , α j form an edge. Then, by reversing the argument on the previous slide, the as- signment X i �→ A i gives rise to a representation of k .

A motivating example (Damour et al., Henneaux et al.) This example extends the spin representation of so (10). Let • V = R 10 with standard basis vectors v i , • q : V → R : x �→ x 2 1 + · · · + x 2 10 , • b : V × V → R : ( x, y ) �→ 2( x 1 y 1 + · · · + x 10 y 10 ) associated bilinear form, • T ( V ) the tensor algebra of V , • C := C ( V, q ) := T ( V ) / � vw + wv − b ( v, w ) � the Clifford algebra . In C we have v 2 i = 1 and v i v j = − v j v i for i � = j . Since C is associative, it becomes a Lie algebra by setting [ A, B ] := AB − BA.

Let the diagram of E 10 be labelled as 123 s s s s s s s s s s 12 23 34 45 56 67 78 89 910 and define a Lie algebra homomorphism ρ : k ( E 10 ) → C using these labels, i.e., via X 1 �→ 1 X 3 �→ 1 X 2 �→ 1 2 v 1 v 2 , 2 v 1 v 2 v 3 , 2 v 2 v 3 , X 4 �→ 1 X 6 �→ 1 X 5 �→ 1 2 v 3 v 4 , 2 v 4 v 5 , 2 v 5 v 6 , X 7 �→ 1 X 9 �→ 1 X 8 �→ 1 2 v 6 v 7 , 2 v 7 v 8 , 2 v 8 v 9 , X 10 �→ 1 2 v 9 v 10 , where X i denotes the Berman generator corresponding to the root α i , enumerated in Bourbaki style.

i = − 1 Observe that each A i := ρ ( X i ) satisfies A 2 4 id. Note that ( v 1 v 2 v 3 ) 2 = ( v 2 v 3 ) 2 = − 1 depends on v 2 i = 1; for parity reasons, this would not be true in the Clifford algebra C ( V, − q ), as then ( v 1 v 2 v 3 ) 2 = − ( v 2 v 3 ) 2 = 1. Using the criterion established above, one checks that ρ indeed is a Lie algebra homomorphism, i.e., that the defining relations of k from Theorem 1 are respected. One needs to establish i = − 1 (i) A 2 4 · id s , (ii) A i A j = A j A i , if α i , α j do not form an edge, (iii) A i A j = − A j A i , if α i , α j form an edge.

We have already observed (i). Assertions (ii) and (iii) are obvious for i, j ∈ { 1 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 } (spin representation). Moreover, one computes ( v 1 v 2 v 3 )( v 3 v 4 ) = − ( v 3 v 4 )( v 1 v 2 v 3 ) and ( v 1 v 2 v 3 )( v k 1 v k 2 ) = ( v k 1 v k 2 )( v 1 v 2 v 3 ) , if { k 1 , k 2 } is a set of two elements that is either a subset of { 1 , 2 , 3 } or disjoint from { 1 , 2 , 3 } .

The extension theorem for generalized spin representations (GSR) Theorem 2 (Hainke, K.) Let 1 ≤ r < n , k ≤ r := � X 1 , . . . , X r � , ρ : k ≤ r → End( C s ) a GSR. (i) If X r +1 centralizes k ≤ r , then ρ extends to a GSR ρ ′ : k ≤ r +1 → End( C s ) via ρ ′ ( X r +1 ) := 1 2 i · id s . (ii) If X r +1 does not centralize k ≤ r , then ρ extends to a GSR ρ ′ : k ≤ r +1 → End( C s ⊕ C s ) as follows. Define � if α i , α r +1 do not form an edge , X i , s 0 ( X i ) := − X i , if α i , α r +1 form an edge , and let � � ρ ′ ( X r +1 ) := 1 0 1 ρ ′ | k ≤ r := ρ ⊕ ρ ◦ s 0 and 2 i · id s ⊗ . 1 0

Proof If X r +1 centralizes k ≤ r : ρ ′ ( X r +1 ) 2 = − 1 4 id s and ρ ′ ( X r +1 ) com- mutes with everything. The criterion above applies. If X r +1 does not centralize k ≤ r : ρ ′ | k ≤ r is a GSR of k ≤ r which extends ρ . (Multiplication with − 1 does not change (anti)com- mutation relation.) Moreover, ρ ′ ( X i ) commutes with ρ ′ ( X r +1 ), if α i , α r +1 not an edge; and ρ ′ ( X i ) anticommutes with ρ ′ ( X r +1 ), if α i , α r +1 an edge: � � � � � � � � � � 1 0 0 i 0 i 0 i 1 0 = − = 0 − 1 0 − i 0 0 0 − 1 i i Again the criterion above applies.

Quotients Corollary 3 k admits ‘many’ compact quotients. Proof: Let ρ be a GSR as constructed in Theorem 2. Considering C ∼ = R 2 , multiplication by i can be realized via the � � − 1 0 skew-symmetric matrix . 1 0 If the representation of k ≤ r is given by skew-symmetric matrices, then step (ii) can be made to involve skew-symmetric matrices only, as � � � � 0 0 1 i and 0 − 1 0 i are C -conjugate (minimum polynomial x 2 + 1).

Quotients, ii Corollary 4 Assume the diagram does not admit any isolated nodes. Then k admits ‘many’ semisimple quotients. Proof: Compact + perfect = ⇒ semisimple. Example: The GSR by Damour et al./Henneaux et al. leads to k ( E 10 ) ։ so 32 .

Part 2: ‘Maximal compact’ subgroups of simply laced Kac–Moody groups as amalgams of Lie groups (Classical facts)

‘Maximal compact’ subgroups Let • G a simply connected simply laced split Kac–Moody group, • T a maximal torus, • ω a Cartan–Chevalley involution fixing T , • K := Fix G ( ω ) ‘maximal compact’ subgroup. Theorem 5 (Iwasawa decomposition; Kac–Peterson 1980ies) Let B be a Borel subgroup of G containing the torus T . Then G = KB.

Presentations arising from group actions on simply connected simplicial complexes Theorem 6 (Simplicial geometric group theory) Let • ∆ simply connected finite-dim. coloured simpl. complex, • G → Aut(∆) colour-preserving simplicial rigid action, transitive on maximal simplices, • c maximal simplex, • I index set for vertices of c , • ( G J ) ∅� = J ⊆ I family of pointwise stabilizers of non-empty sub- simplices of c , → G J ′ canonical embedding for J ⊇ J ′ . • φ J,J ′ : G J ֒ Then � � all relations in the G J plus G ∼ � G J | = . all identifications via the φ J,J ′ ∅� = J ⊆ I Terminology: ( G J ) ∅� = J ⊆ I together with the connecting mor- phisms is a diagram of groups . The group G is called a colimit .

Theorem 7 (Non-simplicial version) Let • X simply connected topological space, • G → Homeo( X ) action, • U an open path-connected weak fundamental domain (i.e., X = G.U), • Σ = { g ∈ G | U ∩ g.U � = ∅} , • R = { xy = ( xy ) | x, y ∈ Σ , U ∩ xU ∩ xyU � = ∅} . Then G ∼ = � Σ | R � . Theorem 7 implies Theorem 6: Define U as an ǫ -neighbourhood of the maximal simplex c .

Example 8 Let Sym 4 act naturally on the barycentric subdivision of a 3- simplex considered as a 2-dimensional simplicial complex. Let c be the maximal simplex consisting of the vertex 1, the barycentre of the edge { 1 , 2 } , and the barycentre of the face { 1 , 2 , 3 } . Then G 1 = Sym { 2 , 3 , 4 } G { 1 , 2 } = Sym { 1 , 2 } × Sym { 3 , 4 } G { 1 , 2 , 3 } = Sym { 1 , 2 , 3 } . The other stabilizers arise as intersections. Theorem 6 states that ∼ Sym 4 � G 1 ∪ G { 1 , 2 } ∪ G { 1 , 2 , 3 } | all relations in these groups � = ∼ i = 1 , ( s i s i +1 ) 3 = 1 , s 1 s 3 = s 3 s 1 � � s 1 , s 2 , s 3 | s 2 = (Think s 1 = (12), s 2 = (23), s 3 = (34).)