A bundle-theoretic perspective to Kac Moody groups

A bundle-theoretic perspective to Kac Moody groups Christoph Wockel May 31, 2007

Outline Kac Moody algebras and groups Central extensions and cocycles Cocycles for twisted loop algebras and -groups Computation of homotopy groups

Kac-Moody algebras Generalisations of simple, finite-dimensional, complex Lie algebras: A Mat n (Z) : generalised Cartan matrix g(a) : assoc. Kac Moody algebra (generators and relations) g(a) finite-dimensional A is positive definite g(a) affine : A positive semidefinite and rk(a) = n 1 Realisation in the affine case: k := g(a) finite-dimensional, simple (A: generic from A) Lk := C[t, t 1 ] k σ Aut(k) with σ m = 1 for m {1, 2, 3} Set ε := exp( 2πi m ) σ Aut(Lk), tj x ε j t j σ(x). Lk σ := (Lk) σ (twisted loop algebra)

Kac Moody groups Theorem (Kac) Lk σ KC dc is a realisation of g(a) and g(a) = Lk σ KC.

Kac Moody groups Theorem (Kac) Lk σ KC dc is a realisation of g(a) and g(a) = Lk σ KC. The Tits functor A G(A) produces groups, associated to Kac-Moody algebras and a partially defined exponential function x exp G (x) for x α g α Proposition Let (V, π) be an integrable g(a) -module. Then there exists a unique homomorphism π : G(A) Aut(V ) satisfying exp π(x) = π exp G (x) for x α g α This fact should (at least) be satisfied by each notion of a Kac Moody group.

Twisted loop algebras and algebras of sections switch form the complex to the real case k : finite-dimensional Lie algebra, ϕ Aut(k) P ϕ : Lie algebra-bundle over S 1 with holonomy ϕ, i.e. (R k)/ where (x, y) (x + 1, ϕ 1 (y)) algebra of sections: ΓP ϕ = {f : R k : f (x + 1) = ϕ 1 (f (x))} algebra of sections ΓP ϕ is a twisted loop algebra

Twisted loop algebras and algebras of sections switch form the complex to the real case k : finite-dimensional Lie algebra, ϕ Aut(k) P ϕ : Lie algebra-bundle over S 1 with holonomy ϕ, i.e. algebra of sections: (R k)/ where (x, y) (x + 1, ϕ 1 (y)) ΓP ϕ = {f : R k : f (x + 1) = ϕ 1 (f (x))} algebra of sections ΓP ϕ is a twisted loop algebra Problem Construct central extensions ΓP ϕ and corresponding Lie groups G( ΓP ϕ ), satisfying the previous integrability condition.

Differential calculus in locally convex spaces Definition (Milnor) X, Y : locally convex spaces, U X open subset. A map f : U Y is differentiable, if the differential quotients df (x, v) := lim h 0 f (x + hv) f (x) h exist for each v X, and (x, v) df (x, v) is jointly continuous. smooth mappings locally convex manifolds an Lie groups differentiable mappings are continuous Example unit groups of Banach algebras mapping groups C (M, G), sections in Lie group bundles ΓK groups of diffeomorphisms Diff(M) for compact M direct limits of finite-dimensional Lie groups

Cohomology groups for central extensions Setting: z: locally convex space, Γ z: discrete subgroup Z := z/γ with π 1 (Z) = Γ G: connected l.c. Lie group with Lie algebra g := L(G).

Cohomology groups for central extensions Setting: z: locally convex space, Γ z: discrete subgroup Z := z/γ with π 1 (Z) = Γ G: connected l.c. Lie group with Lie algebra g := L(G). Lie algebra cocycle: ω : g g z, continuous, bilinear, alternating, satisfying ω([x, y], z) + ω([y, z], x) + ω([z, x], y) = 0 Lie group cocycle: f : G G Z, smooth at (e, e), satisfying...

Central Extensions and smooth Lie group cohomology Central extension of Lie algebras: linearly split exact sequence 0 z ĝ g 0 Central extension of Lie groups: locally trivial exact sequence Theorem (Neeb (02)) 1 Z Ĝ G 1 With the canonical identifications H 2 s (G, Z) = Ext Lie (G, Z) and H 2 c (g, z) = Ext c (g, z) there is an exact sequence Hom(π 1 (G), Z) Ext Lie (G, Z) D Ext c (g, z) P Hom(π 2 (G), Z) Hom(π 1 (G), Lin(g, z)),

Cocycles for mapping algebras (trivial bundles) k: finite-dimensional Lie algebra

Cocycles for mapping algebras (trivial bundles) k: finite-dimensional Lie algebra κ : k k V biliniear, symmetric, κ([x, y], z) = κ(x, [y, z])

Cocycles for mapping algebras (trivial bundles) k: finite-dimensional Lie algebra κ : k k V biliniear, symmetric, κ([x, y], z) = κ(x, [y, z]) M compact, g := C (M, k), z := Ω 1 (M, V )/dc (M, V )

Cocycles for mapping algebras (trivial bundles) k: finite-dimensional Lie algebra κ : k k V biliniear, symmetric, κ([x, y], z) = κ(x, [y, z]) M compact, g := C (M, k), z := Ω 1 (M, V )/dc (M, V ) ω κ : g g z (f, g) [κ(f, dg)] is a Lie algebra cocycle: [κ(f, dg) + κ(g, df )] = [κ(f, dg) + κ(df, g)] = [dκ(f, g)] = 0 [κ([f, g], dh) + κ([g, h], df ) + κ([h, f ], dg)] = [κ([f, g], dh) + κ([df, g], h) + κ([f, dg], h) ] = [dκ([f, g], h)] }{{} =κ(d[f,g],h) Theorem (Pressley, Segal (86); Maier, Neeb (03)) If κ is universal, then P([ω κ ]) vanishes, i.e., the corresponding central extension of C (M, k) integrates to a central extension of C (M, K) 0. Moreover, both central extensions are universal.

Cocycles for algebras of sections P M principal K-bundle, (M compact)

Cocycles for algebras of sections P M principal K-bundle, (M compact) P 0 := P/K 0 squeezed π 0 (K)-bundle (regular cover of M)

Cocycles for algebras of sections P M principal K-bundle, (M compact) P 0 := P/K 0 squeezed π 0 (K)-bundle (regular cover of M) π 0 (K)-module (V, ρ) ( K-module with K 0 ker(ρ))

Cocycles for algebras of sections P M principal K-bundle, (M compact) P 0 := P/K 0 squeezed π 0 (K)-bundle (regular cover of M) π 0 (K)-module (V, ρ) ( K-module with K 0 ker(ρ)) κ : k k V bilinear, symmetric, K-equivariant : connection on P (and on associated vector bundles) associated vector bundles K := P Ad k and V := P 0 ρ V induced maps between sections Γκ : ΓK ΓK ΓV g := ΓK, z := Ω 1 (M, V)/d(ΓP V ) ω κ, : g g z (f, g) [Γκ(f, g)] is a Lie algebra cocycle

Integrating the cocycle ω κ P 0 := P/K 0 is a π 0 (K)-principal bundle, i.e., a regular cover of M with π 0 (K) = π 1 (M)/π 1 (P 0 )

Integrating the cocycle ω κ P 0 := P/K 0 is a π 0 (K)-principal bundle, i.e., a regular cover of M with π 0 (K) = π 1 (M)/π 1 (P 0 ) π 0 (K) acts on P 0 and accordingly on H 1 (P 0 ) (canonical action of π 1 (M)/π 1 (P 0 ) on π 1 (P 0 ) ab = H1 (P 0 ) by conjugation) Theorem (W. (07)) The group of sections ΓK in the Lie group bundle K := P conj K ( gauge group ) is an infinite-dimensional Lie group with Lie algebra L(ΓK) = ΓK. Theorem (Neeb, W. (07)) If κ is universal, π 0 (K) is finite and acts trivially on H 1 (P 0 ), then P([ ω κ ]) vanishes, i.e., the corresponding central extension of ΓK integrates to a central extension of (ΓK) 0.

Further Generalisations approach generalises to algebras/group of sections in Lie algebra- and Lie group bundles additional ingredient: smooth action ρ : K H H Lie group bundle H and Lie algebra bundle H. associated group and algebra of sections ΓH and ΓH all other ingredients compatible with this additional structure Example K = T = S 1, H = C (S 1, G) for finite-dim. Lie group G action: ρ(t, γ)(t ) = γ(t t ) ΓH = C (P, G) more generally, for N compact, Diff(N) acts canonically on C (N, G) and a Diff(N)-principal bundle yields an associated fibre bundle P N = P Diff(N) N with ΓH = C (P N, G)

The route back to Kac Moody groups Theorem (Milnor (84)) If H is a regular Lie group, G is a simply connected Lie gorup, then each morphism ϕ : L(G) L(H) lifts to a unique morphism Φ : G H.

The route back to Kac Moody groups Theorem (Milnor (84)) If H is a regular Lie group, G is a simply connected Lie gorup, then each morphism ϕ : L(G) L(H) lifts to a unique morphism Φ : G H. Corollary G simply connected, V Banach space each representation ϕ : L(G) End(V ) lifts to a unique representation Φ : G GL(V )

Homotopy groups of twisted loop groups k finite-dimensional, simple and compact K simply connected Lie group with Lie algebra L(K) = k ϕ Aut(k), lifting to Φ Aut(K)

Homotopy groups of twisted loop groups k finite-dimensional, simple and compact K simply connected Lie group with Lie algebra L(K) = k ϕ Aut(k), lifting to Φ Aut(K) P Φ bundle (over S 1 ) with holonomy Φ, ΓP Φ = {f : R K : f (x + 1) = Φ 1 (f (x))} twisted loop group P ϕ bundle with holonomy ϕ, ΓP ϕ twisted loop algebra Fibration ev 0 : ΓP Φ K, γ γ(0) gives a long exact sequence... π 1 (ker(ev 0 )) π }{{} 1 (ΓP Φ ) π 1 (K)... ΩK

Homotopy groups of central extensions Setting as on the previous slide, g := ΓP ϕ, G := (ΓP Φ ) 0

Homotopy groups of central extensions Setting as on the previous slide, g := ΓP ϕ, G := (ΓP Φ ) 0 κ : k k R Cartan Killing form (universal, for k is compact) yields a cocycle ω κ and a central extension 0 z ĝ g 0 integrates, by the above theorem, to a central extension 1 Z Ĝ G 1 for some Z = z/γ and gives rise to an exact sequence... π 2 (G) δ π 1 (Z) π 1 (Ĝ) π 1(G)...

Lifting property for Kac Moody groups Proposition For the cocycle ΓP ϕ ΓP ϕ Ω 1 (S 1, V)/d(ΓV), (f, g) [Γκ(f, g)], the corresponding central extension ĝ of g := ΓP ϕ integrates to a central extension Ĝ of G := (ΓP Φ) 0. Moreover, each representation of ĝ on a Banach space lifts to a unique representation of Ĝ. Ĝ has the lifting property for the Kac Moody algebra ĝ call Ĝ the Kac Moody group associated to ϕ Aut(k) generalisations as central extensions of ΓK for K = P Ad k for flat principal bundles P seems to be appropriate access to Aut(ΓK) as the group Aut(P) of bundle automorphisms ( leads to Lie group structures on Aut(ΓK))