Bundles over quantum weighted projective spaces Tomasz Swansea University Lancaster, September 2012 Joint work with Simon A Fairfax References: TB & SAF, Quantum teardrops, Comm. Math. Phys. in press (arxiv:1107.1417) TB & SAF, Bundles over quantum real weighted projective spaces, Axioms in press (arxiv:1207.2313) TB, Circle actions on a quantum Seifert manifold, Proceedings of Science (CORFU2011) (arxiv:1203.6801)
Themes Examples of quantum orbifolds. Every n-orbifold is a quotient of a manifold by an almost free action of a subgroup of O(n). Every orbifold Riemannian surface is a quotient of a Seifert 3-manifold by an action of S 1. Illustration of noncommutative resolution of singularities.
Quantum spheres Let q be a real number, 0 < q < 1. The coordinate algebra O(Sq 2n+1 ) of the odd-dimensional quantum sphere is the unital complex -algebra with generators z 0, z 1,..., z n subject to the following relations: z i z j = qz j z i for i < j, z i z j = qz j z i for i j, z i z i = z i z i + (q 2 1) n m=i+1 z m z m, n z m zm = 1. m=0 The coordinate algebra O(S 2n q ) of the even-dimensional quantum sphere is the unital complex -algebra with generators z 0, z 1,..., z n and above relations supplemented with z n = z n.
For any n + 1 pairwise coprime numbers l 0,..., l n, O(U(1)) = C[u, u ] coacts on O(Sq 2n+1 ), as ϱ l0,...,l n : z i z i u l i, i = 0, 1,..., n. The quantum weighted projective space is the subalgebra of O(Sq 2n+1 ) containing all elements invariant under the coaction ϱ l0,...,l n, i.e. O(WP q (l 0, l 1,..., l n )) = {x O(Sq 2n+1 ) ϱ l0,...,l n (x) = x 1}. In the case l 0 = l 1 = = 1 one obtains the algebra of functions on the quantum complex projective space CP n q defined in [YaS Soibel man & LL Vaksman 90].
Quantum weighted projective lines The weighted projective line O(WP q (k, l)) is the -algebra generated by a and b subject to the following relations l 1 a = a, ab = q 2l ba, bb = q 2kl a k (1 q 2m a), Embedding in O(S 3 q): b b = a k l (1 q 2m a). m=1 m=0 a z 1 z 1, b zl 0 z k 1.
Quantum weighted projective lines.special cases O(WP q (1, 1)) is the standard Podleś sphere. O(WP q (1, l)) is the quantum teardrop. No repeated roots in polynomial relations. Classical singularities are resolved.
Representations Theorem Up to a unitary equivalence, the following is the list of all bounded irreducible -representations of O(WP q (k, l)). (i) One dimensional representation ϕ 0 : a 0, b 0. (ii) Infinite dimensional faithful representations ϕ s : O(WP q (k, l)) End(V s ), s = 1, 2,..., l. V s l 2 (N) has orthonormal basis e s p, p N, ϕ s (b)e s 0 = 0 and ϕ s (b)e s p = q k(lp+s) ϕ s (a)e s p = q 2(lp+s) e s p, l r=1 ( 1 q 2(lp+s r)) 1/2 e s p 1.
C(WP q (k, l)) and K -groups Theorem Let K s denote the algebra of all compact operators on the Hilbert space V s. There is a split-exact sequence of C -algebra maps 0 l s=1 K s C(WP q (k, l)) C 0. Consequently, K 0 (C(WP q (k, l))) = Z l+1, K 1 (C(WP q (k, l))) = 0.
Quantum principal bundles Definition Let H be a Hopf algebra with bijective antipode and let A be a right H-comodule algebra with coaction ϱ : A A H. Let B = A coh := {b A ϱ(b) = b 1}. A is a principal H-comodule algebra if: (a) the coaction ϱ is free, that is the canonical map can : A B A A H, a a aϱ(a ), is bijective (the Hopf-Galois condition); (b) there exists a strong connection in A, that is B A A, b a ba, splits as a left B-module and right H-comodule map (the equivariant projectivity).
Strong connection forms A right H-comodule algebra A with coaction ϱ : A A H is principal if and only if it admits a strong connection form, that is if there exists a map ω : H A A, such that ω(1) = 1 1, µ ω = η ε, (ω id) = (id ϱ) ω, (S ω) = (σ id) (ϱ id) ω. Here µ : A A A denotes the multiplication map, η : C A is the unit map, : H H H is the comultiplication, ε : H C counit and S : H H the (bijective) antipode of the Hopf algebra H, and σ : A H H A is the flip.
Freeness and almost freeness Theorem The algebra O(S 3 q) is a principal O(U(1))-comodule algebra over O(WP q (k, l)) by the coaction ϱ k,l if and only if k = l = 1 (the quantum Hopf fibration). Remark The coaction ϱ 1,l is almost free, meaning that the cokernel of the canonical map can is a finitely generated left O(S 3 q)-module.
Carving out the total space C[Z l ] coacts on O(S 3 q) by z 0 z 0 w, z 1 z 1 1. Coinvariants are the quantum lens space O(L q (l; 1, l)) [JH Hong & W Szymański 03]. It is generated by c = z l 0 and d = z 1, and admits an O(U(1))-coaction c c u, d d u. O(L q (l; 1, l)) is a non-cleft principal O(U(1))-comodule algebra over O(WP q (1, l)). Strong connection form: ω(u n )=c ω(u n 1 )c l ( ) l ( ) m q m(m+1) d m d m 1 ω(u n 1 )d. m q 2 m=1 (Non-cleft because multiples of 1 are the only invertible elements of O(S 3 q).)
Line bundles For any integer n, modules of sections on line bundles over O(WP q (1, l)) (associated to the principal comodule algebra O(L q (l; 1, l))) are defined as L[n] := {x O(L q (l; 1, l)) ϱ l (x) = x u n }. Each L[n] is a projective left O(WP q (1, l))-module with an idempotent E[n] given as follows. Write ω(u n ) = i ω(v n ) [1] i ω(u n ) [2] i. Then E[n] ij = ω(u n ) [2] iω(u n ) [1] j O(WP q (1, l)).
Index pairing Theorem For every s = 1, 2,..., l, (ϕ s, ϕ 0 ) is a 1-summable Fredholm module over O(WP q (k, l)) with the Chern character τ s (a m b n ) = for n = 0, m 0, and 0 otherwise. q2ms 1 q 2ml, Theorem For all s = 1, 2,..., l, τ s (Tr E[1]) = 1. Consequently, the left O(WP q (1, l))-module L[1] is not free.
Prolongations of quantum spheres Quantum spheres are right comodule algebras over the Hopf algebra O(Z 2 ) generated by a self-adjoint grouplike element v satisfying v 2 = 1. The coaction is z i z i v. O(U(1)) is a left O(Z 2 )-comodule algebra with coaction: u v u. Quantum spheres can be prolonged to algebras O(Σ m+1 q ) := O(Sq m ) O(Z2 )O(U(1)).
The structure of O(Σ m q ) Theorem (TB & BP Zieliński) 1 O(Σ 2n+2 q ) is an algebra isomorphic to O(Sq 2n+1 ) O(U(1)). 2 O(Σ 2n+1 q ) is isomorphic to a polynomial -algebra generated by ζ 0, ζ 1,..., ζ n and a central unitary ξ subject to the following relations ζ i ζ j = qζ j ζ i for i < j, ζ i ζ j = qζ j ζ i for i j, ζ i ζ i = ζ i ζ i + (q 2 1) 3 For all m > 1, Σ m+1 q bundles over RP m q. n m=i+1 ζ n = ζ n ξ. ζ m ζ m, n ζ m ζm = 1, m=0 are non-trivial quantum U(1)-principal
Weighted U(1)-actions on O(Σ 2n+1 q ) Choose n + 1 pairwise coprime numbers l 0,..., l n. O(Σ 2n+1 q ) is a right O(U(1))-comodule algebra with coaction Define ϱ l0,...,l n : ξ ξ u 2ln, ζ i ζ i u l i. O(RP q (l 0, l 1,..., l n )) := {x O(Σ 2n+1 q ) ϱ l0,...,l n (x) = x 1}.
Circle actions on the quantum Seifert manifold Σ 3 q The even case. O(RP q (2s, l)) is generated by a = ζ 2 1 ξ and c + = ζ l 0 ξs, which satisfy the following relations: a = a, ac + = q 2l c + a, c + c + = l 1 (1 q 2m a), c+c + = m=0 l (1 q 2m a). m=1 A -algebra generated by a, c + and above relations (with l odd) is denoted by O(RP 2 q(l; +)).
Circle actions on the quantum Seifert manifold Σ 3 q The odd case. O(RP q (2s 1, l)) is generated by a = ζ 2 1 ξ, b = ζl 0 ζ 1ξ s and c = ζ 2l 0 ξk, which satisfy the following relations: a = a, ab = q 2l ba, ac = q 4l c a, b 2 = q 3l ac, bc = q 2l c b, l 1 bb = q 2l a (1 q 2m a), b b = a b c = q l m=0 l (1 q 2m a), m=1 l l 1 (1 q 2m a)b, c b = q l b (1 q 2m a), m=1 2l 1 c c = (1 q 2m a), c c = m=0 2l m=1 m=0 (1 q 2m a). A -algebra generated by a, b, c and above relations is denoted by O(RP 2 q(l; )).
Observations RP q (1; +) is the quantum disc. RP q (1; ) is the quantum real projective plane RP 2 q [PM Hajac, R Matthes, W Szymański]. No multiple poles in polynomial relations for RP q (l; ±); the classical singularities are resolved.
Bounded representations of O(RP q (l; +)) There is a family of one-dimensional representations of O(RP 2 q(l; +)) labelled by θ [0, 1) and given by π θ (a) = 0, π θ (c + ) = e 2πiθ. All other representations are infinite dimensional, labelled by r = 1,..., l, and given by π r (a)e r n = q 2(ln+r) e r n, π r (c + )e r n = l m=1 ( 1 q 2(ln+r m)) 1/2 e r n 1, where e r n, n N is an orthonormal basis for the representation space H r = l 2 (N).
Bounded representations of O(RP q (l; )) A family of one-dimensional representations labelled by θ [0, 1): π θ (a) = 0, π θ (b) = 0, π θ (c ) = e 2πiθ. All other representations are infinite dimensional, labelled by r = 1,..., l: π r (a)e r n = q 2(ln+r) e r n, π r (c )e r n = π r (b)e r n = q ln+r l m=1 2l m=1 ( 1 q 2(ln+r m)) 1/2 e r n 2, ( 1 q 2(ln+r m)) 1/2 e r n 1, where e r n, n N, is an orthonormal basis for the representation space H r = l 2 (N).
K -theory of C(RP q (l; ±)) Let J ± be ideals of C(RP q (l; ±)) generated by a; p ± : C(RP q (l; ±)) C(RP q (l; ±))/J ±. C(RP q (l; ±))/J ± is generated by p ± (c ± ) and C(RP q (l; ±))/J ± = C(S 1 ). There are short exact sequences: 0 l r=1 K r i ± C(RP q (l; ±)) p ± C(S 1 ) 0. The six-term sequences of the K -groups lead to 0 K 1 (C(RP q (l; ±))) Z δ ± Z l K 0 (i ± ) K 0 (C(RP q (l; ±))) K 0(p ± ) Z 0.
K -theory of C(RP q (l; ±)) The connecting map δ + comes out as δ + : Z Z l, m (m, m,..., m). Since δ + is injective, K 1 (C(RP q (l; +))) = 0. Furthermore, K 0 (C(RP q (l; +))) = Im(K 0 (i + )) Z = coker(δ + ) Z = Z l, The connecting map δ comes out as δ : Z Z l, m (2m, 2m,..., 2m). Since δ is injective, K 1 (C(RP q (l; ))) = 0. Furthermore, K 0 (C(RP q (l; ))) = coker(δ ) Z = Z 2 Z l,
The structure of C(RP q (l; +)) Starting from 0 K T σ C(S 1 ) 0, one finds C(RP q (l; +)) = T σ T σ σ T }{{} l times = C(D q ) σ C(D q ) σ σ C(D q ). }{{} l times The symbol map σ can be understood as a projection induced by the inclusion of the classical circle as the boundary of the quantum disc. C(RP q (l; +)) is a quantum double suspension of l points [Hong-Szymański].
The structure of C(RP q (l; )) Starting from 0 K C(RP 2 q) σ C(S 1 ) 0 [Hajac-Matthes-Szymański], one finds C(RP q (l; )) = C(RP 2 q) σ C(RP 2 q) σ σ C(RP 2 q), }{{} l times where the map σ is induced from the projection that corresponds to the inclusion of the classical circle as the equator of the quantum sphere S 2 q.
Freeness and almost freeness Theorem The algebra O(Σ 3 q) is a principal O(U(1))-comodule algebra over O(RP q (k, l)) by the coaction ϱ k,l if and only if k = l = 1 (the circle bundle over RP 2 q). Remark For O(RP q (l; ±)), the cokernel of the canonical map can : O(Σ 3 q) O(RPq(l;±)) O(Σ 3 q) O(Σ 3 q) O(U(1)), (corresponding to ϱ k,l, k = 1, 2) is a finitely generated left O(Σ 3 q)-module.
Carving out total spaces C[Z l ] coacts on O(Σ 3 q) by ζ 0 ζ 0 w, ζ 1 ζ 1 1, ξ ξ 1. Algebra of coinvariants O(Σ 3 q,l ) is generated by α = ζl 0 and ζ 1, ξ, and admits an O(U(1))-coaction α α u, ζ 1 ζ 1 u, ξ ξ u 2. O(Σ 3 q,l ) is a non-cleft principal O(U(1))-comodule algebra over O(RP q (l; )). Strong connection form: ω(u n )=α ω(u n 1 )α l ( ) l ( ) m q m(m+1) ζ 2m 1 m 1 ξ m ω(u n 1 )ζ 1 q 2 m=1
Carving out total spaces C[Z 2l ] (l-odd) coacts on O(Σ 3 q) by ζ 0 ζ 0 w 2, ζ 1 ζ 1 w l, ξ ξ 1. Algebra of coinvariants O(Σ 3 q,l+ ) is generated by α = ζl 0, β = ζ1 2 and ξ and admits an O(U(1))-coaction α α u, β β u, ξ ξ u. O(Σ 3 q,l+ ) is a cleft (trivial) principal O(U(1))-comodule algebra over O(RP q (l; )). The cleaving map is: O(U(1)) O(Σ 3 q,l+ ), u ξ.
In progress... Weighted circle actions on the Heegaard-type quantum sphere. Interpretation of weighted projective spaces in terms of graph C -algebras. Development of differential structures on weighted projective spaces. Spectral geometry of quantum weighted projective spaces.