We construct explicit generators of the K-theory and K-homology of the coordinate algebra of `functions' on quantum projective spaces. We also sketch a construction of unbounded Fredholm modules, that is to say Dirac-like operators and spectral triples of any positive real dimension.
Deep Dive into Bounded and unbounded Fredholm modules for quantum projective spaces.
We construct explicit generators of the K-theory and K-homology of the coordinate algebra of `functions’ on quantum projective spaces. We also sketch a construction of unbounded Fredholm modules, that is to say Dirac-like operators and spectral triples of any positive real dimension.
For quantum projective spaces CP n q we generalize some ideas of [5] and give 'polynomial' generators of its K-theory, that is projections whose entries are in the corresponding coordinate algebra A(CP n q ). Dually, we give generators of its K-homology via even Fredholm modules (A(CP n q ), H (k) , γ (k) , F (k) ), for k = 0, . . . , n. The 'top' Fredholm module -the only one for which the representation of A(CP n q ) is faithful -can be realized as the 'conformal class' of a spectral triple (A(CP n q ), H (n) , γ (n) , D), namely we realize F (n) := D|D| -1 as the 'sign' of a Dirac-like operator. This procedure allows us to construct spectral triples of any summability d ∈ R + .
In the following, without loss of generalities, the real deformation parameter is restricted to be 0 < q < 1. Also, by * -algebra we shall mean a unital involutive associative C-algebra, and by representation of a * -algebra we always mean a unital * -representation.
The ‘ambient’ algebra for the quantum projective space is the coordinate algebra A(S 2n+1 q ) of the unit quantum spheres. This * -algebra is generated by 2n + 2 elements {z i , z * i } i=0,…,n with relations [8]:
The * -subalgebra generated by p ij := z * i z j will be denoted A(CP n q ), and identified with the algebra of ‘polynomial functions’ on the quantum projective space CP n q . The algebra A(CP n q ) is made of (co)invariant elements for the U(1) (co)action z i → λz i for λ ∈ U(1). From the relations of A(S 2n+1 q ) one gets analogous relations for A(CP n q ):
with sign(0) := 0. The elements p ij are the matrix entries of a projection P = (p ij ), that is P 2 = P = P * or n j=0 p ij p jk = p ik and p * ij = p ji . This projection has q-trace:
The original notations of [8] are obtained by setting q = e h/2 ; the generators of [3] correspond to the replacement z i → z n+1-i , while the generators x i used in [6] are related to ours by x i = z * n+1-i and by the replacement q → q -1 . Generators for the K-theory and K-homology of the spheres S 2n+1 q are in [6]; unfortunately, there is no canonical way to obtain generators of the K-theory and/or K-homology of subalgebras, unless they are dense subalgebras (for a pair C * -algebra/pre-C * -algebra, the K-groups coincide). That S 2n+1 q and CP n q are truly different can be seen from the fact that the K-groups of the odd-dimensional spheres are (Z, Z), regardless of the dimension, while for CP n q they are (Z n+1 , 0). That K 0 (C(CP n q )) Z n+1 can be proved by viewing the corresponding C * -algebra C(CP n q ), the universal C * -algebra of CP n q , as the Cuntz-Krieger algebra of a graph [7]. The group K 0 is given as the cokernel of the incidence matrix canonically associated with the graph. The dual result for K-homology is obtained using the same techniques: the group K 0 is now the kernel of the transposed matrix [1]; this leads to K 0 (C(CP n q )) = Z n+1 . In [7], somewhat implicitly, there appear generators of the K 0 groups of C(CP n q ) as projections in C(CP n q ) itself. Here, we give generators of K 0 (C(CP n q )) in the form of ‘polynomial functions’, so they represent elements of K 0 (A(CP n q )) as well. They are also equivariant, i.e. representative of elements in K Uq(su(n+1)) 0 (A(CP n q )). Besides, we give n + 1 Fredholm modules that are generators of the homology group K 0 (C(CP n q )).
A useful (unital * -algebra) morphism A(S 2n+1 q ) → A(S 2n-1 q ) given by the map z n → 0, restricts to a morphism A(CP n q ) → A(CP n-1 q
) and is heavily used in what follows. We also stress that representations of A(S 2n+1 q ), with z i in the kernel for all i > k, and for a fixed 0 ≤ k < n, are the pullback of representations of A(S 2k+1 q ). Here we seek Fredholm modules (and then irreducible * -representations) for CP n q that are not the pullback of Fredholm modules (resp. irreps) for CP n-1 q . So, we look for irreps of S 2n+1 q for which z n is not in the kernel. These are given in [6], and classified by a phase: in particular, they are inequivalent as representation of S 2n+1 q but give the same representation of CP n q (whatever the value of the phase is). In fact, we start with representations that are not exactly irreducible, but close to it: they are the direct sum of an irreducible representation with copies of the trival a → 0 representation. Definition 1. We use the following multi-index notation. We let m = (m 1 , . . . , m n ) ∈ N n and, for 0 ≤ i < k ≤ n, we denote by ε k i ∈ {0, 1} n the array
, is defined as follows (all the representations are on the same Hilbert space). We set π
with m 0 := 0 -, and they are zero on the orthogonal subspace. When k = 0, we define π andπ (n) 0 (z 0 ) |m = 0 in all the other cases.
As a * -algebra, A(CP n q ) is generated by the elements p ij , with i ≤ j since p ji = p * ij ; in fact, from the tracial relation (2) one of the generators on the diagonal, say p nn , is redundant. The computation of π
k is an irreducible * -representation of b
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