Cohomology of higher rank graphs, an interim report.

Transcription

1 Cohomology of higher rank graphs, an interim report. Alex Kumjian 1, David Pask 2, Aidan Sims 2 1 University of Nevada, Reno 2 University of Wollongong GPOTS, Arizona State University Tempe, May 2011

2 Introduction k-graphs Remarks Introduction. We introduce a homology and a cohomology theory for higher rank graphs and define the C -algebra of a k-graph twisted by a two-cocycle. Our definition of the homology of a higher rank graph Λ is modeled on Massey s definition of the cubical singular homology of a topological space (see [Mas91, VII.2]). Our definition is equivalent to that of Grandis for a cubical set (see [Gr05]). Given a 2-cocycle ϕ with values in T, we define the twisted C -algebra C (Λ, ϕ). All non-commutative tori may be realized as examples of this construction. This is an interim report on joint work with David Pask and Aidan Sims.

3 Introduction k-graphs Remarks k-graphs. Definition (see [KP00]) Let Λ be a countable small category and let d : Λ N k be a functor. Then (Λ, d) is a k-graph, if it satisfies the factorization property: For every λ Λ and m, n N k such that there exist unique µ, ν Λ satisfying: d(µ) = m and d(ν) = n, λ = µν. d(λ) = m + n Notation: With notation as above we denote µ = λ(0, m), ν = λ(m, d(λ)). Set Λ n := d 1 (n) and identify Λ 0 = Obj (Λ), the set of vertices. An element λ Λ ei is called an edge.

4 Introduction k-graphs Remarks Remarks and Examples. Let Λ be a k-graph. If k = 0, then d is trivial and Λ is just a set. If k = 1, then Λ is the path category of a directed graph. If k 2, think of Λ as generated by k graphs of different colors that share the same set of vertices Λ 0. Commuting squares form an essential piece of structure for k 2. Simple example: Λ = N k. Example of a 2-graph: Only the morphisms of minimal degree, Λ e1 and Λ e2, are given. f a u e v b

5 Cubes and Faces Homology Künneth Theorem Cohomology Cubes and Faces. Let Λ be a k-graph. For 0 n k we let Q n (Λ) denote the set of n-cubes in Λ, that is, the set of elements λ for which d(λ) = e i1 + + e in where i 1 < < i n. If n > 0, the 2n faces of λ are the n 1-cubes: where 1 j n. F 0 j (λ) = λ(0, d(λ) e ij ) F 1 j (λ) = λ(e ij, d(λ)) Fact: If i < j, then F l i F m j = F m j 1 Fl i. The 0-cubes are simply the vertices, the 1-cubes are the edges of Λ and the 2-cubes are commuting squares.

6 Cubes and Faces Homology Künneth Theorem Cohomology Homology complex. For 0 n k let C n (Λ) = ZQ n (Λ) (and set C n (Λ) = 0 otherwise). Then for n > 0, define n : C n (Λ) C n 1 (Λ) on λ Q n (Λ) by n (λ) = n ( 1) j (Fj 1 (λ) Fj 0 (λ)) j=1 and define 0 to be the zero map. It is straightforward to show that n 1 n = 0. Hence, (C (Λ), ) is a complex and we define the homology of Λ by H n (Λ) = ker n / Im n+1. The assignment Λ H (Λ) is a covariant functor (see [Mas91, VII.2]). Let C m denote the directed cycle with m vertices viewed as 1-graph. Then H n (C m ) = Z for n = 0, 1 (and 0 otherwise).

7 Cubes and Faces Homology Künneth Theorem Cohomology The Künneth Theorem. Using basic homological algebra (cf. [Mac67]) we prove: Theorem (Künneth Formula) For i = 1, 2, let Λ i be a k i -graph. Then Λ 1 Λ 2 is a (k 1 + k 2 )-graph. For n 0 there is a split exact sequence 0 p 1+p 2=r H p1 (Λ 1 ) H p2 (Λ 2 ) α β H r (Λ 1 Λ 2 ) p 1+p 2=r In the above 2-graph example, Λ = C 2 C 1 and so Tor(H p1 (Λ 1 ), H p2 (Λ 2 )) 0. H 0 (Λ) = Z, H 1 (Λ) = Z 2, H 2 (Λ) = Z.

8 Cubes and Faces Homology Künneth Theorem Cohomology Cohomology. Let Λ be a k-graph and let A be an abelian group. For n N set and define C n (Λ, A) = Hom(ZQ n (Λ), A) δ n : C n (Λ, A) C n+1 (Λ, A) by δ n (ϕ) = ϕ n+1. It is straightforward to show that (C (Λ, A), δ ) is a complex and we define the cohomology of Λ by H n (Λ, A) = ker δ n / Im δ n 1. The assignment Λ H (Λ, A) is a contravariant functor (it also depends covariantly on A).

9 The C -algebra C (Λ, ϕ). C (Λ, ϕ). Main Results Suppose that Λ satisfies ( ): For all v Λ 0, n N k, vλ n is finite and nonempty where vλ n := r 1 (v) Λ n. Definition Let ϕ Z 2 (Λ, T). Define C (Λ, ϕ) to be the universal C -algebra generated by a family of operators {t λ : λ Λ ei, i = 1,..., k} and a family of orthogonal projections {p v : v Λ 0 } satisfying: 1 For λ Λ ei, t λ t λ = p s(λ). 2 Suppose µν = ν µ where d(µ) = d(µ ) = e i, d(ν) = d(ν ) = e j and i < j. Then t ν t µ = ϕ(µν)t µ t ν. 3 For v Λ 0 and i = 1,..., k, p v = λ vλ e i t λ t λ.

10 C (Λ, ϕ). Main Results Main Results. Fact: The isomorphism class of C (Λ, ϕ) only depends on [ϕ] H 2 (Λ, T). There is a gauge action γ of T k on C (Λ, ϕ): For all z T k γ z (p v ) = p v for all v Λ 0, γ z (t λ ) = z i t λ for all λ Λ ei, i = 1,..., k. Moreover, the fixed point algebra C (Λ, ϕ) γ is AF. In fact, it is isomorphic to the gauge invariant subalgebra of the untwisted C -algebra, see [KP00]. Theorem (Gauge Invariant Uniqueness Theorem) Let π : C (Λ, ϕ) B be an equivariant -homomorphism. Then π is injective iff π(p v ) 0 for all v Λ 0. Fact: There is a T-valued groupoid 2-cocycle σ ϕ on G Λ such that C (Λ, ϕ) = C (G Λ, σ ϕ ).

11 References. Preliminaries References Thanks More Stuff [Gr05] M. Grandis, Directed combinatorial homology and noncommutative tori, [KP00] A. Kumjian and D. Pask, Higher rank graph C -algebras, [KPS] A. Kumjian, D. Pask and A. Sims, Homology and cohomology of higher rank graphs, in preparation. [Mac67] S. Mac Lane, Homology [Mas91] W. Massey, A Basic Course in Algebraic Topology [Ren80] J. Renault, A groupoid approach to C*-algebras, 1980.

12 References Thanks More Stuff Thanks! Any questions?

13 References Thanks More Stuff More Theorems. Theorem (The Universal Coefficient Theorem) Let Λ be a k-graph and let A be an abelian group. Then for n 0, there is a short exact sequence 0 Ext(H n 1 (Λ), A) H n (Λ, A) Hom(H n (Λ), A) 0. By a standard argument, a short exact sequence of coefficient groups gives rise to a long exact sequence 0 A B C 0 0 H 0 (Λ, A) H 0 (Λ, B) H 0 (Λ, C) H 1 (Λ, A) H n 1 (Λ, C) H n (Λ, A) H n (Λ, B) H n (Λ, C)

14 References Thanks More Stuff Categorical Cocycle Cohomology. The categorical cocycle cohomology, H cc(λ, A), is just the usual cocycle cohomology for groupoids (see [Ren80]) extended to small categories. We have proven that for n = 0, 1, 2 H n (Λ, A) = H n cc(λ, A). A map c : Λ Λ A is a categorical 2-cocycle if for any composable triple (λ 1, λ 2, λ 3 ) we have c(λ 1, λ 2 ) + c(λ 1 λ 2, λ 3 ) = c(λ 1, λ 2 λ 3 ) + c(λ 2, λ 3 ) and c is a categorical 2-coboundary if there is b : Λ A such that c(λ 1, λ 2 ) = b(λ 1 ) b(λ 1 λ 2 ) + b(λ 2 ). H 2 cc(λ, A) is the quotient group (2-cocycles modulo 2-coboundaries).

15 References Thanks More Stuff The C*-algebra C (Λ; c). Suppose Λ satisfies ( ) and let c be a T-valued categorical 2-cocycle. Definition (see [KPS]) Let C (Λ; c) be the universal C*-algebra generated by the set {t λ : λ Λ} satisfying: 1 {t v : v Λ 0 } is a family of orthogonal projections. 2 For λ Λ, t s(λ) = t λ t λ. 3 If s(λ) = r(µ), then t λ t µ = c(λ, µ)t λµ. 4 For v Λ 0, n N k s v = λ vλ n s λ s λ If [ϕ] is mapped to [c] in the identification H 2 (Λ, A) = H 2 cc(λ, A), then. C (Λ, ϕ) = C (Λ; c).