k-graphs and 2-cocycles with C∗-algebrasKatharine Adamyk, Gereltuya Erdenejargal, Nicholas Noone, Oliver Orejola, and Sarah Salmon

Directed by Elizabeth Gillaspy, PhD

MotivationOur research examines the properties of C∗-algebras fromgraph theory perspective.C∗-algebras are objects in functional analysis.k-graphs with 2-cocycles give rise to examples of C∗-algebras.K-Theory is an invariant of C∗-algebras.

k-graph2-cocycles C∗-algebra K-Theory

What are k-graphs and 2-cocycles?A k-graph is a generalized directed graph. Here k- is a naturalnumber and indicates how many color of paths are involved in theskeleton.

k-graph skeleton factorization

directedgraph withk differentcolors

of edges

a set ofequivalenceclasses ofk-colorpaths

= +

2-cocycles are functions from the set of 2-color path equivalenceclasses to the unit circle, satisfyingc(front)c(left)c(top) = c(back)c(right)c(bottom), ∀µ ∈ P.

What Are We Interested In?Two properties of 2-cocycles are of special interest for us.Ihomotopy: c1 and c2 are homotopic if there exists h : P→ Rsuch thatc1(µ) = c2(µ) e2πih(µ) ,∀ µ ∈ P, and for every cubeh(front) + h(left) + h(top) = h(back) + h(right) + h(bottom)

Icohomology: c1 and c2 are cohomologous if there existsβ : E→ T such that for each µ ∈ P

β(µ10)β(µ00)c1(µ) = β(µ01)β(µ11)c2(µ)

µ =

x

y

µ01

µ00

µ10µ11

Example k-graph

Skeleton:

A Bx1

b1

r1

b2

x2

r2

Factorization Squares:

B B

BB

µ1

b2

b2

x2x2

B B

BB

µ2

r2

r2

b2b2

B A

BB

µ3

r1

r2

x1x2

B A

BB

µ4

b1

b2

x1x2

B A

BB

µ5

r1

r2

b1b2

B B

BB

µ6

r2

r2

x2x2

Factorization, P:

B B

BB

B A

BB

x1

x2

x2

x2

b1b2

b2b2

r1r2

r2r2

σ1 :

front = µ1left = µ2top = µ3back = µ4right = µ5

bottom = µ6

B B

BB

B B

BB

x2

x2

x2

x2

b2b2

b2b2

r2r2

r2r2

σ2 :

front = µ1left = µ2top = µ6back = µ1right = µ2

bottom = µ6

Example of a 2-cocycle on σ1:

π4

π33π

4

π

5π3

2π

c(µ5)c(µ2)

c(µ4)

c(µ6)

c(µ3)

c(µ1)

Cohomology conditions on the edges of the each squares:

µ1 : β(b2)β(x2)c1(µ1) = β(x2)β(b2)c2(µ1)µ2 : β(r2)β(b2)c1(µ2) = β(b2)β(r2)c2(µ2)µ3 : β(r1)β(x2)c1(µ3) = β(x1)β(r2)c2(µ3)µ4 : β(b1)β(x2)c1(µ5) = β(x1)β(b2)c2(µ4)µ5 : β(r1)β(b2)c1(µ5) = β(b1)β(r2)c2(µ5)µ6 : β(r2)β(x2)c1(µ6) = β(x2)β(r2)c2(µ6)

The commuting squares are in yellow .

Non-Valid k-graph

x y

red1

black

red2

What does it mean inthe C∗-algebra level?

IHomotopy =⇒ same K-Theory (Gillaspy)ICohomology =⇒ isomorphic C∗-algebra

ConjecturesI. If two cocycles on a given k-graph are cohomologous, then they are homotopic.An related conjecture has previously been proven in terms ofK-theory, but we would like to prove it in terms of 2-cocycles on k-graphs.II. All 2-cocycles are homotopic on a given k-graph.

Ongoing ResearchWe mainly pursued the conjecture II in our research.Matrix formulation: If all 2-cocycles on the k-graph is homotopic, then we canwrite the following linear equation,Ψ ~J = ~K, where

Ψ is from our k-graph,~J ∈ [0, 1] is logarithm of 2-cocycles, and~K ∈ Z is the integer difference of the same points on the unit circle.

squares

↓cubes→ ψ11 ψ12 . . . ψ1n

ψ21 ψ22 . . . ψ2n...

.... . .

...ψm1 ψm2 . . . ψmn

j(µ1)j(µ2)...

j(µn)

=

k1k2...km

.

Ψ Matrix ExampleFrom the k = 3 graph example, we can build the Ψ matrix from the factorizationand ~J vector from the assigned 2-cocycle.

µ1 µ2 µ3 µ4 µ5 µ6σ1 1 1 1 −1 −1 −1σ2 0 0 0 0 0 0

01/65/63/81/81/2

=

−10

.

Given that j(µ) ∈ [0, 1], is it true for Ψ ~J ∈ Z?

New Conjecture

Ψ matrix is totally unimodular matrix if and only if all 2-cocycles on the k-graphare homotopic.