Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Graded K -theory, filtered K -theory and theclassification of Leavitt path algebras

Pere Ara

Universitat Autonoma de Barcelona

Abend Seminars 2020Western Sydney University

June 25, 2020Joint work with Roozbeh Hazrat (WSU) and Huanhuan Li

(Anhui)

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Definition of Leavitt path algebras. Examples.

Definition

Let E = (E 0,E 1, r , s) be a directed graph (where r : E 1 → E 0 isthe range map and s : E 1 → E 0 is the source map). Let k be afield.The Leavitt path algebra Lk(E ) is the k-algebra given bygenerators {v , e, e∗ | v ∈ E 0, e ∈ E 1}, subject to the followingrelations:

(V) vw = δv ,wv and v = v∗ for all v ,w ∈ E 0 ,

(E) s(e)e = er(e) = e for all e ∈ E 1 ,

(CK1) e∗f = δe,f r(e) for all e, f ∈ E 1, and

(CK2) v =∑

e∈E1:s(e)=v ee∗ whenever 0 < |s−1({v})| <∞.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Examples

The n-line quiver, An, is the following quiver:

•v1f1 // •v2

f2 // •v3 // . . . // •vn−1fn−1 // •vn

We have Lk(An) ∼= Mn(k).

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Examples

The n-line quiver, An, is the following quiver:

•v1f1 // •v2

f2 // •v3 // . . . // •vn−1fn−1 // •vn

We have Lk(An) ∼= Mn(k).

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Examples

The n-line quiver, An, is the following quiver:

•v1f1 // •v2

f2 // •v3 // . . . // •vn−1fn−1 // •vn

We have Lk(An) ∼= Mn(k).

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

The n-rose quiver Rn:

•v e1ff

e2

ss

e3

��

en

QQ···

We have that Lk(Rn) ∼= Lk(1, n) =

= 〈X1, . . . ,Xn,X∗1 , . . . ,X

∗n | X ∗i Xj = δij1,

n∑i=1

XiX∗i = 1〉

is the Leavitt algebra of type (1, n), i.e., L ∼= Ln.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

The n-rose quiver Rn:

•v e1ff

e2

ss

e3

��

en

QQ···

We have that Lk(Rn) ∼= Lk(1, n) =

= 〈X1, . . . ,Xn,X∗1 , . . . ,X

∗n | X ∗i Xj = δij1,

n∑i=1

XiX∗i = 1〉

is the Leavitt algebra of type (1, n), i.e., L ∼= Ln.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Lk(E ) is linearly spanned by terms γν∗ for γ and ν paths in E withr(γ) = r(ν).

Definition

Lk(E ) is a graded algebra over Z:

Lk(E ) =⊕n∈Z

Lk(E )n,

where Lk(E )n is spanned by elements γν∗ with |γ| − |ν| = n.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Lk(E ) is linearly spanned by terms γν∗ for γ and ν paths in E withr(γ) = r(ν).

Definition

Lk(E ) is a graded algebra over Z:

Lk(E ) =⊕n∈Z

Lk(E )n,

where Lk(E )n is spanned by elements γν∗ with |γ| − |ν| = n.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

The involution

Given any involution ∗ on k , there is a unique involution on Lk(E )such that

(rλν∗)∗ = r∗νλ∗.

Note that Lk(E )∗n = Lk(E )−n for n ∈ Z.

We will always assume (k, ∗) is given and Lk(E ) has its canonicalgraded ∗-algebra structure.

Remark

When k = C, the Leavitt path algebra LC(E ) embeds as a dense∗-subalgebra of C ∗(E ), the graph C ∗-algebra of E .

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

The involution

Given any involution ∗ on k , there is a unique involution on Lk(E )such that

(rλν∗)∗ = r∗νλ∗.

Note that Lk(E )∗n = Lk(E )−n for n ∈ Z.

We will always assume (k, ∗) is given and Lk(E ) has its canonicalgraded ∗-algebra structure.

Remark

When k = C, the Leavitt path algebra LC(E ) embeds as a dense∗-subalgebra of C ∗(E ), the graph C ∗-algebra of E .

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

The involution

Given any involution ∗ on k , there is a unique involution on Lk(E )such that

(rλν∗)∗ = r∗νλ∗.

Note that Lk(E )∗n = Lk(E )−n for n ∈ Z.

We will always assume (k, ∗) is given and Lk(E ) has its canonicalgraded ∗-algebra structure.

Remark

When k = C, the Leavitt path algebra LC(E ) embeds as a dense∗-subalgebra of C ∗(E ), the graph C ∗-algebra of E .

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

We denote by K gr0 (Lk(E )) the graded K0-group of Lk(E ). It is the

group of differences [P]− [Q], where P and Q are graded f.g.projective Lk(E )-modules.

• There is a Z-action on K gr0 (Lk(E )) given by

n[P] = [P(n)]

where P(n)m = Pn+m.

• K gr0 (Lk(E )) is also an ordered group, with positive cone

K gr0 (Lk(E ))+ = {[P] : P graded f.g. proj. module }.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

We denote by K gr0 (Lk(E )) the graded K0-group of Lk(E ). It is the

group of differences [P]− [Q], where P and Q are graded f.g.projective Lk(E )-modules.

• There is a Z-action on K gr0 (Lk(E )) given by

n[P] = [P(n)]

where P(n)m = Pn+m.

• K gr0 (Lk(E )) is also an ordered group, with positive cone

K gr0 (Lk(E ))+ = {[P] : P graded f.g. proj. module }.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

We denote by K gr0 (Lk(E )) the graded K0-group of Lk(E ). It is the

group of differences [P]− [Q], where P and Q are graded f.g.projective Lk(E )-modules.

• There is a Z-action on K gr0 (Lk(E )) given by

n[P] = [P(n)]

where P(n)m = Pn+m.

• K gr0 (Lk(E )) is also an ordered group, with positive cone

K gr0 (Lk(E ))+ = {[P] : P graded f.g. proj. module }.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Conjecture

(Hazrat’s Conjecture) Let E and F be two finite graphs and let kbe a field. Assume there is a Z-equivariant isomorphism of orderedgroups

K gr0 (Lk(E )) ∼= K gr

0 (Lk(F ))

sending [Lk(E )] to [Lk(F )]. Then Lk(E ) ∼= Lk(F ) as gradedk-algebras.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Hazrat proved his conjecture for all acyclic, comet, or moregenerally, polycephalic graphs.

Conjecture

(Hazrat’s Conjecture II) Let E and F be arbitrary graphs and let kbe a field. Assume there is a Z-equivariant isomorphism of orderedgroups

K gr0 (Lk(E )) ∼= K gr

0 (Lk(F )).

Then Lk(E ) and Lk(F ) are graded Morita-equivalent.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Hazrat proved his conjecture for all acyclic, comet, or moregenerally, polycephalic graphs.

Conjecture

(Hazrat’s Conjecture II) Let E and F be arbitrary graphs and let kbe a field. Assume there is a Z-equivariant isomorphism of orderedgroups

K gr0 (Lk(E )) ∼= K gr

0 (Lk(F )).

Then Lk(E ) and Lk(F ) are graded Morita-equivalent.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Let E a finite essential graph, meaning that E has neither sourcesnor sinks. Then we can write

Lk(E ) = Lk(E )0[t+, t−;α],

as graded ∗-algebras, where α : Lk(E )0 → eLk(E )0e is acorner-isomorphism of Lk(E )0.

If α : R → eRe is an isomorphism, where R is a unital k-algebra,the algebra R[t+, t−;α] is the algebra generated by R, t+, t−, withthe relations t−t+ = 1 and t+rt− = α(r) for r ∈ R.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Let E a finite essential graph, meaning that E has neither sourcesnor sinks. Then we can write

Lk(E ) = Lk(E )0[t+, t−;α],

as graded ∗-algebras, where α : Lk(E )0 → eLk(E )0e is acorner-isomorphism of Lk(E )0.

If α : R → eRe is an isomorphism, where R is a unital k-algebra,the algebra R[t+, t−;α] is the algebra generated by R, t+, t−, withthe relations t−t+ = 1 and t+rt− = α(r) for r ∈ R.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Theorem (A-Pardo, 2014)

Let k be a field and let E and F be finite essential graphs. Supposethat there is a Z-equivariant isomorphism of ordered groups

K gr0 (LR(E )) ∼= K gr

0 (LR(F ))

sending [Lk(E )] to [Lk(F )]. Then there exists a locally innerautomorphism g of Lk(E )0 such that

Lgk (E ) ∼=gr Lk(F ),

where Lgk (E ) = Lk(E )0[t+, t−;α ◦ g ].

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Filtered K -theory (Eilers, Restorff, Ruiz, Sørensen)

Definition

Let R be a graded ring. For k ≤ 0 ≤ m, the filtered K -theoryFKk,m(R) in the range [k ,m] is the collection of algebraicK -theory groups

Kn(I ), Kn(I/J), (k ≤ n ≤ m)

for graded ideals J ⊂ I , together with all the exact sequences

Kn(I2/I1)ι∗−−−−→ Kn(I3/I1)

π∗−−−−→ Kn(I3/I2)∂∗−−−−→ Kn−1(I2/I1)

for graded ideals I1 ⊂ I2 ⊂ I3 and k < n ≤ m.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Main result

Theorem (A-Hazrat-Li)

Let E ,F be row-finite graphs and k a field. Suppose that thereexists an order-preserving Z-equivariant isomorphism

ϕ : K gr0 (Lk(E )) −→ K gr

0 (Lk(F )).

Then the lattices of graded ideals of Lk(E ) and Lk(F ) areisomorphic and moreover

FK0,1(Lk(E )) ∼= FK0,1(Lk(F )),

where FK0,1(L(E )) is a certain quotient of FK0,1(Lk(E )) that wewill describe later.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

The talented monoid

A-Moreno-Pardo introduced the graph monoid ME for a row-finitegraph E :

ME = 〈E 0 : v =∑

e∈s−1(v)

r(e)〉.

They showed that

V(Lk(E )) ∼= ME∼= V(C ∗(E )),

where V(A) is the monoid of isomorphism classes of f.g. projectivemodules over A.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

The talented monoid

A-Moreno-Pardo introduced the graph monoid ME for a row-finitegraph E :

ME = 〈E 0 : v =∑

e∈s−1(v)

r(e)〉.

They showed that

V(Lk(E )) ∼= ME∼= V(C ∗(E )),

where V(A) is the monoid of isomorphism classes of f.g. projectivemodules over A.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

For a graded algebra A, let Vgr (A) be the monoid ofgraded-isomorphism classes of graded f.g. projective A-modules.

Hazrat has introduced the name ’talented monoid’ for the monoid

MgrE =

⟨v(i), v ∈ E 0, i ∈ Z : v(i) =

∑e∈s−1(v)

r(e)(i − 1)⟩

associated to a row-finite graph E .

Theorem (A-Hazrat-Li-Sims)

For a row-finite graph E we have

(a) MgrE∼= Vgr (Lk(E )), and

(b) MgrE is a cancellative monoid, and thus it is isomorphic to

K gr0 (Lk(E ))+.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

For a graded algebra A, let Vgr (A) be the monoid ofgraded-isomorphism classes of graded f.g. projective A-modules.

Hazrat has introduced the name ’talented monoid’ for the monoid

MgrE =

⟨v(i), v ∈ E 0, i ∈ Z : v(i) =

∑e∈s−1(v)

r(e)(i − 1)⟩

associated to a row-finite graph E .

Theorem (A-Hazrat-Li-Sims)

For a row-finite graph E we have

(a) MgrE∼= Vgr (Lk(E )), and

(b) MgrE is a cancellative monoid, and thus it is isomorphic to

K gr0 (Lk(E ))+.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

For a graded algebra A, let Vgr (A) be the monoid ofgraded-isomorphism classes of graded f.g. projective A-modules.

Hazrat has introduced the name ’talented monoid’ for the monoid

MgrE =

⟨v(i), v ∈ E 0, i ∈ Z : v(i) =

∑e∈s−1(v)

r(e)(i − 1)⟩

associated to a row-finite graph E .

Theorem (A-Hazrat-Li-Sims)

For a row-finite graph E we have

(a) MgrE∼= Vgr (Lk(E )), and

(b) MgrE is a cancellative monoid, and thus it is isomorphic to

K gr0 (Lk(E ))+.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

It turns out that the talented monoid MgrE contains a lot of

information about Lk(E ). For instance, it contains full informationon its lattice Lgr (Lk(E )) of graded ideals:

Theorem (A-Hazrat-Li-Sims)

For a row-finite graph E we have

Lgr (Lk(E )) ∼= TE ∼= L(Vgr (Lk(E ))) ∼= L(MgrE )

Here TE is the lattice of hereditary saturated subsets of E 0.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

It turns out that the talented monoid MgrE contains a lot of

information about Lk(E ). For instance, it contains full informationon its lattice Lgr (Lk(E )) of graded ideals:

Theorem (A-Hazrat-Li-Sims)

For a row-finite graph E we have

Lgr (Lk(E )) ∼= TE ∼= L(Vgr (Lk(E ))) ∼= L(MgrE )

Here TE is the lattice of hereditary saturated subsets of E 0.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

It turns out that the talented monoid MgrE contains a lot of

information about Lk(E ). For instance, it contains full informationon its lattice Lgr (Lk(E )) of graded ideals:

Theorem (A-Hazrat-Li-Sims)

For a row-finite graph E we have

Lgr (Lk(E )) ∼= TE ∼= L(Vgr (Lk(E ))) ∼= L(MgrE )

Here TE is the lattice of hereditary saturated subsets of E 0.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

An exact sequence in K -theory

We now want to recover K0(L(E )) and K1(L(E )) from K gr0 (LE ) in

a functorial way.

We establish an exact sequence:

K1(L(E ))T // K gr

0 (L(E ))φ // K gr

0 (L(E ))U // K0(L(E )) // 0,

where φ[v(i)] = [v(i + 1)]− [v(i)], and where U([v(i)]) = [v ]forgets the grading.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

An exact sequence in K -theory

We now want to recover K0(L(E )) and K1(L(E )) from K gr0 (LE ) in

a functorial way.We establish an exact sequence:

K1(L(E ))T // K gr

0 (L(E ))φ // K gr

0 (L(E ))U // K0(L(E )) // 0,

where φ[v(i)] = [v(i + 1)]− [v(i)], and where U([v(i)]) = [v ]forgets the grading.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

An exact sequence in K -theory

We now want to recover K0(L(E )) and K1(L(E )) from K gr0 (LE ) in

a functorial way.We establish an exact sequence:

K1(L(E ))T // K gr

0 (L(E ))φ // K gr

0 (L(E ))U // K0(L(E )) // 0,

where φ[v(i)] = [v(i + 1)]− [v(i)], and where U([v(i)]) = [v ]forgets the grading.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

The (algebraic) K1-group K1(Lk(E )) of Lk(E ) was computed byA-Brustenga-Cortinas.

Let E be a row-finite graph. Let AE be the adjacency matrix of E ,with the zero rows (corresponding to the sinks of E ) suppressed.Let I be the matrix obtained from the identity E 0 × E 0 matrixsuppressing the columns corresponding to the sinks. We considerthe matrix At

E − I as a homomorphism

AtE − I : ZR → ZE0

,

where R = E 0 \ Sink(E ) are the non-sinks of E and ZX indicatesthe free abelian Z-module on X for each set X .

Indeed, the matrix AtE − I induces a homomorphism

AtE − I : GR → GE0

for every abelian group G .

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

The (algebraic) K1-group K1(Lk(E )) of Lk(E ) was computed byA-Brustenga-Cortinas.

Let E be a row-finite graph. Let AE be the adjacency matrix of E ,with the zero rows (corresponding to the sinks of E ) suppressed.Let I be the matrix obtained from the identity E 0 × E 0 matrixsuppressing the columns corresponding to the sinks. We considerthe matrix At

E − I as a homomorphism

AtE − I : ZR → ZE0

,

where R = E 0 \ Sink(E ) are the non-sinks of E and ZX indicatesthe free abelian Z-module on X for each set X .

Indeed, the matrix AtE − I induces a homomorphism

AtE − I : GR → GE0

for every abelian group G .Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

There is a non-canonical splitting

K1(Lk(E )) ∼=coker(AtE − I : (k×)R → (k×)E

0)

⊕ker(AtE − I : ZR → ZE0

)

where k× = k \ {0} is the multiplicative group of the units of k .

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

There is a non-canonical splitting

K1(Lk(E )) ∼=coker(AtE − I : (k×)R → (k×)E

0)

⊕ker(AtE − I : ZR → ZE0

)

where k× = k \ {0} is the multiplicative group of the units of k .

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

In the case of graph C ∗-algebras we have an isomorphism

K top1 (C ∗(E )) ∼= ker

(AtE − I : ZR → ZE0

),

and there is a formula, obtained by Carlsen, Eilers and Tomforde,giving the isomorphism

χ : ker(AtE − I : ZR → ZE0

)→ K top

1 (C ∗(E )).

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

In the case of graph C ∗-algebras we have an isomorphism

K top1 (C ∗(E )) ∼= ker

(AtE − I : ZR → ZE0

),

and there is a formula, obtained by Carlsen, Eilers and Tomforde,giving the isomorphism

χ : ker(AtE − I : ZR → ZE0

)→ K top

1 (C ∗(E )).

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

The formula for χ can be adapted to the case of Leavitt pathalgebras and gives a map

χ′ : ker(AtE − I : ZR → ZE0

)→ K1(Lk(E )).

However this map χ′ is not a group homomorphism. The problemis that the map χ′ depends on the choice of certain permutations.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

The formula for χ can be adapted to the case of Leavitt pathalgebras and gives a map

χ′ : ker(AtE − I : ZR → ZE0

)→ K1(Lk(E )).

However this map χ′ is not a group homomorphism. The problemis that the map χ′ depends on the choice of certain permutations.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Theorem

Let GE be the subgroup of K1(Lk(E )) generated by[(−v) + (1− v)]1, for all v ∈ E 0. Then there is a well-definedgroup homomorphism

χ : ker(AtE − I : ZR → ZE0

)→ K1(Lk(E ))/GE .

Moreover this map is functorial with respect to the maps inducedby graded subquotients of Lk(E ), and it is a section of a canonical

homomorphism ξ : K1(Lk(E ))/GE → ker(AtE − I : ZR → ZE0

)in

algebraic K -theory (an index map).

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

We can now define the quotient FK of filtered K -theory.

Definition

Let E be a row-finite graph and k be a field. Define the algebraicfiltered K -theory FK0,1(Lk(E )) as the collection

{Kn(J/I )}0≤n≤1

where (I , J) ranges over all the graded ideals of Lk(E ) such thatI ⊆ J,

K 0(J/I ) = K0(J/I )

andK 1(J/I ) = K1(J/I )/GJ/I

where the subgroup GJ/I of K1(J/I ) is defined by using that J/I iscanonically a Leavitt path algebra of a suitable graph.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Shift equivalence

Let A,B be square matrices with coefficients in Z+.

We say that A and B are shift equivalent if there are matrices R,Sover Z+ and a positive integer ` such that

SR = A`, RS = B`, BR = RA, AS = SB.

If E and F are finite graphs without sinks, we say that E and Fare shift equivalent if their adjacency matrices AE and AF are shiftequivalent.It can be shown that E and F are shift equivalent if and only ifK gr

0 (LE ) and K gr0 (LF ) are equivariantly order-isomorphic.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Shift equivalence

Let A,B be square matrices with coefficients in Z+.

We say that A and B are shift equivalent if there are matrices R,Sover Z+ and a positive integer ` such that

SR = A`, RS = B`, BR = RA, AS = SB.

If E and F are finite graphs without sinks, we say that E and Fare shift equivalent if their adjacency matrices AE and AF are shiftequivalent.

It can be shown that E and F are shift equivalent if and only ifK gr

0 (LE ) and K gr0 (LF ) are equivariantly order-isomorphic.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Shift equivalence

Let A,B be square matrices with coefficients in Z+.

We say that A and B are shift equivalent if there are matrices R,Sover Z+ and a positive integer ` such that

SR = A`, RS = B`, BR = RA, AS = SB.

If E and F are finite graphs without sinks, we say that E and Fare shift equivalent if their adjacency matrices AE and AF are shiftequivalent.It can be shown that E and F are shift equivalent if and only ifK gr

0 (LE ) and K gr0 (LF ) are equivariantly order-isomorphic.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Theorem (A-Hazrat-Li)

Let E and F be finite graphs without sinks. If E and F are shiftequivalent, then the graph C ∗-algebras C ∗(E ) and C ∗(F ) areMorita equivalent.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

The proof uses:

Theorem (Eilers, Restorff, Ruiz, Sørensen)

Let E and F be graphs. If there is an isomorphism

FK0,1(LC(E )) ∼= FK0,1(LC(F )),

then there is an isomorphism

FKtop(C ∗(E )) ∼= FKtop(C ∗(F )).

This result can be easily adapted to show that the same holdsreplacing FK0,1 with FK0,1.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

The proof uses:

Theorem (Eilers, Restorff, Ruiz, Sørensen)

Let E and F be graphs. If there is an isomorphism

FK0,1(LC(E )) ∼= FK0,1(LC(F )),

then there is an isomorphism

FKtop(C ∗(E )) ∼= FKtop(C ∗(F )).

This result can be easily adapted to show that the same holdsreplacing FK0,1 with FK0,1.

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

The proof:Let E and F be finite graphs without sinks.

If E and F are shift equivalent, then K gr0 (LE ) ∼=Z K gr

0 (LF ).

By our main theorem, we then have FK0,1(LE ) ∼= FK0,1(LF ).

By [ERRS], FKtop(C ∗(E )) ∼= FKtop(C ∗(F ))

By a main result of ERRS, C ∗(E ) ∼M C ∗(F ).

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

AEoo shift equivalence of matrices //KS

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AF

K gr0 (Lk(E )) oo

iso. of Kgr0 -group

//

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K gr0 (Lk(F ))

FK0,1(Lk(E )) ooiso. of alg. fil. K -groups //

for k=C

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FK0,1(Lk(F ))

FK0,1(C ∗(E )) ooiso. of fil. K -groups //KS

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Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

Thank you very much for your attention!!!

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras

Leavitt path algebras Graded K -theory Filtered K -theory The proof of the main theorem An application to symbolic dynamics

The simplest example in which χ′1 depends on the choice ofbijections is when E has only one vertex v and one edge e. In thiscase, Lk(E ) = k[t, t−1] andK1(LK (E )) = K0(k)⊕ K1(k) = Z⊕ k×. Considerx = 2 ∈ Ker(At

E − I ) = Z. Then L+x = {(v , 1), (v , 2)} and

L−x = {(e, 1), (e, 2)}. Now taking [v , i ] = i and 〈e, i〉 = i for

i = 1, 2, we obtain χ′1(2) =[(e 0

0 e

)]1. Now taking [v , i ] = i for

i = 1, 2 and 〈e, 1〉′ = 2, 〈e, 2〉′ = 1, we get

χ′1(2) =[(0 e

e 0

)]16= χ′1(2) (if the characteristic of k is different

from 2)

Pere Ara Universitat Autonoma de Barcelona

Graded K -theory, filtered K -theory and the classification of Leavitt path algebras