s. paul smith · s. paul smith university of washington seattle, wa 98195....

Graded modules over path algebras

S. Paul Smith

University of WashingtonSeattle, WA 98195.

[email protected]

April 13, 2012

ARTIN (Algebra and Ring Theory in the North)

Newcastle (UK)

S. Paul Smith Graded modules over path algebras

References for this talk

Category equivalences involving graded modules over pathalgebras of quivers, Adv. in Math., in press. arXiv:1107.3511

(with C. Holdaway) An equivalence of categories for gradedmodules over monomial algebras and path algebras of quivers,J. Algebra, 353 (2011) 249-260. arXiv: 1109.4387

“Degenerate” 3-dimensional Sklyanin algebras are monomialalgebras, J. Algebra, 358 (2012) 74-86. arXiv: 1112.5809

Path algebras

Let Q be a directed graph with finitely many arrows and vertices.Call Q a quiver.

Loops and multiple arrows between vertices are allowed, e.g.,

��~~~~~~~

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__@@@@@@@

A path is a concatenation of arrows.

Path algebras I

For example, reading from right to left, as with composition offunctions, p = cbgffeb is a path in

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��

•99c<<

__@@@@@@@fee

Fix a field k , usually unimportant.

The path algebra of Q is the k-algebra

kQ := the linear span of all paths in Q

with multiplication

{pq if q ends where p begins

0 otherwise.

Path algebras II

WriteI = the set of vertices in Q

andei = the empty path at vertex i , i ∈ I .

The ei s are mutually orthogonal idempotents.and ∑

ei = 1,

the identity in kQ, because

eipej =

{p if p is a path from j to i

0 otherwise.

Path algebras II

WriteI = the set of vertices in Q

andei = the empty path at vertex i , i ∈ I .

The ei s are mutually orthogonal idempotents.and ∑

ei = 1,

the identity in kQ, because

eipej =

{p if p is a path from j to i

0 otherwise.

Grading and QGr(kQ)

Make kQ a graded ring by declaring its degree-n component is

kQn := the linear span of length-n paths.

Define

QGr(kQ) :=Gr(kQ)

Fdim(kQ)=

Z-graded left kQ-modules

{M = sum of its finite diml submods}

Grading and QGr(kQ)

Make kQ a graded ring by declaring its degree-n component is

kQn := the linear span of length-n paths.

Define

QGr(kQ) :=Gr(kQ)

Fdim(kQ)=

Z-graded left kQ-modules

{M = sum of its finite diml submods}

Philosophy of nc projective algebraic geometry

Certain abelian and triangulated categories behave as if they are“quasi-coherent sheaves” on “non-commutative schemes”.

Examples:

If R is a ring ModR is the category of “quasi-coherent sheaveson the non-commutative affine scheme Specnc R”.

QGr(kQ) is the category of “quasi-coherent sheaves on thenon-commutative projective scheme Projnc(kQ)”.

The geometric object is defined implicitly, cf., definition of stacks.

Today: just treat QGr(kQ) as an interesting abelian category.

Philosophy of nc projective algebraic geometry

Certain abelian and triangulated categories behave as if they are“quasi-coherent sheaves” on “non-commutative schemes”.

Examples:

If R is a ring ModR is the category of “quasi-coherent sheaveson the non-commutative affine scheme Specnc R”.

QGr(kQ) is the category of “quasi-coherent sheaves on thenon-commutative projective scheme Projnc(kQ)”.

The geometric object is defined implicitly, cf., definition of stacks.

Today: just treat QGr(kQ) as an interesting abelian category.

Theorem (S)

There are category equivalences

QGr(kQ) ≡ ModS(Q) ≡ GrL(Q◦) ≡ ModL(Q◦)0 ≡ QGr(kQ(n)).

S(Q) = lim−→EndkI (kQ⊗n1 ) is a direct limit of finite dimensional

semisimple algebras;

Q◦ is the quiver without sources or sinks that is obtained byrepeatedly removing all sinks and sources from Q;

L(Q◦) is the Leavitt path algebra of Q◦;

L(Q◦)0 is its degree zero component;

Q(n) is the quiver whose incidence matrix is the nth power ofthat for Q.

All short exact sequences in qgr(kQ), the full subcategory offinitely presented objects in QGr(kQ), split.

Theorem (S)

There are category equivalences

QGr(kQ) ≡ ModS(Q) ≡ GrL(Q◦) ≡ ModL(Q◦)0 ≡ QGr(kQ(n)).

S(Q) = lim−→EndkI (kQ⊗n1 ) is a direct limit of finite dimensional

semisimple algebras;

Q◦ is the quiver without sources or sinks that is obtained byrepeatedly removing all sinks and sources from Q;

L(Q◦) is the Leavitt path algebra of Q◦;

L(Q◦)0 is its degree zero component;

Q(n) is the quiver whose incidence matrix is the nth power ofthat for Q.

All short exact sequences in qgr(kQ), the full subcategory offinitely presented objects in QGr(kQ), split.

Hazrat+Dade =⇒ GrL(Q◦) ≡ ModL(Q◦)0

Theorem (R. Hazrat, arXiv:1005.1900)

L(Q)nL(Q)−n = L(Q)0 for all n ∈ Z if and only if Q does not havea sink.

Theorem (Dade)

If A is a Z-graded ring such that AnA−n = A0 for all n ∈ Z, then

GrA ≡ ModA0

via M M0.

UHF example

Q := one vertex and n arrows.CQ = C〈x1, . . . , xn〉 = the free algebra.

S(Q) is the direct limit (in the category of C-algebras) of

Mn(C)→ Mn(C)⊗2 → Mn(C)⊗3 → · · ·

where all the maps are f 7→ 1⊗ f .

The direct limit in the category of C∗-algebras is

Mn∞(C) = a uniformly hyperfinite C∗-algebra,

a von Neumann algebra.

S(Q) is a dense subalgebra of Mn∞(C).Equivalently, Mn∞(C) is the norm closure of S(Q).

UHF example

Mn(C)→ Mn(C)⊗2 → Mn(C)⊗3 → · · ·

UHF example

Mn(C)→ Mn(C)⊗2 → Mn(C)⊗3 → · · ·

UHF example

Mn(C)→ Mn(C)⊗2 → Mn(C)⊗3 → · · ·

UHF example

Mn(C)→ Mn(C)⊗2 → Mn(C)⊗3 → · · ·

The CAR (canonical anti-commutation relations) algebra

M2∞(C) is important in quantum mechanics

Theorem/Definition: Let B(H) be the bounded operators on aninfinite dimensional separable Hilbert space H. Let α : H → B(H)be any continuous linear map such that

α(g)α(h) + α(h)α(g) = 0 and

α(g)∗α(h) + α(h)α(g)∗ = 〈h, g〉 idH for all g , h ∈ H.

The C∗-subalgebra of B(H) generated by {α(h) | h ∈ H} isisomorphic to M2∞(C) and is called the CAR algebra.

•99 ee)

= C〈x , y〉 QGr M2∞(C)

•99 ee)

= C〈x , y〉 QGr M2∞(C)

•99 ee)

= C〈x , y〉 QGr M2∞(C)

•99 ee)

= C〈x , y〉 QGr M2∞(C)

The Veronese subalgebra kQ(m)

Vertices in Q(m) = the vertices in Q

Arrows in Q(m) = the paths of length m in Q.

Corollary

QGr(kQ) ≡ QGr(kQ(m))

Proof: S(Q(m)) = lim−→nEndkI kQmn is a subsystem of the directed

system defining S(Q) so

S(Q) = S(Q(m)).

The Veronese subalgebra kQ(m)

Vertices in Q(m) = the vertices in Q

Arrows in Q(m) = the paths of length m in Q.

Corollary

QGr(kQ) ≡ QGr(kQ(m))

Proof: S(Q(m)) = lim−→nEndkI kQmn is a subsystem of the directed

system defining S(Q) so

S(Q) = S(Q(m)).

An example by Raf Bocklandt

Q := vertices the elements of the group Z/n and arrows i → i + 1for all i . There is a unique path of length n starting at each i andthat path ends at i , so Q(n) is

•��

· · · •��

n vertices

and kQ(n) ∼= k[x ]⊕n. Now,

Proj(k[x ]⊕n

)= n points = Spec(k⊕n)

so QGr(kQ) ≡ QGr(kQ(n)) ≡ QGr(k[x ]⊕n

)but, using Serre [FAC],

QGr(k[x ]⊕n

)≡Qcoh

(k[x ]⊕n

))≡ Qcoh(n points) ≡ Mod(k⊕n).

It is the case that S(Q) ∼= k⊕n.

•��

· · · •��

n vertices

Proj(k[x ]⊕n

QGr(k[x ]⊕n

)≡Qcoh

(k[x ]⊕n

•��

· · · •��

n vertices

Proj(k[x ]⊕n

QGr(k[x ]⊕n

)≡Qcoh

(k[x ]⊕n

•��

· · · •��

n vertices

Proj(k[x ]⊕n

QGr(k[x ]⊕n

)≡Qcoh

(k[x ]⊕n

•��

· · · •��

n vertices

Proj(k[x ]⊕n

QGr(k[x ]⊕n

)≡Qcoh

(k[x ]⊕n

•��

· · · •��

n vertices

Proj(k[x ]⊕n

QGr(k[x ]⊕n

)≡Qcoh

(k[x ]⊕n

Make qgr(kQ) a triangulated category

S(Q) is von Neumann regular

all short exact sequences in modS(Q) split

all short exact sequences in qgr(kQ) split

Make qgr(kQ) a triangulated category:

suspension functor Σ = the Serre degree twist (−1)

declare that the distinguished triangles in qgr(kQ) are alldirect sums of the following triangles:

M→ 0→ ΣM id−→ ΣM,

M id−→M→ 0→ ΣM,

0→M id−→M→ 0,

Make qgr(kQ) a triangulated category

S(Q) is von Neumann regular

all short exact sequences in modS(Q) split

all short exact sequences in qgr(kQ) split

Make qgr(kQ) a triangulated category:

suspension functor Σ = the Serre degree twist (−1)

declare that the distinguished triangles in qgr(kQ) are alldirect sums of the following triangles:

M→ 0→ ΣM id−→ ΣM,

M id−→M→ 0→ ΣM,

0→M id−→M→ 0,

The singularity category of a finite dimensional algebra

Λ = a finite dimensional algebramod(Λ) = finite dimensional left Λ-modulesDb(modΛ) = the bounded derived categoryDperf(modΛ) = the full subcategory of perfect complexes

Dsing(Λ) :=Db(modΛ)

Dperf(modΛ)

The singularity category of a radical-square-zero algebra

Theorem (X.-W. Chen)

Let Λ be a finite dimensional k-algebra and J its Jacobson radical.Suppose J2 = 0. Define

S(Λ) := lim−→EndΛ(J⊗n)

where the maps in the directed system are f 7→ idJ ⊗f and

B := lim−→HomΛ(J⊗n, J⊗n−1).

B is an invertible S(Λ)-bimodule

J is a progenerator in Dsg(Λ) with endomorphism ring S(Λ);

HomDsing(Λ)(J,−) is an equivalence of triangulated categories(Dsing(Λ), [1]

)≡(modS(Λ),−⊗S(Λ) B

)where proj S(Λ) is the category of finitely presented rightS(Λ)-modules.

Chen + S

Let Λ = kQ/kQ≥2.

Easy: S(Λ) = S(Q).

Theorem (Chen + S)

qgr(kQ) ≡ modS(Q) = modS(Λ) ≡ Dsing(Λ)

as triangulated categories.

Chen + S

Let Λ = kQ/kQ≥2.

Easy: S(Λ) = S(Q).

Theorem (Chen + S)

Chen + S

Let Λ = kQ/kQ≥2.

Easy: S(Λ) = S(Q).

Theorem (Chen + S)

Monomial algebras

A monomial algebra is an algebra of the form

A =k〈x1, . . . , xg 〉(w1, . . . ,wr )

where w1, . . . ,wr are words in the letters x1, . . . , xg .

Theorem (Holdaway-S)

Let A be a monomial algebra and Q its Ufnarovskii graph. There isa homomorphism of graded algebras f : A→ kQ such thatkQ ⊗A − induces an equivalence of categories

QGr A ≡ QGr kQ.

More general monomial algebras: This theorem also holds whenA = the path algebra of a quiver modulo a finite number ofrelations of the form “path= 0”.

Monomial algebras

A monomial algebra is an algebra of the form

A =k〈x1, . . . , xg 〉(w1, . . . ,wr )

where w1, . . . ,wr are words in the letters x1, . . . , xg .

Theorem (Holdaway-S)

Let A be a monomial algebra and Q its Ufnarovskii graph. There isa homomorphism of graded algebras f : A→ kQ such thatkQ ⊗A − induces an equivalence of categories

QGr A ≡ QGr kQ.

More general monomial algebras: This theorem also holds whenA = the path algebra of a quiver modulo a finite number ofrelations of the form “path= 0”.

Sklyanin algebras

If (a, b, c) ∈ P2 define Sa,b,c =k〈x , y , z〉(f1, f2, f3)

f1 =ayz + bzy + cx2

f2 =azx + bxz + cy 2

f3 =axy + byx + cz2.

Call Sa,b,c a 3-dimensional Sklyanin algebra if (a, b, c) ∈ P2 −D

(1, 0, 0), (0, 1, 0), (0, 0, 1)}t{

(a, b, c) | a3 = b3 = c3}

⊂ P2k .

Artin, Tate, and Van den Bergh:

Sa,b,c is like the polynomial ring k[X ,Y ,Z ]

QGr Sa,b,c is like QcohP2.

“Degenerate” Sklyanin algebras: assume (a, b, c) ∈ D

Theorem (C. Walton)

Sa,b,c has infinite global dimension, is not noetherian, hasexponential growth, and has zero divisors.

Theorem (S)

Let k be a field having a primitive cube root of unity ω.

1 If a = b, then

Sa,b,c∼=

k〈u, v ,w〉(u2, v 2,w 2)

2 If a 6= b, then

Sa,b,c∼=

k〈u, v ,w〉(uv , vw ,wu)

3 Gr(Sa,b,c) ≡ Gr(Sa′,b′,c ′) for all (a, b, c), (a′, b′, c ′) ∈ D

Theorem (S)

There is a quiver Q, independent of (a, b, c) ∈ D, such that

QGr(Sa,b,c) ≡ QGr kQ ≡ ModS(Q)

where S(Q) = lim−→ Sn and each Sn is a product of three matrix

algebras. There is an action of µ3 = 3√

1 ⊂ k× as automorphismsof the free algebra F = k〈X ,Y 〉 such that

QGr(Sa,b,c) ≡ QGr(F o µ3).

Theorem (S)

There is a quiver Q, independent of (a, b, c) ∈ D, such that

QGr(Sa,b,c) ≡ QGr kQ ≡ ModS(Q)

where S(Q) = lim−→ Sn and each Sn is a product of three matrix

algebras. There is an action of µ3 = 3√

1 ⊂ k× as automorphismsof the free algebra F = k〈X ,Y 〉 such that

QGr(Sa,b,c) ≡ QGr(F o µ3).

The Ufnarovskii graph for A = k〈u, v ,w〉/(u2, v 2,w 2) is the quiver

��@@@

��•

<< •v1

v2qq (2)

The equivalence QGr A ≡ QGr kQ is induced by the k-algebrahomomorphism f : A→ kQ defined by

f (u) = u1 + u2,

f (v) = v1 + v2,

f (w) = w1 + w2.

The Ufnarovskii graph for A′ = k〈u, v ,w〉/(uv , vw ,wu) is thequiver

��~~~~~~~

��

•w1 99 w2

// •

__@@@@@@@v1ee

The equivalence QGr A′ ≡ QGr kQ ′ is induced by the k-algebrahomomorphism f ′ : A′ → kQ ′ defined by

f ′(u) = u1 + u2,

f ′(v) = v1 + v2,

f ′(w) = w1 + w2.

The 3-Veronese quivers of Q and Q ′ are the same, then QGr kQ isequivalent to QGr kQ ′.The incidence matrix of the nth Veronese quiver is the nth power ofthe incidence matrix of the original quiver. The incidence matricesof Q and Q ′ are0 1 1

1 0 11 1 0

1 1 00 1 11 0 1

and the third power of each is2 3 3

3 2 33 3 2

so QGr kQ ≡ QGr kQ ′.

Examples of interest to C∗-algebraists

Take C = the base field.

S(Q) = lim−→EndCI (CQn) in the category of C-algebras.

S(Q) = lim−→EndCI (CQn) in the category of C∗-algebras.

S(Q) is a dense subalgebra of S(Q).

S(Q) is an AF algebra. (AF=almost finite dimensional)AF-algebras are an important class of C∗-algebras.Some interesting AF-algebras arise as S(Q):

Mn∞(C)

Penrose tilings

Compact operators + multiples of the identity

Mn∞(C)

Penrose tilings

Mn∞(C)

Penrose tilings

Mn∞(C)

Penrose tilings

Penrose tiling example

Connes associates to the space of Penrose tilings of R2 theC∗-algebra with Bratteli diagram

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��>>>>>>>> 3 //

��>>>>>>>> 5 //

��>>>>>>>> 8 //

AAAAAAAA · · ·

@@��1

@@��2

@@��3

@@��5

>>}}}}}}}}· · ·

This is also the Bratteli diagram for S(Q) where

Q = • **99 •jj

ModS(Q) ≡ QGr CQ ≡ QGrC〈x , y〉

(y 2).

Compact operators + scalars

K(H) = the C∗-algebra of compact operators on an infinitedimensional separable Hilbert space H.

Let Q = •$$

•oozz

The Bratteli diagram for S(Q) is

��>>>>>>>> 1 //

AAAAAAAA · · ·

1 // 2 // 3 // 4 // 5 // · · ·

soS(Q) ∼= K(H)⊕ C idH .

Annoying pop-up

Projnc

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)is the space of Penrose tilings.

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Annoying pop-up

Projnc

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THE END

Annoying pop-up

Projnc

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THE END

Annoying pop-up

Projnc

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THE END

s. paul smith · s. paul smith university of washington seattle, wa 98195....

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