Connected gradings and fundamental groups Mar a Julia Redondo - - PowerPoint PPT Presentation

Connected gradings and fundamental groups

Mar´ ıa Julia Redondo

Joint work with Claude Cibils and Andrea Solotar

ICTP, Trieste - February 1-5, 2010

The intrinsic fundamental group of a linear category,

• C. Cibils, M. J. Redondo, A. Solotar,

arXiv:0706.2491 π1(A) = automorphism group of the fibre functor Main goal: compute explicitely π1(A) for several algebras A Main tool: the relation between gradings and Galois coverings of the algebra (considered as a k-linear category with one object).

Galois coverings A k-category B is a small category such that

◮ each set of morphisms yBx is a k-module, and ◮ composition of morphisms is k-bilinear.

Each xBx is a k-algebra and yBx is a yBy-xBx-bimodule. A k-algebra A can be viewed as a k-category with one object.

Let C and B be k-categories. A k-functor F : C → B is a covering if it is surjective on objects and if for each b ∈ B0 and each x in F −1(b), the map F x

b : StxC → StbB,

provided by F, is a k-isomorphism, where the star StbB at an

• bject b of a k-category B is the direct sum of all k-modules of

morphisms with source or target b.

A morphism from a covering F : C → B to a covering G : D → B is a pair of k-linear functors (H, J) where C

H

• F
• D

G

• B

J ∼ =

B

conmutes, with J an isomorphism. The morphism (H, J) induces a unique group epimorphism λH : Aut1 F → Aut1 G verifying Hf = λH(f )H, for all f ∈ Aut1 F. C

f

• F
• C

H

• F
• D

G

• λH(f ) D

G

• B

B

J ∼ =

B

B

Aut1F = {(H, id) : H invertible} Let b ∈ B0 and let F −1(b) be the corresponding fibre.

◮ F −1(b) = ∅ by definition of covering. ◮ Aut1F acts freely on F −1(b).

Definition

A covering F : C → B of k-categories is a Galois covering if C is connected and if Aut1F acts transitively on some fibre. One can prove that for a Galois covering F, the group Aut1F acts transitively on any fibre.

The fundamental group Given a k-category B and a fixed object b0 of B, Gal(B, b0) denotes the category of Galois coverings of B with morphisms (H, J) such that J(b) = b, for any b in B0. Consider Φ : Gal(B, b0) → Sets given by Φ(F) = F −1(b0)

Definition

π1(B, b0) = Aut Φ

A universal covering U : U → B is an object in Gal(B) such that for any Galois covering F : C → B, and for any u0 ∈ U0, c0 ∈ C0 with U(u0) = F(c0), there exists a unique morphism (H, id) from U to F verifying H(u0) = c0. U

H

• U
• C

F

• u0
• c0
• B

B U(u0) F(c0)

Theorem

If a connected k-category B admits a universal covering U then π1(B, b0) ≃ Aut1U.

Examples of Galois coverings A grading X of a k-category B by a group Γ is given by a direct sum decomposition of each k-module of morphisms

yBx =

• s∈Γ

X s (yBx) such that X t (zBy) X s (yBx) ⊂ X ts (zBx). The homogeneous component of degree s from x to y is the k-module X s (yBx). A grading is said to be connected if given any two objects in B, and any element g ∈ Γ, they can be joined by a non-zero homogeneous walk of degree g.

Let X be a Γ-grading of the k-category B. The smash product category B#Γ is given by:

◮ (B#Γ)0 = B0 × Γ, ◮ the module of morphisms from (b, g) to (c, h) is X h−1g cBb.

Morphisms are provided by homogeneous components, and composition in B#Γ is given by the original composition in B.

The smash product construction provides examples of Galois coverings. If X is a Γ-grading of the k-category B, the functor B#Γ

B

(b, g)

b

(c,h)B#Γ(b,g) = X h−1g cBb

cBb

is a Galois covering with Γ as group of automorphisms.

The action of Γ on the smash product category B#Γ is given as follows. The action on objects is given by the left action of Γ on itself: s(b, g) = (b, sg). A morphism (b, g) → (c, h) is a homogeneous morphism from b to c of degree h−1g. So it is also a morphism from (b, sg) to (c, sh) since (sh)−1sg = h−1g.

Let F : B#Γ1 → B and G : B#Γ2 → B be Galois coverings associated to connected gradings X1 and X2 of B with groups Γ1 and Γ2, and let b0 ∈ B. Let (H, J) : F → G be a morphism of coverings in Gal(B, b0). Then there exists a map h : Γ1 → Γ2 such that H(b0, g) = (b0, h(g)) for all g ∈ Γ1. Moreover, h(g) = λH(g)h(1), where λH : Γ1 → Γ2 is the group morphism associated to H. Given σ ∈ Aut Φ and F : B#Γ → B, then the corresponding map σF : Γ → Γ is given by σF(g) = gσF(1).

Connection between gradings and coverings Let Gal#(B, b0) be the full subcategory of Gal(B, b0) whose

• bjects are the smash product Galois coverings F : B#Γ → B.

Theorem (Cibils, Marcos - 2006)

The categories Gal#(B, b0) and Gal(B, b0) are equivalent. The proof follows from the fact that any Galois covering F : C → B is isomorphic to the Galois covering B#Aut1F → B.

Corollary

Let Φ# : Gal#(B, b0) → Sets be the functor given by Φ#(F : B#Γ → B) = F −1(b0) = Γ. Then π1(B, b0) ∼ = Aut Φ#.

The matrix algebra Let k be a field containing a primitive n-th root of unity q. The matrix algebra Mn(k) has a well-known presentation as follows: Mn(k) = k{x, y}/xn = 1, yn = 1, yx = qxy where x =        · · · 1 1 · · · 1 · · · . . . ... . . . · · · 1        , y =        q · · · q2 · · · q3 · · · . . . ... . . . · · · qn        .

If we set deg(x) = (t, 1) and deg(y) = (1, t), for t a generator of Cn, we obtain a connected grading of k{x, y} such that the ideal

• f relations is homogeneous.

Theorem

The algebra Mn(k) admits a simply connected grading by the group Cn × Cn.

Another presentation of Mn(k) is given by the following quiver with relations (Q, I). Q : 1

α1 2 β1

• α2 3

β2

• · · ·

n − 1

αn−1

n

βn−1

• I =< βiαi − ei, αiβi − ei+1|1 ≤ i < n > .

Then kQ/I is isomorphic to Mn(k).

Let Fn−1 be the free group on n − 1 generators s1, . . . , sn−1, and let

◮ deg ei = 1 for any i with 1 ≤ i ≤ n, and ◮ deg αi = si and deg βi = (si)−1 for any i with 1 ≤ i ≤ n − 1.

This provides a well defined grading of kQ/I, hence of Mn(k). Moreover, it is also simply connected.

Corollary

The matrix algebra Mn(k) has no universal covering.

Definition

The quotient of a Γ-grading X of a category B by a normal subgroup N of Γ is a Γ/N-grading X/N of B, where the homogeneous component of degree α is (X/N)αcBb =

• g∈α

X g cBb. The corresponding functor between the smash product coverings is precisely the canonical projection obtained through the quotient of B#Γ → B by N: B#Γ

• B

B#Γ/N

Proposition

Let k be a field containing a primitive n-th root of unity. The grading by Cn × Cn and the grading by the free group Fn−1 have a maximal common quotient Cn-grading, which is unique. Mn(k)#Cn × Cn

• Mn(k)#Fn−1
• Mn(k)#Cn
• Mn(k)

Next we use the description of gradings of matrix algebras given by several authors and we obtain:

Theorem

◮ (Boboc, D˘

asc˘ alescu and Khazal, 2003) If char(k) = 2,then π1M2(k) ≃ Z × C2.

◮ (Boboc and D˘

asc˘ alescu, 2007) If char(k) = 3,then π1M3(k) ≃ F2 × C3.

◮ (Bahturin and Zaicev, 2002)

If k is an algebraically closed field, char(k) = 0 and p a prime then π1Mp(k) ≃ Fp−1 × Cp.

Triangular matrices Using a description of gradings of triangular algebras given by Valenti and Zaicev (2007),

Theorem

Let k be a field and let Tn(k) be the algebra of triangular matrices

• f size n. Then

π1Tn(k) ≃ Fn−1.

Truncated polynomial algebra

Theorem

Let k be a field of characteristic p and let A = k[x]/(xp). There are two types of connected gradings of A, with no common quotient except the trivial one:

◮ the natural grading given by Cp since k[x]/(xp) is isomorphic

to the group algebra KCp.

◮ the grading given by Z or any of its quotients.

Corollary

Let k be a field of characteristic p. Then π1k[x]/(xp) = Z × Cp.

The diagonal algebra kn Let E be a finite set with n elements and k a field. The diagonal algebra kn is the vector space of maps from E to k with pointwise multiplication.

Proposition

Let E be a finite set with cardinality n and let k be a field with enough n-th roots of unity. Let G be any abelian group of order n. Then there is a simply connected G-grading of kn.

Corollary

Let n be a non-square free positive integer and let k be a field as

• above. The algebra kn does not admit a universal covering.

The following theorem is based on results due to D˘ asc˘ alescu.

Theorem

◮ Let k be a field of characteristic different from 2. Then

π1(k × k) = C2.

◮ Let k be a field containing all roots of unity of order 2 and 3.

Then π1(k3) = C3 × C2.

◮ Let k be a field containing all roots of unity of order 2, 3 and

• 4. Then π1(k4) = C4 × (C2 × C2) × C3 × (C2 ∗ C2).