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 π 1 ( A ) = automorphism group of the fibre functor Main goal : compute explicitely π 1 ( 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 y B x is a k -module, and ◮ composition of morphisms is k -bilinear. Each x B x is a k -algebra and y B x is a y B y - x B x -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 ∈ B 0 and each x in F − 1 ( b ), the map F x b : St x C → St b B , provided by F , is a k -isomorphism, where the star St b B at an object 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 H C D F G J � B B ∼ = conmutes, with J an isomorphism. The morphism ( H , J ) induces a unique group epimorphism λ H : Aut 1 F → Aut 1 G verifying Hf = λ H ( f ) H , for all f ∈ Aut 1 F . λ H ( f ) � D f H C C D F F G G J � B B B B ∼ =

Aut 1 F = { ( H , id) : H invertible } Let b ∈ B 0 and let F − 1 ( b ) be the corresponding fibre. ◮ F − 1 ( b ) � = ∅ by definition of covering. ◮ Aut 1 F 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 Aut 1 F acts transitively on some fibre. One can prove that for a Galois covering F , the group Aut 1 F acts transitively on any fibre.

The fundamental group Given a k -category B and a fixed object b 0 of B , Gal( B , b 0 ) denotes the category of Galois coverings of B with morphisms ( H , J ) such that J ( b ) = b , for any b in B 0 . Consider Φ : Gal( B , b 0 ) → Sets given by Φ( F ) = F − 1 ( b 0 ) Definition π 1 ( B , b 0 ) = 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 u 0 ∈ U 0 , c 0 ∈ C 0 with U ( u 0 ) = F ( c 0 ), there exists a unique morphism ( H , id) from U to F verifying H ( u 0 ) = c 0 . H u 0 c 0 U C � � � � � � � � U F U ( u 0 ) F ( c 0 ) B B Theorem If a connected k-category B admits a universal covering U then π 1 ( B , b 0 ) ≃ Aut 1 U .

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 X s ( y B x ) � y B x = s ∈ Γ such that X t ( z B y ) X s ( y B x ) ⊂ X ts ( z B x ). The homogeneous component of degree s from x to y is the k -module X s ( y B x ). 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 = B 0 × Γ, ◮ the module of morphisms from ( b , g ) to ( c , h ) is X h − 1 g c B b . 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 B #Γ � b ( b , g ) � ( c , h ) B #Γ ( b , g ) = X h − 1 g c B b � � � c B b 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 − 1 g . So it is also a morphism from ( b , sg ) to ( c , sh ) since ( sh ) − 1 sg = h − 1 g .

Let F : B #Γ 1 → B and G : B #Γ 2 → B be Galois coverings associated to connected gradings X 1 and X 2 of B with groups Γ 1 and Γ 2 , and let b 0 ∈ B . Let ( H , J ) : F → G be a morphism of coverings in Gal( B , b 0 ). Then there exists a map h : Γ 1 → Γ 2 such that H ( b 0 , g ) = ( b 0 , h ( g )) for all g ∈ Γ 1 . Moreover, h ( g ) = λ H ( g ) h (1), where λ H : Γ 1 → Γ 2 is the group morphism associated to H . Given σ ∈ Aut Φ and F : B #Γ → B , then the corresponding map σ F : Γ → Γ is given by σ F ( g ) = g σ F (1).

Connection between gradings and coverings Let Gal # ( B , b 0 ) be the full subcategory of Gal( B , b 0 ) whose objects are the smash product Galois coverings F : B #Γ → B . Theorem (Cibils, Marcos - 2006) The categories Gal # ( B , b 0 ) and Gal( B , b 0 ) are equivalent. The proof follows from the fact that any Galois covering F : C → B is isomorphic to the Galois covering B #Aut 1 F → B . Corollary Let Φ # : Gal # ( B , b 0 ) → Sets be the functor given by Φ # ( F : B #Γ → B ) = F − 1 ( b 0 ) = Γ . Then π 1 ( B , b 0 ) ∼ = Aut Φ # .

The matrix algebra Let k be a field containing a primitive n -th root of unity q . The matrix algebra M n ( k ) has a well-known presentation as follows: M n ( k ) = k { x , y } / � x n = 1 , y n = 1 , yx = qxy � where     0 0 · · · 0 1 q 0 0 · · · 0 q 2 1 0 · · · 0 0 0 0 · · · 0         q 3 0 1 · · · 0 0 0 0 · · · 0 x = , y = .      . .   . .  ... ... . . . .     . . . .     q n 0 0 · · · 1 0 0 0 0 · · ·

If we set deg( x ) = ( t , 1) and deg( y ) = (1 , t ), for t a generator of C n , we obtain a connected grading of k { x , y } such that the ideal of relations is homogeneous. Theorem The algebra M n ( k ) admits a simply connected grading by the group C n × C n .

� � � Another presentation of M n ( k ) is given by the following quiver with relations ( Q , I ). α n − 1 α 1 � 2 α 2 � 3 � n Q : 1 · · · n − 1 β 1 β 2 β n − 1 I = < β i α i − e i , α i β i − e i +1 | 1 ≤ i < n > . Then kQ / I is isomorphic to M n ( k ).

Let F n − 1 be the free group on n − 1 generators s 1 , . . . , s n − 1 , and let ◮ deg e i = 1 for any i with 1 ≤ i ≤ n , and ◮ deg α i = s i and deg β i = ( s i ) − 1 for any i with 1 ≤ i ≤ n − 1. This provides a well defined grading of kQ / I , hence of M n ( k ). Moreover, it is also simply connected. Corollary The matrix algebra M n ( 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 ) α c B b = X g c B b . g ∈ α The corresponding functor between the smash product coverings is precisely the canonical projection obtained through the quotient of B #Γ → B by N : � B B #Γ � � � � � � � � � � � � � � � � � � B #Γ / N

� � Proposition Let k be a field containing a primitive n-th root of unity. The grading by C n × C n and the grading by the free group F n − 1 have a maximal common quotient C n -grading, which is unique. M n ( k )# C n × C n M n ( k )# F n − 1 � � � ������������ � � � � � � � � � � M n ( k )# C n M n ( 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 π 1 M 2 ( k ) ≃ Z × C 2 . ◮ (Boboc and D˘ asc˘ alescu, 2007) If char ( k ) � = 3 ,then π 1 M 3 ( k ) ≃ F 2 × C 3 . ◮ (Bahturin and Zaicev, 2002) If k is an algebraically closed field, char ( k ) = 0 and p a prime then π 1 M p ( k ) ≃ F p − 1 × C p .

Triangular matrices Using a description of gradings of triangular algebras given by Valenti and Zaicev (2007), Theorem Let k be a field and let T n ( k ) be the algebra of triangular matrices of size n. Then π 1 T n ( k ) ≃ F n − 1 .

Truncated polynomial algebra Theorem Let k be a field of characteristic p and let A = k [ x ] / ( x p ) . There are two types of connected gradings of A, with no common quotient except the trivial one: ◮ the natural grading given by C p since k [ x ] / ( x p ) is isomorphic to the group algebra KC p . ◮ the grading given by Z or any of its quotients. Corollary Let k be a field of characteristic p. Then π 1 k [ x ] / ( x p ) = Z × C p .

The diagonal algebra k n Let E be a finite set with n elements and k a field. The diagonal algebra k n 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 k n . Corollary Let n be a non-square free positive integer and let k be a field as above. The algebra k n does not admit a universal covering.