Representations of Quivers

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1 MINGLE 2012 Simon Peacock 4th October, 2012

2 Outline 1 Quivers Representations 2 Path Algebra Modules 3 Modules Representations

3 Quiver A quiver, Q, is a directed graph.

4 Quiver A quiver, Q, is a directed graph. Example α ζ ε β γ δ 4 5

5 Representation A representation of a quiver over a field k is an assignment of a k-vector space, α V i, to each vertex i and a k-linear map, f α : V i V j to each edge i j.

6 Representation A representation of a quiver over a field k is an assignment of a k-vector space, α V i, to each vertex i and a k-linear map, f α : V i V j to each edge i j. Example f γ V 4 f α V 1 V 2 V 3 f ζ f ε f β f δ V 5

7 Morphism of representations If V = (V i, f α ) and W = (W i, g α ) are two representations of a quiver Q, then a morphism φ : V W is a set of k-linear maps {φ i : V i W i i a vertex} α such that for each edge i j the square V i f α V j φ i φ j W i g α W j commutes; that is g α φ i = φ j f α.

8 Morphism of representations If V = (V i, f α ) and W = (W i, g α ) are two representations of a quiver Q, then a morphism φ : V W is a set of k-linear maps {φ i : V i W i i a vertex} α such that for each edge i j the square V i f α V j φ i φ j W i g α W j commutes; that is g α φ i = φ j f α. If every φ i is invertible then φ is an isomorphism.

9 Examples Example

10 Examples Example Basic linear algebra

11 Examples Example Basic linear algebra Example

12 Examples Example Basic linear algebra Example Jordan-Normal form

13 Algebra is x sighting A ring, A, is an algebra over a field k, if A also has the structure of a k-vector space. Abelian group under addition, with a (not necessarily commutative) multiplication that distributes over addition. I always assume rings have a unit.

14 Algebra is x sighting A ring, A, is an algebra over a field k, if A also has the structure of a k-vector space. Example C is a 2-dimensional algebra over R. Abelian group under addition, with a (not necessarily commutative) multiplication that distributes over addition. I always assume rings have a unit.

15 Paths A path in a quiver is a sequence of zero or more edges, starting from a given vertex. The product of two path is defined to be the concatenation if this makes sense or zero otherwise.

16 Paths A path in a quiver is a sequence of zero or more edges, starting from a given vertex. The product of two path is defined to be the concatenation if this makes sense or zero otherwise. Example α β The paths are e 1, e 2, e 3, α, β and αβ.

17 Paths A path in a quiver is a sequence of zero or more edges, starting from a given vertex. The product of two path is defined to be the concatenation if this makes sense or zero otherwise. Example α β e 1 α = α The paths are e 1, e 2, e 3, α, β and αβ.

18 Paths A path in a quiver is a sequence of zero or more edges, starting from a given vertex. The product of two path is defined to be the concatenation if this makes sense or zero otherwise. Example α β The paths are e 1, e 2, e 3, α, β and αβ. e 1 α = α α β = αβ

19 Paths A path in a quiver is a sequence of zero or more edges, starting from a given vertex. The product of two path is defined to be the concatenation if this makes sense or zero otherwise. Example α β The paths are e 1, e 2, e 3, α, β and αβ. e 1 α = α α β = αβ α e 3 = 0

20 Path algebra For a quiver Q, the path algebra over a field k, denoted kq, is the vector space with basis all paths. The ring multiplication is then extended linearly from path multiplication.

21 Path algebra For a quiver Q, the path algebra over a field k, denoted kq, is the vector space with basis all paths. The ring multiplication is then extended linearly from path multiplication. Example Q = α

22 Path algebra For a quiver Q, the path algebra over a field k, denoted kq, is the vector space with basis all paths. The ring multiplication is then extended linearly from path multiplication. Example Q = α kq = ke 1 kα kα 2

23 Path algebra For a quiver Q, the path algebra over a field k, denoted kq, is the vector space with basis all paths. The ring multiplication is then extended linearly from path multiplication. Example Q = α kq = ke 1 kα kα 2 = k[x]

24 Path algebra For a quiver Q, the path algebra over a field k, denoted kq, is the vector space with basis all paths. The ring multiplication is then extended linearly from path multiplication. Example Q = α kq = ke 1 kα kα 2 = k[x] e i is the identity element of kq.

25 Module A module over a ring generalises the idea of a vector space over a field. M is a module over a ring R if M is an abelian group, there is a map M R M that is compatible with the ring and group operations.

26 Module A module over a ring generalises the idea of a vector space over a field. M is a module over a ring R if M is an abelian group, there is a map M R M that is compatible with the ring and group operations. Example Z n is a module over Z with the obvious action: (a 0,..., a n )x (a 0 x,..., a n x)

27 Modules Representations Modules of kq are representations of Q over k.

28 Modules Representations Modules of kq are representations of Q over k. Given a module M:

29 Modules Representations Modules of kq are representations of Q over k. Given a module M: For a vertex i, define V i = Me i.

30 Modules Representations Modules of kq are representations of Q over k. Given a module M: For a vertex i, define V i = Me i. α For an edge i j, define f α to be multiplication by α me i me i α = mαe j Me j

31 Modules Representations Modules of kq are representations of Q over k. Given a module M: For a vertex i, define V i = Me i. α For an edge i j, define f α to be multiplication by α me i me i α = mαe j Me j Example M α Me 1 Me 2 Me 3 ζ β γ ε δ Me 4 Me 5

32 Modules Representations Representations of Q over k are modules of kq.

33 Modules Representations Representations of Q over k are modules of kq. Given a representation (V i, f α ):

34 Modules Representations Representations of Q over k are modules of kq. Given a representation (V i, f α ): M = V i as an abelian group.

35 Modules Representations Representations of Q over k are modules of kq. Given a representation (V i, f α ): M = V i as an abelian group. M kq M is extended from the action

36 Modules Representations Representations of Q over k are modules of kq. Given a representation (V i, f α ): M = V i as an abelian group. M kq M is extended from the action v i e j { vi if i = j 0 otherwise for v i V i

37 Modules Representations Representations of Q over k are modules of kq. Given a representation (V i, f α ): M = V i as an abelian group. M kq M is extended from the action v i e j v i α { vi if i = j 0 otherwise { α fα (v i ) if i 0 otherwise for v i V i

38 Modules Representations These associations are inverses of one another. M (Me i, α)

39 Modules Representations These associations are inverses of one another. M (Me i, α) Me i

40 Modules Representations These associations are inverses of one another. M (Me i, α) Me i = M e i

41 Modules Representations These associations are inverses of one another. M (Me i, α) Me i = M e i = M

42 Modules Representations These associations are inverses of one another. M (Me i, α) Me i = M e i = M (V i, f α ) V i

43 Modules Representations These associations are inverses of one another. M (Me i, α) Me i = M e i = M (V i, f α ) V i ( ) ( ) jv j ei, f α i,α

44 Modules Representations These associations are inverses of one another. M (Me i, α) Me i = M e i = M (V i, f α ) V i ( ) ( ) jv j ei, f α = (V i, f α ) i,α

45 Yay MINGLE!

Representations of quivers

Representations of quivers Gwyn Bellamy October 13, 215 1 Quivers Let k be a field. Recall that a k-algebra is a k-vector space A with a bilinear map A A A making A into a unital, associative ring. Notice