Why quivers? Quiver representations Gabriel's theorem But really, why quivers? Quiver representations and ADE Sira Gratz Sira Gratz Quiver representations “Definition” A representation of a quiver associates to every vertex a vector space, and to every arrow a compatible linear map. Why quivers? Quiver representations Gabriel's theorem But really, why quivers? Quiver Definition A quiver is a directed graph, where loops and multiple edges between two vertices are allowed. Sira Gratz Quiver representations Why quivers? Quiver representations Gabriel's theorem But really, why quivers? Quiver Definition A quiver is a directed graph, where loops and multiple edges between two vertices are allowed. “Definition” A representation of a quiver associates to every vertex a vector space, and to every arrow a compatible linear map. Sira Gratz Quiver representations Let A be a square matrix. When does there exist an invertible matrix S and a diagonal matrix D such that SAS−1 = D? Why quivers? Quiver representations Gabriel's theorem But really, why quivers? A question from linear algebra Question: When is a square matrix diagonalisable? Sira Gratz Quiver representations Why quivers? Quiver representations Gabriel's theorem But really, why quivers? A question from linear algebra Question: When is a square matrix diagonalisable? Let A be a square matrix. When does there exist an invertible matrix S and a diagonal matrix D such that SAS−1 = D? Sira Gratz Quiver representations SAS−1 = D , SA = DS n A K S S n n D K D K Why quivers? Quiver representations Gabriel's theorem But really, why quivers? The same question, in a diagram n A K Sira Gratz Quiver representations n A K S n D K Why quivers? Quiver representations Gabriel's theorem But really, why quivers? The same question, in a diagram SAS−1 = D , SA = DS n A K S n D K Sira Gratz Quiver representations Why quivers? Quiver representations Gabriel's theorem But really, why quivers? The same question, in a diagram SAS−1 = D , SA = DS n n A K A K S S n n D K D K Sira Gratz Quiver representations Why quivers? Quiver representations Gabriel's theorem But really, why quivers? The same question, in a diagram study representations of A n K the loop: S n D K • Sira Gratz Quiver representations Why quivers? Quiver representations Gabriel's theorem But really, why quivers? Another question from linear algebra Question: Let (A; B) and (A0; B0) be pairs of matrices, all of the same dimension. When do there exist invertible matrices S; T such that SAT −1 = A0; SBT −1 = B0? Sira Gratz Quiver representations A A m n m n K K K K B B S T S T A0 A0 m n m n K K K K B0 B0 A A m n m n K K K K B B S T S T A0 A0 m n m n K K K K B0 B0 Why quivers? Quiver representations Gabriel's theorem But really, why quivers? The same question, in a diagram A m n K K B A0 m n K K B0 Sira Gratz Quiver representations Why quivers? Quiver representations Gabriel's theorem But really, why quivers? The same question, in a diagram A A m n m n K K K K B B S T S T A0 A0 m n m n K K K K B0 B0 A A m n m n K K K K B B S T S T A0 A0 m n m n K K K K B0 B0 Sira Gratz Quiver representations Why quivers? Quiver representations Gabriel's theorem But really, why quivers? The same question, in a diagram A m n study representations of K K the 2-Kronecker quiver: B S T A m n K K • • B Sira Gratz Quiver representations More precisely, a quiver consists of the following data: a set of vertices Q0; a set of arrows Q1; a map s : Q1 ! Q0 that maps an arrow to its source; a map t : Q1 ! Q0 that maps an arrow to its target. Why quivers? Definitions and Examples Quiver representations A category of quiver representations Gabriel's theorem Indecomposables But really, why quivers? Quivers Definition A quiver is a directed graph, where loops and multiple edges between two vertices are allowed. Sira Gratz Quiver representations Why quivers? Definitions and Examples Quiver representations A category of quiver representations Gabriel's theorem Indecomposables But really, why quivers? Quivers Definition A quiver is a directed graph, where loops and multiple edges between two vertices are allowed. More precisely, a quiver consists of the following data: a set of vertices Q0; a set of arrows Q1; a map s : Q1 ! Q0 that maps an arrow to its source; a map t : Q1 ! Q0 that maps an arrow to its target. Sira Gratz Quiver representations We have Q0 = f1; 2; 3; 4g; Q1 = fα; β; γ; δ; "; ζg s(α) = 1; t(α) = 2; s(ζ) = t(ζ) = 1; etc. Why quivers? Definitions and Examples Quiver representations A category of quiver representations Gabriel's theorem Indecomposables But really, why quivers? Example 3 " δ γ α ζ 1 2 4 β Sira Gratz Quiver representations Why quivers? Definitions and Examples Quiver representations A category of quiver representations Gabriel's theorem Indecomposables But really, why quivers? Example 3 " δ γ α ζ 1 2 4 β We have Q0 = f1; 2; 3; 4g; Q1 = fα; β; γ; δ; "; ζg s(α) = 1; t(α) = 2; s(ζ) = t(ζ) = 1; etc. Sira Gratz Quiver representations Throughout, we will assume that both Q0 and Q1 are finite. Why quivers? Definitions and Examples Quiver representations A category of quiver representations Gabriel's theorem Indecomposables But really, why quivers? Quivers Definition A quiver is a directed graph. More precisely, a quiver consists of the following data: a set of vertices Q0; a set of arrows Q1; a map s : Q1 ! Q0 that maps an arrow to its source; a map t : Q1 ! Q0 that maps an arrow to its target. Sira Gratz Quiver representations Why quivers? Definitions and Examples Quiver representations A category of quiver representations Gabriel's theorem Indecomposables But really, why quivers? Quivers Definition A quiver is a directed graph. More precisely, a quiver consists of the following data: a set of vertices Q0; a set of arrows Q1; a map s : Q1 ! Q0 that maps an arrow to its source; a map t : Q1 ! Q0 that maps an arrow to its target. Throughout, we will assume that both Q0 and Q1 are finite. Sira Gratz Quiver representations Definition Let Q be a quiver. A representation (Vi ; Mα)i2Q0,α2Q1 of Q is a collection of vector spaces Vi of vector spaces over K, indexed by Q0, along with a collection Mα of linear maps, indexed by Q1, such that for all α 2 Q1 we have Mα : Vs(α) ! Vt(α): Why quivers? Definitions and Examples Quiver representations A category of quiver representations Gabriel's theorem Indecomposables But really, why quivers? Quiver representations Throughout we work over an algebraically closed field K. Sira Gratz Quiver representations Why quivers? Definitions and Examples Quiver representations A category of quiver representations Gabriel's theorem Indecomposables But really, why quivers? Quiver representations Throughout we work over an algebraically closed field K. Definition Let Q be a quiver. A representation (Vi ; Mα)i2Q0,α2Q1 of Q is a collection of vector spaces Vi of vector spaces over K, indexed by Q0, along with a collection Mα of linear maps, indexed by Q1, such that for all α 2 Q1 we have Mα : Vs(α) ! Vt(α): Sira Gratz Quiver representations Why quivers? Definitions and Examples Quiver representations A category of quiver representations Gabriel's theorem Indecomposables But really, why quivers? Quiver representations Throughout we work over an algebraically closed field K. Definition Let Q be a quiver. A (finite dimensional) representation (Vi ; Mα)i2Q0,α2Q1 of Q is a collection of (finite dimensional) vector spaces Vi of vector spaces over K, indexed by Q0, along with a collection Mα of linear maps, indexed by Q1, such that for all α 2 Q1 we have Mα : Vs(α) ! Vt(α): Sira Gratz Quiver representations Why quivers? Definitions and Examples Quiver representations A category of quiver representations Gabriel's theorem Indecomposables But really, why quivers? Example C 1 1 3 5 1 0 1 1 2 100 0 0 C C C 2 1 Sira Gratz Quiver representations Answer What do you mean by “different”? Why quivers? Definitions and Examples Quiver representations A category of quiver representations Gabriel's theorem Indecomposables But really, why quivers? Finding representations Question Can we find all different representations of a given quiver? Sira Gratz Quiver representations Why quivers? Definitions and Examples Quiver representations A category of quiver representations Gabriel's theorem Indecomposables But really, why quivers? Finding representations Question Can we find all different representations of a given quiver? Answer What do you mean by “different”? Sira Gratz Quiver representations Why quivers? Definitions and Examples Quiver representations A category of quiver representations Gabriel's theorem Indecomposables But really, why quivers? Morphisms Definition Let V = (Vi ; Mα)i2Q0,α2Q1 and W = (Wi ; Nα)i2Q0,α2Q1 be representations of a quiver Q.A morphism of quiver representations ': V!W is a collection of linear maps ' = ('i : Vi ! Wi )i2Q0 , such that for each α 2 Q1 the diagram Mα Vs(α) / Vt(α) 's(α) 't(α) Nα Ws(α) / Wt(α) commutes. Sira Gratz Quiver representations 1 0 3 1 1 2 0 0 C C C 2 5 1 Why quivers? Definitions and Examples Quiver representations A category of quiver representations Gabriel's theorem Indecomposables But really, why quivers? Identity morphism 1 0 3 1 1 2 0 0 C C C 2 5 1 Sira Gratz Quiver representations Why quivers? Definitions and Examples Quiver representations A category of quiver representations Gabriel's theorem Indecomposables But really, why quivers? Identity morphism 1 0 3 1 1 2 0 0 C C C 2 5 1 1 0 3 1 1 2 0 0 C C C 2 5 1 Sira Gratz Quiver representations −1 0 2 0 5 C −2 1 1 1 1 4 2 2 3 C −2 1 −1 0 2 5 1 4 −2 1 = = 1 1 0 5 −1 5 2 3 1 1 Why quivers? Definitions