QUIVERS, REPRESENTATIONS, ROOTS AND LIE ALGEBRAS. Volodymyr Mazorchuk. (Uppsala University)

Transcription

1 QUIVERS, REPRESENTATIONS, ROOTS AND LIE ALGEBRAS Volodymyr Mazorchuk (Uppsala University)

2 . QUIVERS Definition: A quiver is a quadruple Q = (V, A, t, h), where V is a non-empty set; A is a set; t and h are maps from A to V. The elements of V are called vertices. The elements from A are called arrows. An arrow α A has tail t(α) and head h(α), which means that it goes from t(α) to h(α). Detailed example: For the quiver α we have V = {, }. A = {α}. t(α) =, h(α) =.

3 More complicated picture:. REPRESENTATIONS OF QUIVERS associate a (complex) vec- To each vertex v V torspace M v. To each arrow α A associate a linear map M(α) : M t(α) M h(α). (x) x 3x is a representation of α

4 Definition. Tow representations M and N of some quiver Q are said to be isomorphic provided that for each v V there exists an isomorphism F v : M v N v such that for every α A the following diagram is commutative (i.e. F h(α) M(α) = N(α)F t(α) ): M t(α) M(α) M h(α) F t(α) N t(α) N(α) F h(α) N h(α) Example: (x) (x) (x) (x) (x) (x) (x) (x) is an isomorphism of representations of α 3

5 3. INDEOMPOSABLE REPRESENTATIONS Definition. The trivial representation of Q is the representation M for which M v = 0 for all v V. Automatically M(α) = 0 for all α A. Definition. If M and N are representations of Q then the direct sum of M and N is the representation M N defined as follows: (M N) v = M v N v for all v V ; (M N)(α) = M(α) N(α) for all α A. Definition. A representation is called indecomposable if it is not isomorphic to a direct sum of two non-trivial representations. Example: (x) (x) is an indecomposable representation of α 4

6 Example: (x) (0) is a decomposable representation of α because it is isomorphic to the direct sum of representations (x) (0) 0 and 0 (x) (0) Krull-Schmidt Theorem: Each representation is isomorphic to a direct sum of indecomposable summands, which are uniquely determined up to isomorphism and permutation. 5

7 4. GABRIEL S THEOREM BASI QUESTION: Which quivers have only finitely many isomorphism classes of indecomposable representation (i.e. are or finite type)? Example: α has THREE indecomposable representations, namely: (x) (x) (x) (0) 0 0 (x) (0) 6

8 Example: α has INFINITELY MANY indecomposable representations, namely: n J n (λ) where n N, λ and λ λ λ J n (λ) = λ λ λ is the Jordan cell of size n and eigenvalue λ. 7

9 GABRIEL S THEOREM: A quiver is of finite type if and only if after forgetting the orientation of arrows it reduces to one of the following graphs: A n :... D n :... E 6 : E 7 : E 8 : The above graphs are known as simply laced Dynkin diagrams. 8

10 5. LIE ALGEBRAS Dynkin diagrams (not only simply laced) are known to classify simple finite-dimensional complex Lie algebras. If D is a simply laced Dynkin diagram then the universal enveloping algebra of the positive part of the corresponding Lie algebra is defined as follows: generators: X v, where v is a vertex; relation [X v, X w ] = 0 if v and w are not connected; [X v, [X v, X w ]] = 0 if v and w are connected. For example, starting from the quiver of type A n the above relations define the universal enveloping algebra of the Lie algebra of all strictly upper triangular complex n n matrices. 9

11 6. HALL ALGEBRA OF A QUIVER Q quiver; M representation of Q; Definition: A subrepresentation N M of M is a collection of N v M v, closed with respect to the action of all M(α), α A. The restriction of M(α) to N is then N(α). Definition: If N M are representation of Q, then their quotient M/N is defined in the natural way. F q finite field with q elements; H(Q) Fq F q -Hall algebra of Q:. basis: isomorphism classes [N] of (finite-dimensional) representation of Q;. multiplication: [X] [Y ] = [M] (F q) M X,Y [M], where (F q ) M X,Y is the number of subrepresentation of M such that the subrepresentation itself is isomorphic to X, while the quotient is isomorphic to Y. 0

12 Proposition: Let Q be a Dynkin quiver, and X, Y, M be finite-dimensional representations of Q. Then there exist polynomials f M X,Y (t) Z[t] such that (F q) M X,Y = f M X,Y (q). Q Dynkin quiver H(Q) (specialized) Hall algebra of Q:. -basis: isomorphism classes [N] of finite-dimensional representation of Q;. multiplication: [X] [Y ] = [M] f M X,Y ()[M]. Ringel s Theorem: Let Q be Dynkin. Then H(Q) is isomorphic to the universal enveloping algebra of the positive part of the Lie algebra, associated to Q. Everything extends even to quantum groups!