Quiver Representation

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  • Harm Derksen and Jerzy Weyman, NOTICES OF THE AMS VOLUME 52, NUMBER 2, Quiver Representation: Quiver is Directed graph \(Q\) is pair \((Q_0, Q_1)\). Represent \(Q\) by a finite dimensional Vector space at each vertex and K-linear map from tail to head of each arrow. QUiver represntation classify as finite or tame or wild. Gabriel’s theorem I: Quiver \(Q\) is finite if and only if undirected graph \((Q_0, Q_1)\) is union of Dynkin graph \(A_n, D_n, E_6, E_7, E_8\). Theorem 10: Quiver \(Q\) is Tame if and only if undirected graph \((Q_0, Q_1)\) is union of Dynkin graphs and extended Dynkin graphs \(\hat{A_n}, \hat{D_n}, \hat{E_6}, \hat{E_7}, \hat{E_8}\). Gabriel’s Theorem II: Finite Quiver indecomposable representation one-to-one correspondence with positive roots of corresponding root system. Proof: Euler form (biliniear asymmetric form), Cartan form (bilinear symmetric form) and Tits form \(q(a)\) on Quiver may be represented in basis. By dimension of continuous parameter bounded by \(1-q(a)\), for finite quiver, nonzero dimension representation, Tits form \(\ge 1\) implies Cartan form is positive definite. Kac and Moody generalize: Kac-1980- The set of dimension vectors and indecomposable representations does not depend on orientation of arrows, dimension vectors of indecomposable represetnations correspond to positive root of root system. Connection between quiver representation and canonical basis of Quantum group is active research.
  • Will Dana - 2021 05 31 University of Michigan - Summer Minciourse Representation of Quiver, Lie Algebra Slide, Notes.
    • Diagram linear map
    • Krull-Schmidt Reduction of every quiver representation to direct sum of indecomposable ones.
    • Root system: vectors closed under reflection, contain no other multiple, crystallographic.
    • Simple roots: closest to a hyperplane, span V. Simple roots form a basis.
    • Cartan matrix for simply laced (same length roots) \(C_{ij}= <\alpha_i, \alpha_j>\)
    • Cartan matrix determines the root system.
      • Example \(A_3\) Diagonal entry is \(2\), off-diagonal is either \(0\) or \(-1\).
    • Dynkin diagram has simple roots as vertices, edge \(i \sim j\) when \(C_{ij}=-1\).
    • Classification of finite simply laced root systems:\(\hat{A_n}, \hat{D_n}, \hat{E_6}, \hat{E_7}, \hat{E_8}\). Also \(B_n, C_n\), exceptional \(F_4, G_2\). Proof: \(C_{ij}\) represents the inner product \(<,>\) in basis \(\alpha_1,\ldots, \alpha_n\) hence positive definite.
    • Gabriel’s Theorem: Dimension vector \({(dimV(x))}_{x \in Q_0}\) of a quiver representation \(V\) (seen as a Dynkin diagram) is one-to-one with positive root.

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