Ringel’s Theorem

Please find linked: Quiver Representation

  • How many ways to extend the realization of representation. Ringel hall algebra given by basis structure constants.
  • Ringel Hall Algebra is associative but not commutative: one way product gives both direct sum and a non-trivial but other way is just a direct sum.
  • Which finite field? No preference. We can treat all together. Define a universal Ringel-Hall Algebra.
  • For Scott Neville’s question: The limit specializing to \(q=1\), the geometric interpretation is via complex numbers (see Kirillov’s book).
  • Universal Enveloping Algebra: ‘free’-est possible algebra where the Lie Algebra commutator holds (Rep theory of Lie Algebras into Rep theory of Associative Algebras).
  • The map \(\phi\) is well defined.
    • Sln had strictly upper triangular and strictly lower triangular. Decomposition into Positive, Cartan and Negative.
    • Corollary Serre relations: Positive part isomorphism with generators \(x_1, \ldots, x_n\) for simply laced Dynkin diagrams, these fall into two classes.
    • If \(S_i, S_j\) not adjacent in the quiver, then the extension is two of one dimensional: \(S_i \oplus S_j\), hence they commute in the Lie Algebra.
    • For adjacent, compute for the quiver \(1 \to 2\).
  • The map \(\phi\) is surjective.
    • Want to show that \([S_i]\) generate the Ringel Hall Lie Algebra.
    • Show the Ringel Hall is generated by the indecomposables. We have their listing by Coxeter groups.
    • There is a ordered list of vertices which exhausts all indecomposables exactly once.
    • Simple rep of acyclic quiver are precisely \(S_v\).
    • Any representations admit a filtration with the whose quotients are representations \(S_v\).
  • The map \(\phi\) is injective. Once surjection is done, count dimension to show injective. Cannot do for infinite but break down and do it part by part to finite dimensional pieces.
    • Poincaré-Birkhoff-Witt basis of any given universal enveloping Lie algebra: basis like polynomial ring. So we have a basis of unordered tuples of positive roots. Using that the quiver is finite type: index basis by weights (to dimension vector). Each finite dimensional piece maps to the correspinding one.

Further

  • Quantum Enevleoping algebra keep \(q\) as a parameter of deformation.
  • 1990 Lusztig canonical basis for enveloping algebra.
  • Infinite dimensional quivers and Kac-Moody algebras generalize (Kirillov’s book).
  • 2013 Bridgeland Derived categories shifted versions to capture universal enveloping algebras.
  • Approach Faculty: Thomas Lam.