Derived noncommutative algebraic geometry - Quivers help encode the data of derived noncommutative schemes.The indecomposable representations are in a one-to-one correspondence with the positive roots of the root system of the Dynkin diagram.ĭlab & Ringel (1973) found a generalization of Gabriel's theorem in which all Dynkin diagrams of finite dimensional semisimple Lie algebras occur.A (connected) quiver is of finite type if and only if its underlying graph (when the directions of the arrows are ignored) is one of the ADE Dynkin diagrams: A n, D n, E 6, E 7, E 8.More precisely, Gabriel's theorem states that: Gabriel (1972) classified all quivers of finite type, and also their indecomposable representations. These give quiver varieties, as constructed by King (1994).Ī quiver is of finite type if it has only finitely many isomorphism classes of indecomposable representations. Given the dimensions of the vector spaces assigned to every vertex, one can form a variety which characterizes all representations of that quiver with those specified dimensions, and consider stability conditions. The quotient K Γ / I is the path algebra of ( Q, I).
FIELD QUIVER FREE
Its left adjoint is a free functor which, from a quiver, makes the corresponding free category. That is, there is a forgetful functor from Cat to Quiv. In category theory, a quiver can be understood to be the underlying structure of a category, but without composition or a designation of identity morphisms. They are commonly used in representation theory: a representation V of a quiver assigns a vector space V( x) to each vertex x of the quiver and a linear map V( a) to each arrow a. In graph theory, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, i.e.
Directed graph which is also a multigraph
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