CATEGORY EQUIVALENCES INVOLVING GRADED MODULES OVER PATH ALGEBRAS OF QUIVERS

By S. P. Smith. 2011

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Abstract Let Q be a finite quiver with vertex set I and arrow set Q 1 , k a field, and k Q its path algebra with its standard grading. This paper proves some category equivalences involving the quotient category QGr ( k Q ) ≔ Gr ( k Q ) / Fdim ( k Q ) of graded k Q -modules modulo those that are the sum of their finite dimensional submodules, namely QGr ( k Q ) ≡ Mod S ( Q ) ≡ Gr L ( Q ∘ ) ≡ Mod L ( Q ∘ ) 0 ≡ QGr ( k Q ( n ) ) . Here S ( Q ) = lim ⟶ End k I ( k Q 1 ⊗ n ) is a direct limit of finite dimensional semisimple algebras; Q ∘ is the quiver without sources or sinks that is obtained by repeatedly removing all sinks and sources from Q ; L ( Q ∘ ) is the Leavitt path algebra of Q ∘ ; L ( Q ∘ ) 0 is its degree zero component; and Q ( n ) is the quiver whose incidence matrix is the n th power of that for Q . It is also shown that all short exact sequences in qgr ( k Q ) , the full subcategory of finitely presented objects in QGr ( k Q ) , split. Consequently qgr ( k Q ) can be given the structure of a triangulated category with suspension functor the Serre degree twist ( − 1 ) ; this triangulated category is equivalent to the “singularity category” D b ( Λ ) / D perf ( Λ ) where Λ is the radical square zero algebra k Q / k Q ≥ 2 , and D b ( Λ ) is the bounded derived category of finite dimensional left Λ -modules.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c4f98d8cb3124e0a1b24cfedfaf3ee1ccb772805

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Citations in the corpus

Finitely presented simple modules over Leavitt path algebras (2013, background): “It is as well interesting to observe that Chen’s work is related to some constructions in non-commutative algebraic geometry (see [22] and [21]).”
Graded Rings and Graded Grothendieck Groups (2014, methodology): “The motivation to write this note came from the recent activities which adopt the graded Grothendieck group as an invariant to classify the Leavitt path algebras [46, 47, 48, 80].”
Irreducible representations of Leavitt path algebras (2011, background): “This graded kQ-module Mp is known as the point module associated to p; see [24, 25].”