Path A_∞algebras of positively graded quivers

Abstract

Let A be a path A_∞-algebra over a positively graded quiver Q. We prove that the derived category of A is triangulated equivalent to the derived category of kQ, which is viewed as a DG algebra with trivial differential. The main technique used in the proof is Koszul duality for DG algebras.

Keywords

A_∞-algebras / Koszul dual / DG-algebras

References

[1] Félix Y, Halperin S, Thomas J C. Rational Homotopy Theory. New York: Springer-Verlag, 2001

[2] He J W, Lu D M. Higher Koszul algebras and A-infinity algebras. J Algebra, 2005, 293(2): 335–362

[3] Kadeishvili T V. The algebraic structure in the homology of an A(∞)-algebra. Soobshch Akad Nauk Gruzin SSR, 1982, 108(2): 249–252

[4] Keller B. Deriving DG categories. Ann Sci Éc Norm Supér (4), 1994, 27(1): 63–102

[5] Keller B. Introduction to A-infinity algebras and modules. Homology, Homotopy Appl, 2001, 3(1): 1–35

[6] Lefevre-Hasegawa K. Sur les A∞-catégories. Thèse de doctorat, Université Paris XII, 2005, http://www.math.jussieu.fr/˜keller/lefevre/publ.html

[7] Loday J L, Vallette B. Algebraic operads. Heidelberg: Springer, 2012

[8] Lu D M, Palmieri J H, Wu Q S, Zhang J J. A∞-algebras for ring theorists. Algebra Colloq, 2004, 11(1): 91–128

[9] Lu D M, Palmieri J H, Wu Q S, Zhang J J. Koszul equivalences in A∞-algebras. New York J Math, 2008, 14: 325–378

[10] Lunts V A. Formality of DG algebras (after Kaledin). J Algebra, 2010, 323(4): 878–898

[11] Stasheff J D. Homotopy associativity of H-spaces. I. Trans Amer Math Soc, 1963, 108: 275–292

[12] Stasheff J D. Homotopy associativity of H-spaces. II. Trans Amer Math Soc, 1963, 108: 293–312

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