, The elements in {z 0 , z 8 , z 12 } ? {? i ; i ? Q 0 } form a basis of HK 0 (A)

, If char(F) = 2, the elements in ? ? ; ? = 0, 8, 12 ? {? 3 , ? 7 , ? 15 } form a basis of HK 1 (A). If char(F) = 3, the elements in ? ?, vol.8

, If char(F) = 2, the elements in h j ; j ? Q 0 ? {? 4 , ? 8 , ? 16 } form a basis of HK 2 (A). If char(F) = 3, the elements in h j

D. Baer, W. Geigle, and H. Lenzing, The preprojective algebra of a tame hereditary Artin algebra, Comm. Algebra, vol.15, pp.425-457, 1987.

A. Beilinson, V. Ginzburg, and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc, vol.9, pp.473-527, 1996.

J. Bell and J. J. Zhang, An isomorphism lemma for graded rings, Proc. Amer. Math. Soc, vol.145, pp.989-994, 2017.

D. J. Benson, Representations and cohomology I, 1991.

R. Berger, Dimension de Hochschild des algèbres graduées (French) [Hochschild dimension for graded algebras, C. R. Math. Acad. Sci. Paris, vol.341, pp.597-600, 2005.

R. Berger, Koszul calculus for N-homogeneous algebras, J. Algebra, vol.519, pp.149-189, 2019.
URL : https://hal.archives-ouvertes.fr/hal-01918544

R. Berger, T. Lambre, and A. Solotar, Koszul calculus, Glasg. Math. J, vol.60, pp.361-399, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01687814

J. Bialkowski, K. Erdmann, and A. Skowro?ski, Deformed preprojective algebras of generalized Dynkin type Ln: classification and symmetricity, J. Algebra, vol.345, pp.150-170, 2011.

R. Bocklandt, Graded Calabi Yau algebras of dimension 3, J. Pure Appl. Algebra, vol.212, pp.14-32, 2008.

S. Brenner, M. C. Butler, and A. King, Periodic algebras which are almost Koszul, Algebr. Represent. Theory, vol.5, pp.331-367, 2002.

K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics, vol.87, 1982.

C. Cibils, 18-42. 13. W. Crawley-Boevey, DMV lectures on Representations of quivers, preprojective algebras and deformations of quotient singularities, Adv. Math, vol.79, 1990.

W. Crawley-boevey, P. Etingof, and V. Ginzburg, Noncommutative geometry and quiver algebras, Adv. Math, vol.209, pp.274-336, 2007.

W. Crawley-boevey and M. P. Holland, Noncommutative deformations of Kleinian singularities, Duke Math. Journal, vol.92, pp.605-635, 1998.

W. Crawley-boevey, M. Van-den, and . Bergh, Absolutely indecomposable representations and Kac-Moody Lie algebras. With an appendix by Hiraku Nakajima, Invent. Math, vol.155, pp.537-559, 2004.

V. Dlab and C. M. Ringel, The preprojective algebra of a modulated graph, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ, vol.832, pp.216-231, 1979.

K. Erdmann and N. Snashall, Hochschild cohomology of preprojective algebras, I, J. Algebra, vol.205, pp.391-412, 1998.

K. Erdmann and N. Snashall, Hochschild cohomology of preprojective algebras, J. Algebra, vol.II, pp.413-434, 1998.

K. Erdmann and N. Snashall, Preprojective algebras of Dynkin type, periodicity and the second Hochschild cohomology, CMS Conf. Proc, vol.24, 1998.

P. Etingof and C. Eu, Hochschild and cyclic homology of preprojective algebras of ADE quivers, Mosc. Math. J, vol.7, pp.601-612, 2007.

C. Eu, The product in the Hochschild cohomology ring of preprojective algebras of Dynkin quivers, J. Algebra, vol.320, pp.1477-1530, 2008.

C. Eu, The calculus structure of the Hochschild homology/cohomology of preprojective algebras of Dynkin quivers, J. Pure Appl. Algebra, vol.214, pp.28-46, 2010.

C. Eu and T. Schedler, Calabi-Yau Frobenius algebras, J. Algebra, vol.321, pp.774-815, 2009.

J. Gaddis, Isomorphisms of graded path algebras

C. Geiss, B. Leclerc, and J. Schröer, Preprojective algebras and cluster algebras, Trends in representation theory of algebras and related topics, pp.253-283, 2008.

I. M. Gelfand and V. A. Ponomarev, Model algebras and representations of graphs (Russian), vol.13, pp.1-12, 1979.

V. Ginzburg, Calabi-Yau algebras

J. Grant and O. Iyama, Higher preprojective algebras, Koszul algebras, and superpotentials

E. L. Green, Introduction to Koszul algebras, Representation theory and algebraic geometry, vol.238, pp.45-62, 1995.

A. Hatcher, Algebraic topology, 2002.

T. Lambre, Dualité de Van den Bergh et structure de Batalin-Vilkoviski? sur les algèbres de Calabi-Yau, J. Noncommut. Geom, vol.4, pp.441-457, 2010.

R. Martínez-villa, Applications to Koszul algebras: the preprojective algebra, vol.18, pp.487-504, 1994.

R. Martínez-villa, Introduction to Koszul Algebras, Revista Union Matematica Argentina, vol.48, pp.67-95, 2007.

A. Polishchuk and L. Positselski, Quadratic algebras, University Lecture Series, vol.37, 2005.

S. Priddy, Koszul resolutions, Trans. Amer. Math. Soc, vol.152, pp.39-60, 1970.

M. L. Reyes, D. Rogalski-a-twisted, and C. Toolkit,

P. H. Shaw, Generalisations of preprojective algebras, 2005.

D. Tamarkin and B. Tsygan, The ring of differential operators on forms in noncommutative calculus, Proc. Sympos. Pure Math, vol.73, pp.105-131, 2005.

M. Van-den and . Bergh, Noncommutative homology of some three-dimensional quantum spaces,K-Theory, vol.8, pp.213-230, 1994.

M. Van-den and . Bergh, A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Amer. Math. Soc, vol.126, pp.2809-2810, 1998.

M. Van-den and . Bergh, Double Poisson algebras, Trans. Amer. Math. Soc, vol.360, pp.5711-5769, 2008.

M. Van-den and . Bergh, Calabi-Yau algebras and superpotentials, Selecta Math. (N.S.), vol.21, pp.555-603, 2015.

C. A. Weibel, An introduction to homological algebra, 1994.

A. Yekutieli, Derived Categories

A. Zimmermann, On the use of Külshammer type invariants in representation theory, Bull. Iran. Math. Soc, vol.37, pp.291-341, 2011.