Higher Koszul algebras and \(A\)-infinity algebras. (English) Zbl 1143.16027
Summary: We study a class of \(A_\infty\)-algebras, named \((2,p)\)-algebras, which is related to the class of \(p\)-homogeneous algebras, especially to the class of \(p\)-Koszul algebras. A general method to construct \((2,p)\)-algebras is given. Koszul dual of a connected graded algebra is defined in terms of \(A_\infty\)-algebra. It is proved that a \(p\)-homogeneous algebra \(A\) is \(p\)-Koszul if and only if the Koszul dual \(E(A)\) is a reduced \((2,p)\)-algebra and generated by \(E^1(A)\). The \((2,p)\)-algebra structure of the Koszul dual \(E(A)\) of a \(p\)-Koszul algebra \(A\) is described explicitly. A necessary and sufficient condition for a \(p\)-homogeneous algebra to be a \(p\)-Koszul algebra is also given when the higher multiplications on the Koszul dual are ignored.
MSC:
| 16S37 | Quadratic and Koszul algebras |
| 16E65 | Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) |
| 16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
| 16W50 | Graded rings and modules (associative rings and algebras) |
Keywords:
\(A\)-infinity algebras; Koszul algebras; connected graded algebras; Ext algebras; tensor algebras; Koszul dualsReferences:
| [1] | Artin, M.; Schelter, W. F., Graded algebras of global dimension 3, Adv. Math., 66, 171-216 (1987) · Zbl 0633.16001 |
| [2] | Beilinson, A. A.; Ginzburg, V.; Soergel, W., Koszul duality patterns in representation theory, J. Amer. Math. Soc., 9, 473-527 (1996) · Zbl 0864.17006 |
| [3] | Berger, R., Koszulity for nonquadratic algebras, J. Algebra, 239, 705-734 (2001) · Zbl 1035.16023 |
| [4] | Berger, R., Koszulity for nonquadratic algebras II · Zbl 1035.16023 |
| [5] | Berger, R.; Dubois-Violette, M.; Wambst, M., Homogeneous algebras, J. Algebra, 261, 172-185 (2003) · Zbl 1061.16034 |
| [6] | Berger, R.; Marconnet, N., Koszul and Gorenstein properties for homogeneous algebras · Zbl 1125.16017 |
| [7] | Green, E. L.; Martinez-Villa, R., Koszul and Yoneda algebras I, (Repr. Theory of Algebras. Repr. Theory of Algebras, CMS Conf. Proc., vol. 18 (1996), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 247-306 · Zbl 0860.16009 |
| [8] | Green, E. L.; Martinez-Villa, R. M., Koszul and Yoneda algebras II, (CMS Conf. Proc., vol. 24 (1998), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 227-244 · Zbl 0936.16012 |
| [9] | B. Keller, \(A\); B. Keller, \(A\) |
| [10] | Keller, B., Introduction to \(A\)-infinity algebras and modules, Homology Homotopy Appl., 3, 1-35 (2001), (electronic) · Zbl 0989.18009 |
| [11] | Löfwall, C., On the subalgebra generated by the one-dimensional elements in the Yoneda Ext-algebra, (Algebra, Algebraic Topology and Their Interactions. Algebra, Algebraic Topology and Their Interactions, Stockholm, 1983. Algebra, Algebraic Topology and Their Interactions. Algebra, Algebraic Topology and Their Interactions, Stockholm, 1983, Lecture Notes in Math., vol. 1183 (1986), Springer-Verlag: Springer-Verlag Berlin), 291-338 · Zbl 0595.16020 |
| [12] | D.-M. Lu, J.H. Palmieri, Q.-S. Wu, J.J. Zhang, Regular algebras of dimension 4 and their \(A_{\infty;}\); D.-M. Lu, J.H. Palmieri, Q.-S. Wu, J.J. Zhang, Regular algebras of dimension 4 and their \(A_{\infty;}\) · Zbl 1193.16014 |
| [13] | Lu, D.-M.; Palmieri, J. H.; Wu, Q.-S.; Zhang, J. J., \(A_\infty \)-algebras for ring theorists, Algebra Colloq., 11, 1, 91-128 (2004) · Zbl 1066.16049 |
| [14] | MacLane, S., Homology (1963), Springer-Verlag: Springer-Verlag Berlin · Zbl 0133.26502 |
| [15] | Manin, Y. I., Quantum Groups and Non-Commutative Geometry (1988), Université de Montréal, Montreal, QC · Zbl 0724.17006 |
| [16] | Martinez-Villa, R., Koszul algebras and the Gorenstein condition, (Lecture Notes in Pure and Appl. Math., vol. 224 (1999), Dekker: Dekker New York), 135-156 · Zbl 1056.16011 |
| [17] | Merkulov, S. A., Strong homotopy algebras of a Kähler manifold, Int. Math. Res. Not., 3, 153-164 (1999) · Zbl 0995.32013 |
| [18] | Priddy, S., Koszul resolutions, Trans. Amer. Math. Soc., 152, 39-60 (1970) · Zbl 0261.18016 |
| [19] | Smith, S. P., Some finite-dimensional algebras related to elliptic curves, (CMS Conf. Proc., vol. 19 (1996), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 315-348 · Zbl 0856.16009 |
| [20] | Van den Bergh, M., Non-commutative homology of some three-dimensional quantum spaces, K-Theory, 8, 213-220 (1994) · Zbl 0814.16006 |
| [21] | Ye, Y.; Zhang, P., Higher Koszul complexes, Sci. China Ser. A, 46, 118-128 (2003) · Zbl 1217.16023 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
