Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-09T01:36:22.538Z Has data issue: false hasContentIssue false
  • English
  • Français

Hopfological algebra

Part of: Commutative algebra: Homological methods Hopf algebras, quantum groups and related topics

Published online by Cambridge University Press:  07 October 2013

You Qi
You Qi*
Affiliation:
Department of Mathematics, Columbia University, Room 509, MC 4406 2990 Broadway, NY 10027, USA email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We develop some basic homological theory of hopfological algebra as defined by Khovanov [Hopfological algebra and categorification at a root of unity: the first steps, Preprint (2006), arXiv:math/0509083v2]. Several properties in hopfological algebra analogous to those of usual homological theory of DG algebras are obtained.

Keywords

Hopfological algebrap-DG algebracategorificationroots of unity

MSC classification

Secondary: 13D09: Derived categories 13D15: Grothendieck groups, $K$-theory 16T05: Hopf algebras and their applications
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 1 , January 2014 , pp. 1 - 45
Copyright
© The Author(s) 2013 

References

Angel, M. and Díaz, R., On $N$-differential graded algebras, J. Pure Appl. Algebra 210 (2007), 673–683.CrossRefGoogle Scholar
Benson, D.  J., Representations and cohomology I, Cambridge Studies in Advanced Mathematics, vol. 30 (Cambridge University Press, Cambridge, 1991).Google Scholar
Bichon, J., N-complexes et algébres de Hopf, C. R. Math. Acad. Sci. Paris. 337 (2003), 441–444.Google Scholar
Bernstein, J. and Lunts, V., Equivariant sheaves and functors, Lecture Notes in Mathematics, vol. 1578 (Springer, Berlin, 1994).CrossRefGoogle Scholar
Cibils, C., Solotar, A. and Wisbauer, R., $N$-complexes as functors, amplitude cohomology and fusion rules, Comm. Math. Phys. 272 (2007), 837–849.Google Scholar
Crane, L. and Frenkel, I., Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases, J. Math. Phys. 35 (1994), 5136–5154.CrossRefGoogle Scholar
Dubois-Violette, M., Generalized differential spaces with ${d}^{N} = 0$ and the $q$-differential calculus, Czech. J. Phys. 46 (1996), 1227–1233.Google Scholar
Gelfand, S. I. and Manin, Y. I., Methods of homological algebra, Springer Monographs in Mathematics, second edition (Springer, Berlin, 2003).CrossRefGoogle Scholar
Happel, D., Triangulated categories in the representation theory of finite dimensioal algebras, London Mathematical Society Lecture Note Series, vol. 119 (Cambridge University Press, Cambridge, 1988).Google Scholar
Kaledin, D., Lecture notes for the mini-course ‘Non-commutative geometry from the homological point of views’, Seoul, October 2009, Preprint (2009), http://imperium.lenin.ru/~kaledin/seoul/.Google Scholar
Kapranov, M., On the $q$-analog of homological algebra, Preprint (1996), arXiv:q-alg/9611005v1.Google Scholar
Kassel, C. and Wambst, M., Algèbre homologique des N-complexes et homologie de Hochschild aux racine de l’unité, Publ. Res. Inst. Math. Sci. 34 (1998), 91–114.Google Scholar
Keller, B., Deriving DG categories, Ann. Sci. Éc. Norm. Supér. (4) 27 (1994), 63–102.Google Scholar
Keller, B., On differential graded categories, in International Congress of Mathematicians, Madrid, 22–30 August 2006, Vol. II (European Mathematical Society, Zürich, 2007), 151–190.Google Scholar
Khovanov, M., Hopfological algebra and categorification at a root of unity: the first steps, Preprint (2006), arXiv:math/0509083v2.Google Scholar
Khovanov, M. and Lauda, A., A diagrammatic approach to categorification of quantum groups I, Represent. Theory 13 (2009), 309–347.Google Scholar
Kuperberg, G., Non-involutory Hopf algebras and 3-manifold invariants, Duke Math. J. 84 (1996), 83–129.CrossRefGoogle Scholar
Lam, T. Y., Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189 (Springer, New York, 1999).Google Scholar
Loday, J. L., Cyclic homology, Grundlehren der mathematischen Wissenschaften, vol 301, second edition (Springer, Berlin, 1998).CrossRefGoogle Scholar
Majid, S., Some comments on bosonisation and biproducts, Czech. J. Phys. 47 (1997), 151–171.CrossRefGoogle Scholar
Mayer, W., A new homology theory I, Ann. of Math. (2) 43 (1942), 370–380.Google Scholar
Mayer, W., A new homology theory II, Ann. of Math. (2) 43 (1942), 594–605.CrossRefGoogle Scholar
Montgomery, S., Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82 (American Mathematical Society, Providence, RI, 1993).Google Scholar
Neeman, A., The connection between the $K$-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. Éc. Norm. Supér. (4) 25 (1992), 547–566.Google Scholar
Neeman, A., The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), 205–236.Google Scholar
Pareigis, B., A non-commutative non-cocommutative Hopf algebra in “nature”, J. Algebra 70 (1981), 356–374.Google Scholar
Ravenel, D.  C., Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984), 351–414.CrossRefGoogle Scholar
Sarkaria, S., Some simplicial (co)homologies, Institut des Hautes Études Scientifiques, Preprint (1995), http://kssarkaria.org/.Google Scholar
Schlichting, M., Negative $K$-theory of derived categories, Math. Z. 253 (2006), 97–134.Google Scholar
Sitarz, A., On the tensor product construction for $q$-differential algebras, Lett. Math. Phys. 44 (1998), 17–21.Google Scholar
Spanier, E.  H., The Mayer homology theory, Bull. Amer. Math. Soc. 55 (1949), 102–112.Google Scholar
Thomason, R.  W. and Trobaugh, T., Higher algebraic K-theory of schemes and of derived categories, in The Grothendieck Festschrift: a collection of articles written in honor of the 60th birthday of Alexander Grothendieck, vol. III, Progress in Mathematics, vol. 88, eds Cartier, P. et al. (Birkhäuser, Boston, MA, 2007), 247–435.Google Scholar
Toën, B., The homotopy theory of dg-categories and derived Morita theory, Invent. Math. 167 (2007), 615–667.Google Scholar