From factorizations of noncommutative polynomials to combinatorial topology

• Autorzy: Vladimir Retakh
• Języki publikacji: EN

Abstrakty

EN

This is an extended version of a talk given by the author at the conference “Algebra and Topology in Interaction” on the occasion of the 70th Anniversary of D.B. Fuchs at UC Davis in September 2009. It is a brief survey of an area originated around 1995 by I. Gelfand and the speaker.

Słowa kluczowe

EN

Noncommutative polynomials; Directed graphs; Koszul algebras; Hilbert series; Combinatorial topology

Kategorie tematyczne

• 57N99: None of the above, but in this section
• 16S37: Quadratic and Koszul algebras

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2010
• Tom: 8
• Numer: 2
• Strony: 235-243

Daty

wydano

2010-04-01

online

2010-04-14

Twórcy

autor

Vladimir Retakh

• Rutgers University

Bibliografia

• [1] Björner A., Topological Methods, In: Handbook of Combinatorics, MIT Press, Cambridge, MA, 1995, 1819–1872
• [2] Cassidy T., Phan C., Shelton B., Noncommutative Koszul algebras from combinatorial topology, J. Reine Angew. Math., to appear
• [3] Gelfand I., Gelfand S., Retakh V., Serconek S., Wilson R., Hilbert series of quadratic algebras associated with decompositions of noncommutative polynomials, J. Algebra, 2002, 254, 279–299 http://dx.doi.org/10.1016/S0021-8693(02)00081-9
• [4] Gelfand I., Gelfand S., Retakh V., Wilson R., Quasideterminants, Advances in Math., 2005, 193, 56–141 http://dx.doi.org/10.1016/j.aim.2004.03.018
• [5] Gelfand I., Gelfand S., Retakh V., Wilson R., Factorizations of polynomials over noncommutative algebras and sufficient sets of edges in directed graphs, Lett. Math. Physics, 2005, 74, 153–167 http://dx.doi.org/10.1007/s11005-005-0024-8
• [6] Gelfand I., Krob D., Lascoux A., Leclerc B., Retakh V., Thibon J.-Y., Noncommutative symmetric functions, Advances in Math., 1995, 112, 218–348 http://dx.doi.org/10.1006/aima.1995.1032
• [7] Gelfand I., Retakh V., Determinants of matrices over moncommutative rings, Funct. Anal. Appl., 1991, 25, 91–102 http://dx.doi.org/10.1007/BF01079588
• [8] Gelfand I., Retakh V., A theory of noncommutative determinants and characteristic functions of graphs, Funct. Anal. Appl., 1992, 26(4), 1–20
• [9] Gelfand I., Retakh V., A theory of noncommutative determinants and characteristic functions of graphs. I, Publ. LACIM, UQAM, Montreal, 1993, 1–26
• [10] Gelfand I., Retakh V., Noncommutative Vieta theorem and symmetric functions, In: Gelfand Mathematical Seminars 1993–95, Birkhäuser, Boston, 1996, 93–100
• [11] Gelfand I., Retakh V., Quasideterminants, I, Selecta Math., 1997, 3, 517–546 http://dx.doi.org/10.1007/s000290050019
• [12] Gelfand I., Retakh V., Serconek S., Wilson R., On a class of algebras associated to directed graphs, Selecta Math., 2005, 11, 281–295 http://dx.doi.org/10.1007/s00029-005-0005-x
• [13] Gelfand I., Retakh V., Wilson R., Quadratic-linear algebras associated with decompositions of noncommutative polynomials and differential polynomials, Selecta Math., 2001, 7, 493–523 http://dx.doi.org/10.1007/s00029-001-8096-5
• [14] Osofsky B., Quasideterminants and right roots of polynomials over division rings, In: Algebras, Rings and Their Representations, World Sci. Publ., Singapore, 2006, 241–263 http://dx.doi.org/10.1142/9789812774552_0015
• [15] Osofsky B., Noncommutative linear algebra, Contemp. Math., 2006, 419, 231–254
• [16] Piontkovski D., Algebras associated to pseudo-roots of noncommutative polynomials are Koszul, Intern. J. Algebra Comput., 2005, 15, 643–648 http://dx.doi.org/10.1142/S0218196705002396
• [17] Polischuk A., Positselski L., Quadratic Algebras, Amer. Math. Soc., Providence, RI, 2005
• [18] Retakh V., Serconek S., Wilson R., On a class of Koszul algebras associated to directed graphs, J. Algebra, 2006, 304, 1114–1129 http://dx.doi.org/10.1016/j.jalgebra.2005.11.005
• [19] Retakh V., Serconek S., Wilson R., Hilbert series of algebras associated to directed graphs, J. of Algebra, 2007, 312, 142–151 http://dx.doi.org/10.1016/j.jalgebra.2006.06.048
• [20] Retakh V., Serconek S., Wilson R., Construction of some algebras associated to directed graphs and related to factorizations of noncommutative polynomials, Contemp. Math., 442, Amer. Math. Soc., Providence, RI, 2007
• [21] Retakh V., Serconek S., Wilson R., Koszulity of splitting algebras associated with cell complexes, J. Algebra, 2010, 323, 983–999 http://dx.doi.org/10.1016/j.jalgebra.2009.11.039
• [22] Retakh V., Serconek S., Wilson R., Hilbert series of some algebras associated with directed graphs and cohomology (in preparation)
• [23] Retakh V., Wilson R., Advanced course on quasideterminants and universal localization, CRM, Barcelona, 2007
• [24] Retakh V., Wilson R., Algebras associated to directed acyclic graphs, Adv. Appl. Math., 2009, 42, 42–59 http://dx.doi.org/10.1016/j.aam.2008.04.002
• [25] Sadofsky H., Shelton B., The Koszul property as a topological invariant and measure of singularities, preprint available at arXiv:0911.2541
• [26] Serconek S., Wilson R., Quadratic algebras associated with decompositions of noncommutative polynomials are Koszul algebras, J. Algebra, 2004, 278, 473–493 http://dx.doi.org/10.1016/S0021-8693(03)00340-5

Identyfikatory

DOI

10.2478/s11533-010-0008-5