Retakh, Vladimir; Wilson, Robert Lee Algebras associated to acyclic directed graphs. (English) Zbl 1177.05127 Adv. Appl. Math. 42, No. 1, 42-59 (2009). Summary: We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices and edges. Each finite acyclic directed graph admits countably many structures of a generalized layered graph. We construct linear bases in such algebras and compute their Hilbert series. Our interest to generalized layered graphs and algebras associated to those graphs is motivated by their relations to factorizations of polynomials over noncommutative rings. Cited in 6 Documents MSC: 05E05 Symmetric functions and generalizations 05C20 Directed graphs (digraphs), tournaments 15A15 Determinants, permanents, traces, other special matrix functions 16T30 Connections of Hopf algebras with combinatorics Keywords:generalized layered graphs; directed graphs; ranking function; Hilbert series; factorizations of noncommutative polynomials PDFBibTeX XMLCite \textit{V. Retakh} and \textit{R. L. Wilson}, Adv. Appl. Math. 42, No. 1, 42--59 (2009; Zbl 1177.05127) Full Text: DOI arXiv References: [1] Anick, D. J., Non-commutative graded algebras and their Hilbert series, J. Algebra, 78, 120-140 (1982) · Zbl 0502.16002 [2] Duffy, C., Representations of \(Aut(A(\Gamma))\) acting on homogeneous components of \(A(\Gamma)\) and \(A(\Gamma)^! (2008)\), Adv. in Appl. Math., in press · Zbl 1173.16014 [3] C. Duffy, Graded traces and irreducible representations of \(\mathit{Aut}(A(\operatorname{\Gamma;}))A(\operatorname{\Gamma;})A ( \operatorname{\Gamma;} )^!\); C. Duffy, Graded traces and irreducible representations of \(\mathit{Aut}(A(\operatorname{\Gamma;}))A(\operatorname{\Gamma;})A ( \operatorname{\Gamma;} )^!\) [4] Gelfand, I.; Gelfand, S.; Retakh, V.; Wilson, R., Factorizations of polynomials over noncommutative algebras and sufficient sets of edges in directed graphs, Lett. Math. Phys., 74, 153-167 (2005) · Zbl 1112.05048 [5] Gelfand, I.; Retakh, V.; Serconek, S.; Wilson, R., On a class of algebras associated to directed graphs, Selecta Math. (N.S.), 11, 281-295 (2005) · Zbl 1080.05040 [6] Gelfand, I.; Retakh, V.; Wilson, R., Quadratic-linear algebras associated with decompositions of noncommutative polynomials and noncommutative differential polynomials, Selecta Math., 7, 493-523 (2001) · Zbl 0992.16025 [7] Golod, E. S., Noncommutative complete intersections and the homology of the Shafarevich complex, Russian Math. Surveys, 52, 4, 830-831 (1997) · Zbl 0926.16009 [8] Retakh, V.; Serconek, S.; Wilson, R., On a class of Koszul algebras associated to directed graphs, J. Algebra, 304, 1114-1129 (2006) · Zbl 1127.16022 [9] Retakh, V.; Serconek, S.; Wilson, R., Hilbert series of algebras associated to directed graphs, J. Algebra, 312, 142-151 (2007) · Zbl 1139.16015 [10] Retakh, V.; Serconek, S.; Wilson, R., Construction of some algebras associated to directed graphs and related to factorizations of noncommutative polynomials, (Proceedings of the Conference “Lie Algebras, Vertex Operator Algebras and Their Applications”. Proceedings of the Conference “Lie Algebras, Vertex Operator Algebras and Their Applications”, Contemp. Math., vol. 412 (2007)), 201-219 · Zbl 1138.05324 [11] Stanley, R. P., Enumerative Combinatorics, vol. 1 (1999), Cambridge University Press · Zbl 1247.05003 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.