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On a class of Koszul algebras associated to directed graphs
 MR MR2265508 (2007f:16067
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Construction of some algebras associated to directed graphs and related to factorizations of noncommutative polynomials
 Contemporary Math
, 2006
"... Abstract. This is a survey of recently published results. We introduce and study a wide class of algebras associated to directed graphs and related to factorizations of noncommutative polynomials. In particular, we show that for many wellknown graphs such algebras are Koszul and compute their Hilbe ..."
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Cited by 10 (5 self)
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Abstract. This is a survey of recently published results. We introduce and study a wide class of algebras associated to directed graphs and related to factorizations of noncommutative polynomials. In particular, we show that for many wellknown graphs such algebras are Koszul and compute their Hilbert series. Let R be an associative ring with unit and P(t) = a0t n +a1t n−1 + · · ·+an be a polynomial over R. Here t is an independent central variable. We consider factorizations of P(t) into a product (0.1) P(t) = a0(t − yn)(t − yn−1)...(t − y1) if such factorizations exist. When R is a (commutative) field, there is at most one such factorization up to a permutation of factors. When R is not commutative, the polynomial P(t) may have several essentially different factorizations. The set of factorizations of a polynomial over a noncommutative ring can be rather complicated and studying them is a challenging and useful problem (see, for example, [N,
1 Koszul Algebras
, 2005
"... The algebras Qn describe the relationship between the roots and coefficients of a noncommutative polynomial. I.Gelfand, S.Gelfand, and V. Retakh have defined quotients of these algebras corresponding to graphs. In this work we find the Hilbert series of the class of algebras corresponding to the n ..."
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The algebras Qn describe the relationship between the roots and coefficients of a noncommutative polynomial. I.Gelfand, S.Gelfand, and V. Retakh have defined quotients of these algebras corresponding to graphs. In this work we find the Hilbert series of the class of algebras corresponding to the nvertex path, Pn. We also show this algebra is Koszul. We do this by first looking at class of quadratic algebras we call Partially Generator Commuting. We then find a sufficient condition for a PGCAlgebra to be Koszul and use this to show a similar class of PGC algebras, which we call chPn, is Koszul. Then we show it is possible to extend what we did to the algebras Pn although they are not PGC. Finally we examine the Hilbert Series of the algebras Pn
1 Koszul Algebras
"... describe the relationship between the roots and coefficients of a noncommutative polynomial. I.Gelfand, S.Gelfand, and V. Retakh have defined quotients of these algebras corresponding to graphs. In this work we find the Hilbert series of the class of algebras corresponding to the graph K3. We also ..."
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describe the relationship between the roots and coefficients of a noncommutative polynomial. I.Gelfand, S.Gelfand, and V. Retakh have defined quotients of these algebras corresponding to graphs. In this work we find the Hilbert series of the class of algebras corresponding to the graph K3. We also show this algebra is Koszul.
Koszul Algebras Associated to Graphs
"... Quadratic algebras associated to graphs have been introduced by I. Gelfand, S. Gelfand, and Retakh in connection with decompositions of noncommutative polynomials. Here we show that for each graph with rare triangular subgraphs, the corresponding quadratic algebra is a Koszul domain with global dime ..."
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Quadratic algebras associated to graphs have been introduced by I. Gelfand, S. Gelfand, and Retakh in connection with decompositions of noncommutative polynomials. Here we show that for each graph with rare triangular subgraphs, the corresponding quadratic algebra is a Koszul domain with global dimension equal to the number of vertices of the graph. 1