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Hilbert series of algebras associated to directed graphs
 J. of Algebra
, 2007
"... Abstract. We compute the Hilbert series of some algebras associated to directed graphs introduced recently by I. Gelfand and the authors. 1. ..."
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Abstract. We compute the Hilbert series of some algebras associated to directed graphs introduced recently by I. Gelfand and the authors. 1.
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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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,
NONCOMMUTATIVE KOSZUL ALGEBRAS FROM COMBINATORIAL TOPOLOGY
, 811
"... Abstract. Associated to any uniform finite layered graph Γ there is a noncommutative graded quadratic algebra A(Γ) given by a construction due to Gelfand, Retakh, Serconek and Wilson. It is natural to ask when these algebras are Koszul. Unfortunately, a mistake in the literature states that all such ..."
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Abstract. Associated to any uniform finite layered graph Γ there is a noncommutative graded quadratic algebra A(Γ) given by a construction due to Gelfand, Retakh, Serconek and Wilson. It is natural to ask when these algebras are Koszul. Unfortunately, a mistake in the literature states that all such algebras are Koszul. That is not the case and the theorem was recently retracted. We analyze the Koszul property of these algebras for two large classes of graphs associated to finite regular CW complexes, X. Our methods are primarily topological. We solve the Koszul problem by introducing new cohomology groups HX(n, k), generalizing the usual cohomology groups H n (X). Along with several other results, our methods give a new and primarily topological proof of the main result of [12] and [7]. 1.
Graded traces and irreducible representations of Aut(A(Γ)) acting on grA(Γ) and grA(Γ
, 2008
"... Graded traces and irreducible representations of Aut(A(Γ)) acting on graded A(Γ) and A(Γ)! ..."
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Graded traces and irreducible representations of Aut(A(Γ)) acting on graded A(Γ) and A(Γ)!
Representations of Aut(A(Γ)) acting on homogeneous components of
 A(Γ) and A(Γ) ! , arXiv: 0804.4728
, 2008
"... Abstract. In this paper we will study the structure of algebras A(Γ) associated to two directed, layered graphs Γ. These are algebras associated with Hasse graphs of ngons and the algebras Qn related to pseudoroots of noncommutative polynomials. We will find the filtration preserving automorphism g ..."
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Abstract. In this paper we will study the structure of algebras A(Γ) associated to two directed, layered graphs Γ. These are algebras associated with Hasse graphs of ngons and the algebras Qn related to pseudoroots of noncommutative polynomials. We will find the filtration preserving automorphism group of these algebras and then we will find the multiplicities of the irreducible representations of Aut(A(Γ)) acting on the homogeneous components of A(Γ) and A(Γ) !.
From factorizations of noncommutative polynomials to combinatorial topology
 Cent. Eur. J. Math
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ALGEBRAS ASSOCIATED TO DIRECTED ACYCLIC GRAPHS
, 707
"... Abstract. 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 directed acyclic graph admits a structure of a generalized layered graph. We construct linear bases in such algebras an ..."
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Abstract. 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 directed acyclic graph admits a structure 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. In this paper we construct and study a class of algebras A(Γ) associated to generalized layered graphs Γ, i.e. directed graphs with a ranking function .  on their vertices. Therefore, each edge has a length l; if an edge e goes from a vertex v to a vertex w then l(e) = v  − w. Each
KOSZULITY OF SPLITTING ALGEBRAS ASSOCIATED WITH CELL COMPLEXES
, 810
"... Abstract. We associate to a good cell decomposition of a manifold M a quadratic algebra and show that the Koszulity of the algebra implies a restriction on the Euler characteristic of M. For a twodimensional manifold M the algebra is Koszul if and only if the Euler characteristic of M is two. Let Γ ..."
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Abstract. We associate to a good cell decomposition of a manifold M a quadratic algebra and show that the Koszulity of the algebra implies a restriction on the Euler characteristic of M. For a twodimensional manifold M the algebra is Koszul if and only if the Euler characteristic of M is two. Let Γ = (V, E) be a layered graph (where V is the set of vertices and E is the set of edges). One may define an assoicated algebra, A(Γ), to be the algebra generated by E subject to the relations which state that (t − e1)(t − e2)...(t − ek) = (t − f1)(t − f2)...(t − fk)
ALGEBRAS ASSOCIATED TO ACYCLIC DIRECTED GRAPHS
, 707
"... Abstract. 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 su ..."
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Abstract. 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. By a generalized layered graph we mean a pair Γ = (G, .) where G = (V, E) is a directed graph and .  : V → Z≥0 satisfies v > w whenever v, w ∈ V and there is an edge e ∈ E from v to w. We call .  the rank function of Γ. We write l(e) = v  − w  and call this the
Splitting Algebras II: The Cohomology Algebra. ArXiv eprints
, 2012
"... Abstract. Gelfand, Retakh, Serconek and Wilson, in [3], defined a graded algebra AΓ attached to any finite ranked poset Γ a generalization of the universal algebra of pseudoroots of noncommutative polynomials. This algebra has since come to be known as the splitting algebra of Γ. The splitting al ..."
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Abstract. Gelfand, Retakh, Serconek and Wilson, in [3], defined a graded algebra AΓ attached to any finite ranked poset Γ a generalization of the universal algebra of pseudoroots of noncommutative polynomials. This algebra has since come to be known as the splitting algebra of Γ. The splitting algebra has a secondary filtration related to the rank function on the poset and the associated graded algebra is denoted here by A′Γ. We calculate the cohomology algebra (and coalgebra) of A′Γ explicitly. As a corollary to this calculation we have a proof that A′Γ is Koszul (respectively quadratic) if and only if Γ is CohenMacaulay (respectively uniform). We show by example that the cohomology algebra (resp. coalgebra) of AΓ may be strictly smaller that the cohomology algebra (resp. coalgebra) of A′Γ.