Construction of Some Algebras Associated to Directed Graphs and Related to Factorizations of Noncommutative Polynomials

This is a survey of recently published results. We introduce and study a wide class algebras associated to directed graphs and related to factorizations of noncommutative polynomials. In particular, we show that for many well-known graphs such algebras are Koszul and compute their Hilbert series. Let R be an associative ring with unit and P (t) = a0t +a1t +· · ·+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, GLR, GR1, GR2, GRW, GGRW, LL, O, B, V, W]). In this paper we present an approach relating such factorizations to algebras associated with directed graphs and study properties of such algebras. In the factorization (0.1) the element y1 is called a right root of P (t) and element yn is called a left root of P (t). This terminology can be justified by the following equalities (see, for example, [L]): a0y n 1 + a1y n−1 1 + · · ·+ an−1y1 + an = 0, 1991 Mathematics Subject Classification. 05E05; 15A15; 16W30.