Noncommutative Algebras Associated to Complexes and Graphs
نویسندگان
چکیده
1. Introduction. This is a first of our papers devoted to " noncom-mutative topology and graph theory ". Its origin is the paper [GRW] where a new class of noncommutative algebras Q n was introduced. As explained in [GRW], the algebra Q n is closely related to decompositions of a generic polynomial P (t) of degree n over a division algebra into linear factors. The structure of the algebra Q n seems to be very interesting. It has linearly independent generators u(B), B ⊂ {1,. .. , n}. Here u(∅) = 1 is a unit element of Q n. An important property of Q n is that under any homomorphism of Q n into a commutative integral domain, each element u(B) with |B| ≥ 2 maps to zero. In other words, elements u(B) carry the " noncommutative nature " of Q n. Moreover, the " degree of noncommutativity " carried by u(B) depends on the size of B. The noncommutative nature of Q n can be studied by looking at quotients of Q n by ideals generated by some u(B). These quotients are " more commutative " then Q n. For example, the quotient of Q n by the ideal generated by all u(B) with |B| ≥ 2 is isomorphic to the algebra of commutative polynomials in n variables. To consider more refined cases we need to turn to a " noncommutative combinatorial topology ". In our approach the algebra Q n corresponds to an n-simplex ∆ n and we consider quotients of Q n by ideals generated by some u(B) corresponding to subcomplexes of ∆ n. We describe generators and relations for those quotients. We pay special attention to the quotients of Q n corresponding to 1-dimensional subcomplexes of ∆ n. (They are " next " to algebras of commutative polynomials).
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تاریخ انتشار 2000