The ideals of free differential algebras

[摘要] We consider the free C-algebra b(q) with N generators {xi(i)}(i-1,...,N), together with a set of N differential operators {partial derivative i}(i-1,...,N) that act as twisted derivations on B-q according to the rule partial derivative(i)xi(j) = delta(ij) + q(ij)xi(j)partial derivative(i); that is, For All x is an element of B, partial derivative(i)(xi(j)x) = delta(ij)x + q(ij)xi(i)partial derivative(i)x, and partial derivative(i)C = 0. The subscript q on B-q stands for {q(ij)}(i,j is an element of(1,...,N)) and is interpreted as a point in parameter space, q = (q(ij)) is an element of C-N2. A constant C B-q is a nontrivial clement with the property partial derivative(i)C = 0, i = 1,..., N. To each point in parameter space there corresponds a unique set of constants and a differential complex. There are no constants when the parameters q(ij) are in general position. We obtain some precise results concerning the algebraic surfaces in parameter space on which constants exist. Let J(q) denote the ideal generated by the constants. We relate the quotient algebras B-q' = B-q/J(q) to Yang-Baxter algebras and, in particular, to quantized Kac-Moody algebras. The differential complex is a generalization of that of a quantized Kac-Moody algebra described in terms of Serre generators. Integrability conditions for q-differential equations are related to Hochschild cohomology. It is shown that H-p(B-q',B-q') = 0 for p greater than or equal to 1. The intimate relationship to generalized, quantized Kac-Moody algebras suggests an approach to the problem of classification of these algebras. (C) 2000 Academic Press.

[发布日期] 1999-12-15