Algebraic cycles and the Lie algebra of mixed Tate motives

Abstract

The purpose of this note is to suggest a candidate for the category 5' X5T4 of mixed Tate motives (0& Q) over a field F. We will define a graded pro-Lie algebra ' = I D 2 E * * * defined over Q and depending on F. X%>5~# will be the category of f.d. graded Q-representations of 2. To avoid having to work with pro-objects, we consider the co-Lie algebra X' = Fv (continuous dual). X( is graded in positive degrees and is generated by suitable algebraic cycles. To link X( to the category of mixed Tate motives, we will verify the following properties: (i) Xl = F X 0 Q, where F x is the multiplicative group of F. (ii) X D 0iog, where 0Og is the co-Lie algebra associated with the polylogarithm functions by Deligne [3]. In particular, the polylogarithm motives defined by Beilinson exist in our theory. (iii) Suppose F = C(V), the field of functions on a variety V. Let 'Op be the complex 1-minimal model [6, 7] for the de Rham complex of F. We define a homomorphism of co-Lie algebras