MIXED TATE MOTIVES // DRINFELD ASSOCIATORS // PRO-UNIPOTENT GALOIS // MULTIPLE ZETA VALUES
Mixed Tate motives MT(k) over a field k form the simplest non-trivial Tannakian subcategory of mixed motives. They are built from Tate objects Q(n) = H^{2n}(P^n) by extensions. The category is controlled by algebraic K-theory: Ext^i_{MT}(Q(0), Q(n)) ~ K_{2n-i}(k)_Q.
Over Q (or number fields), the Tannakian group of MT(k) is a semi-direct product of G_m (the weight) and a pro-unipotent group U — the "unipotent motivic Galois group." The Lie algebra of U is free on generators in degrees 3, 5, 7, ... related to odd zeta values.
The pro-unipotent completion of a group G is the inverse limit of its nilpotent quotients, viewed as a pro-unipotent algebraic group. For pi_1(P^1 \ {0,1,inf}), this gives a group whose Lie algebra is the completed free Lie algebra on two generators e_0, e_1.
Iterated integrals int_0^1 w_{i1} ... w_{in} compute matrix entries of the pro-unipotent period map. The monodromy around 0 is exp(e_0), around 1 is exp(e_1). The Drinfeld associator Phi_{KZ} is the regularized parallel transport from 0 to 1.
A Drinfeld associator is a formal power series Phi(A, B) in two non-commuting variables satisfying the pentagon and hexagon relations. The Knizhnik-Zamolodchikov associator Phi_{KZ} is the canonical example — its coefficients are multiple zeta values.
Associators form a torsor under the Grothendieck-Teichmuller group GT, which acts on the tower of moduli spaces M_{0,n}. Drinfeld conjectured GT = Gal(Q)^un (the pro-unipotent motivic Galois group). This remains one of the deepest open problems in arithmetic geometry.
Multiple zeta values zeta(s_1,...,s_k) = sum_{n_1>...>n_k>=1} 1/(n_1^{s_1}...n_k^{s_k}) are periods of mixed Tate motives over Z. They satisfy double shuffle relations, and conjecturally all algebraic relations between them come from motivic relations. The dimension of the space of weight-n MZVs grows like d_n ~ C * 2^n/n.
The motivic Galois group acts on motivic MZVs by the "coaction" formula of Goncharov and Brown. This coaction splits MZVs into their "de Rham" and "Betti" parts, revealing hidden structure. Brown proved all periods of MT(Z) are MZVs, using the coaction to establish linear independence modulo products.