Constructing faithful representations of finitely-generated torsion-free nilpotent groups

We formulate an algorithm for calculating a representation by unipotent matrices over the integers of a nitelygenerated torsionfree nilpotent group given by a polycyclic presentation The algorithm works along a polycyclic series of the group each step extending a representation of an element of that series to the next elemen

The Semisimplicity Conjecture for A-Motives

We prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V_p(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Akio Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G_p(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G_p(M) may be identified naturally with the Zariski closure of the image of the absolute Galois group of K in the automorphism group of V_p(M). The second states that the connected component of G_p(M) is reductive if M is semisimple and has a separable endomorphism algebra.Comment: 47 page

Universal representations of braid and braid-permutation groups

Drinfel'd used associators to construct families of universal representations of braid groups. We consider semi-associators (i.e., we drop the pentagonal axiom and impose a normalization in degree one). We show that the process may be reversed, to obtain semi-associators from universal representations of 3-braids. We view braid groups as subgroups of braid-permutation groups. We construct a family of universal representations of braid-permutation groups, without using associators. All representations in the family are faithful, defined over \bbQ by simple explicit formulae. We show that they give universal Vassiliev-type invariants for braid-permutation groups.Comment: 19 pages, references adde