. R A ] 1 6 N ov 2 00 4 q - DEFORMATION OF WITT - BURNSIDE RINGS

In this paper we construct a q-deformation of the Witt-Burnside ring of a profinite group over a commutative ring, where q ranges over the ring of integers. When q = 1, this coincides with the Witt-Burnside ring introduced by A. To achieve our goal we show that there exists a q-deformation of the necklace ring of a profinite group over a commutative ring. As in the classical case, i.e., the case q = 1, q-deformed Witt-Burnside rings and necklace rings always come equipped with inductions and restrictions. We study their properties. In particular, by investigating the properties of restrictions on the q-deformed necklace ring of the multiplicative infinite cyclic group C, we prove Lenart's conjecture proposed in (J.