$q$-deformation of Witt-Burnside rings

Details $q$-deformation of Witt-Burnside rings $q$-deformation of Witt-Burnside rings

2004-11-16

arxiv.org/abs/math/0411353v3

Mathematics

Rings and Algebras

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Scientific paper

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 set of integers. When $q=1$, it coincides with the Witt-Burnside ring introduced by A. Dress and C. Siebeneicher (Adv. Math. {70} (1988), 87-132). To achieve our goal we first 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 also study their properties. As a byproduct, we prove a conjecture due to Lenart (J. Algebra. 199 (1998), 703-732). Finally, we classify $\mathbb W_G^q$ up to strict natural isomorphism in case where $G$ is an abelian profinite group.

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