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Register a new userGeneralizing Witt vector construction
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Young-Tak Oh
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The purpose of this this paper is to generalize the functors arising from the theory of Witt vectors duto to Cartier. Given a polynomial $g(q)in mathbb Z[q]$, we construct a functor ${overline {W}}^{g(q)}$ from the category of $mathbb Z[q]$-algebras to that of commutative rings. When $q$ is specialized into an integer $m$, it produces a functor from the category of commutative rings with unity to that of commutative rings. In a similar way, we also construct several functors related to ${overline { W}}^{g(q)}$. Functorial and structural properties such as induction, restriction, classification and unitalness will be investigated intensively.
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We describe infinite-dimensional Leibniz algebras whose associated Lie algebra is the Witt algebra and we prove the triviality of low-dimensional Leibniz cohomology groups of the Witt algebra with the coefficients in itself.
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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