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Deformed Laurent series rings and completions of the Weyl division ring

Gang Han, Yulin Chen, Zhennan Pan

Abstract

Let $ L((T^{-1}))$ be the space of (inverse) Laurent serieswith coefficients in some field $L$. It has a standard degree map and the induced topology. With its usual addition and a new product on this space which is continuous and preserves the standard degree map, it will be a complete topological division ring, and called a deformed Laurent series ring. Under mild restrictions, we give the necessary and sufficient conditions for a product on $ L((T^{-1}))$ to make it a deformed Laurent series ring. Then we apply the above theory to construct the completions of the Weyl division ring $D_1$, over some field of characteristic 0, with respect to a class of discrete valuations on it. Such completions are topological division rings with nice properties. For instance, their valuation rings are non-commutative Henselian rings; the centralizer of each element not in the center is commutative.

Deformed Laurent series rings and completions of the Weyl division ring

Abstract

Let be the space of (inverse) Laurent serieswith coefficients in some field . It has a standard degree map and the induced topology. With its usual addition and a new product on this space which is continuous and preserves the standard degree map, it will be a complete topological division ring, and called a deformed Laurent series ring. Under mild restrictions, we give the necessary and sufficient conditions for a product on to make it a deformed Laurent series ring. Then we apply the above theory to construct the completions of the Weyl division ring , over some field of characteristic 0, with respect to a class of discrete valuations on it. Such completions are topological division rings with nice properties. For instance, their valuation rings are non-commutative Henselian rings; the centralizer of each element not in the center is commutative.
Paper Structure (6 sections, 26 theorems, 179 equations)

This paper contains 6 sections, 26 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. A class of degree maps on $D_1$
  3. Deformed Laurent series rings with the standard degree map
  4. Construction of $\widetilde{D}_{r,s}$ and $\check{D}_{r,s}$ and embedding of $\widehat{D}_{r,s}$
  5. Topological generators of $\widehat{D}_{r,s}$
  6. $\widehat{D}_{r, s}$ satisfies the commutative centralizer condition

Key Result

Lemma 2.3

For $(\rho,\sigma)\in \mathbb R^2\setminus (0,0)$ and $\rho+\sigma\ge0$, $\mathsf{v}_{\rho,\sigma}:A_1\rightarrow \mathbb R_{-\infty}$ is a degree map.

Theorems & Definitions (47)

  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Example 2.5
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 37 more