nLab Weyl ring

Contents

Context

Algebra

higher algebra

universal algebra

group theory

Contents

Idea

A $\mathbb{Z}$-Weyl algebra.

Definition

Finitely generated Weyl rings

Given an abelian group $G$, the $n$-th Weyl ring is a ring $A_n(G)$ with

• an abelian group homomorphism $g:G \to A_n(G)$

• a function $x:[1, n] \to A_n(G)$

• a function $\partial:[1, n] \to A_n(G)$

such that

• for every number $i, j \in [1,n]$, $x(i) \cdot x(j) = x(j) \cdot x(i)$

• for every number $i, j \in [1,n]$, $\partial(i) \cdot \partial(j) = \partial(j) \cdot \partial(i)$

• for every number $i \in [1,n]$, $\partial(i) \cdot x(i) - x(i) \cdot \partial(i) = 1$

• for every number $i, j \in [1,n]$, $i \neq j$ implies $\partial(i) \cdot x(j) - x(j) \cdot \partial(i) = 0$

• for every other ring $R$ with abelian group homomorphism $h:G \to R$ with

• a function $x:[1, n] \to R$

• a function $\partial:[1, n] \to R$

where

• for every number $i, j \in [1,n]$, $x(i) \cdot x(j) = x(j) \cdot x(i)$

• for every number $i, j \in [1,n]$, $\partial(i) \cdot \partial(j) = \partial(j) \cdot \partial(i)$

• for every number $i \in [1,n]$, $\partial(i) \cdot x(i) - x(i) \cdot \partial(i) = 1$

• for every number $i, j \in [1,n]$, $i \neq j$ implies $\partial(i) \cdot x(j) - x(j) \cdot \partial(i) = 0$

there is a unique ring homomorphism $i:A_n(G) \to R$ such that $i \circ g = h$.

General Weyl rings

Given an abelian group $G$ and a set $S$ with stable equality, the $S$-generated Weyl ring is a ring $A(S,G)$ with

• an abelian group homomorphism $g:G \to A(S,G)$

• a function $x:S \to A(S,G)$

• a function $\partial:S \to A(S,G)$

such that

• for every number $i, j \in S$, $x(i) \cdot x(j) = x(j) \cdot x(i)$

• for every number $i, j \in S$, $\partial(i) \cdot \partial(j) = \partial(j) \cdot \partial(i)$

• for every number $i \in S$, $\partial(i) \cdot x(i) - x(i) \cdot \partial(i) = 1$

• for every number $i, j \in S$, $i \neq j$ implies $\partial(i) \cdot x(j) - x(j) \cdot \partial(i) = 0$

• for every other ring $R$ with abelian group homomorphism $h:G \to R$ with

• a function $x:S \to R$

• a function $\partial:S \to R$

where

• for every number $i, j \in S$, $x(i) \cdot x(j) = x(j) \cdot x(i)$

• for every number $i, j \in S$, $\partial(i) \cdot \partial(j) = \partial(j) \cdot \partial(i)$

• for every number $i \in S$, $\partial(i) \cdot x(i) - x(i) \cdot \partial(i) = 1$

• for every number $i, j \in S$, $i \neq j$ implies $\partial(i) \cdot x(j) - x(j) \cdot \partial(i) = 0$

there is a unique ring homomorphism $i:A(S,G) \to R$ such that $i \circ g = h$.