nLab N-graded module

Idea

A mathematical structure used to define geometric algebras.

Definition

Given a commutative ring $R$ (usually a field $F$), an $\mathbb{N}$-graded $R$-module is an $R$-module $V$ (usually a vector space $V$) with a binary function $\langle - \rangle_{(-)}: V \times \mathbb{N} \to V$ called the grade projection operator such that

• for all $v:V$, $v = \sum_{n:\mathbb{N}} \langle v \rangle_n$

• for all $a:R$, $b:R$, $v:V$, $w:V$, and $n:\mathbb{N}$, $\langle a v + b w \rangle_n = a \langle v \rangle_n + b \langle w \rangle_n$

• for all $v:V$ and $n:\mathbb{N}$, $\langle \langle v \rangle_n \rangle_n = \langle v \rangle_n$

• for all $v:V$, $m:\mathbb{N}$, and $n:\mathbb{N}$, $(m \neq n) \Rightarrow (\langle \langle v \rangle_m \rangle_n = 0)$

Terms of $V$ are called multivectors.

For a natural number $n:\mathbb{N}$, the image of $\langle - \rangle_n$ under $V$ is called the space of $n$-vectors and is denoted as $\langle V \rangle_n$.

$$\langle V \rangle_n \coloneqq \mathrm{im}(\langle - \rangle_n)$$

The terms of $\langle V \rangle_n$ are called $n$-vectors.

We define the n-truncation operator $\vert - \vert_{(-)}: V \times \mathbb{N} \to V$

$$\vert v \vert_n = \sum_{m = 0}^n \langle v \rangle_m$$

For a natural number $n:\mathbb{N}$, the image of $\vert - \vert_n$ under $V$ is called the space of $n$-multivectors and is denoted as $\vert V \vert_n$.

$$\vert V \vert_n \coloneqq \mathrm{im}(\vert - \vert_n)$$

The terms of $\vert V \vert_n$ are called $n$-multivectors or $n$-truncated multivectors. The $n$-multivectors are exactly those multivectors whose grade projections for all natural numbers $i \gt n$ are all zero, where $\vert V \vert_i$ is the trivial $R$-module for all natural numbers $i \gt n$.

Examples

• Every polynomial ring is an $\mathbb{N}$-graded module, where the $n$-vectors are homogeneous polynomials of degree $n$, and the $n$-multivectors are general polynomials of degree $n$.

• Every geometric algebra is an $\mathbb{N}$-graded module.

• Every exterior algebra is an $\mathbb{N}$-graded module.

See also

• commutative ring

• module

• graded vector space

• geometric algebra

• non-associative geometric algebra

References

• G. Aragón, J.L. Aragón, M.A. Rodríguez (1997), Clifford Algebras and Geometric Algebra, Advances in Applied Clifford Algebras Vol. 7 No. 2, pg 91–102, doi:10.1007/BF03041220, S2CID:120860757