nLab N-graded module

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Algebra

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homotopy theory, (∞,1)-category theory, homotopy type theory

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see also algebraic topology

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A mathematical structure used to define geometric algebras.

Definition

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

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

  • for all a:Ra:R, b:Rb:R, v:Vv:V, w:Vw:V, and n:n:\mathbb{N}, av+bw n=av n+bw n\langle a v + b w \rangle_n = a \langle v \rangle_n + b \langle w \rangle_n

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

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

Terms of VV are called multivectors.

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

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

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

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

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

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

|V| nim(|| n)\vert V \vert_n \coloneqq \mathrm{im}(\vert - \vert_n)

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

Examples

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

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

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

See also

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

Last revised on August 2, 2022 at 11:56:48. See the history of this page for a list of all contributions to it.