Graded algebra

A graded algebra is an algebra generated when an outer product (wedge product) is defined in a vector space <math>V_n</math> over the scalars <math>F</math>.

The outer product generates a set of new entities: the <math>k</math>-vectors. As they are obtained by the outer product of <math>k</math> linearly independent vectors, they are said to be of step or grade <math>k</math>. <math>k</math>-vectors are vectors in nature, so any <math>k</math>-vector is a member of a vector subspace known as subspace of grade <math>k</math>, denoted by ∧^kV_n. Each of this has a dimension of <math>C(n, k)</math> where <math>C(n, k)</math> is the binomial coefficient.

Vectors are said to have step 1, so

<math>\wedge^1 V_n = V_n,</math>

with dimension <math>n</math>, and scalars are considered as the 0-step vector space ∧⁰V_n, and have dimension 1. The <math>n</math>-vectors also generate a 1-dimensional vector space, so all <math>n</math>-vectors are scalar multiples of a arbitrarily-chosen unitary <math>n</math>-vector. Given that essentially behave as scalars, they are often referred to as pseudoscalars. Similarly, <math>(n - 1)</math>-vectors are also called pseudovectors.

In order to achieve closure, all these spaces are combined by considering the direct sum of all of them. The resulting space is a new vector space called the graded algebra:

<math> \wedge V_n

  = \wedge^0 V_n + \wedge^1 V_n + \cdots + \wedge^n V_n</math>

and we call multivectors to its elements.

The dimension of the graded algebra is <math>2^n</math>, and the structure of the grades subspaces is that of the Pascal triangle (see binomial coefficient).