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Graded algebra

A graded algebra is an algebra generated when an outer product (wedge product) is defined in a vector space over the scalars .

The outer product generates a set of new entities: the -vectors. As they are obtained by the outer product of linearly independent vectors, they are said to be of step or grade . -vectors are vectors in nature, so any -vector is a member of a vector subspace known as subspace of grade , denoted by ∧kVn. Each of this has a dimension of where is the binomial coefficient.

Vectors are said to have step 1, so

with dimension , and scalars are considered as the 0-step vector space ∧0Vn, and have dimension 1. The -vectors also generate a 1-dimensional vector space, so all -vectors are scalar multiples of a arbitrarily-chosen unitary -vector. Given that essentially behave as scalars, they are often referred to as pseudoscalars. Similarly, -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:

and we call multivectors to its elements.

The dimension of the graded algebra is , and the structure of the grades subspaces is that of the Pascal triangle (see binomial coefficient).