Encyclopedia > Graded algebra

  Article Content

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 ∧kVn. 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 ∧0Vn, 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).



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Canadian Charter of Rights and Freedoms

... Links Canadian Charter of Rights and Freedoms (http://www.laurentia.com/ccrf/ccrf.htm) Constitutional Law of Canad ...

 
 
 
This page was created in 26.8 ms