Iš Gvildenu svetainės

Book: GeometricAlgebra

See: Math, Clifford algebra, Geometry, Classical Lie Groups

Reach out to Steven Lehar and Math Future.

Clifford algebra, geometric algebra, and applications, D. Lundholm and L. Svensson, 2009.

Study and learn

More questions

Videos

Books

Readings

Websites

Steven Lehar

Ideas

Interpretations

Basis elements

Squares

Terms

Elements

Even subalgebra of Clifford algebra.

Multiplication

Facts

Ideas

Totality = Pseudoscalar

Totality (pseudoscalar) {$e_{1}e_{2}\dots e_{n}$} is an important concept for me and also for Clifford algebras.

The totality commutes with every element precisely when n is odd. That is because to move the element from the right hand side to the left hand side we need to swap with n-1 different elements (switching sign) and 1 element which is the same (thus not switching sign).

This is the basis for a twofold periodicity.

{$e_{1}e_{2}\dots e_{n}\cdot e_{i}=(-1)^{n-1}e_{i}\cdot e_{1}e_{2}\dots e_{n}$}

Squaring the totality yields a fourfold periodicity.

This is because in the sum 0+1+2+3+4... we start with even and then have

Where even means "stay the same" and odd means "switch". Note that here the two are dual!

Classifying Clifford algebras

For complex numbers there is essentially only one quadratic form {$\sum_{i=1}^{n}u_{i}^2$} because squaring a complex number can generate -1 as needed. Whereas the reals discriminate real and imaginary square roots and so they have a signature {$\sum_{i=1}^{p}u_{i}^2 - \sum_{i=1}^{q}u_{i}^2$}.

Parsiųstas iš http://www.ms.lt/sodas/Book/GeometricAlgebra
Puslapis paskutinį kartą pakeistas 2018 kovo 23 d., 12:19