A general definition of an algebra

• A collection of basis vectors
• A field i.e. real, complex, quaternion, etc.

together with three kinds of operations

• vector addition
• scalar multiplication
• vector multiplication

Geometric Algebra

Geometric algebra comes in different flavors.

A discussion of vectors as instances of an algebra class leads to how easy it is for rotors to transform, dilate, rotate or reflect these vector in their spaces with object oriented programming. Consider a new kind of product for two vectors, a combination of the dot and the wedge. Using this goemetric product cleans up many derivations and representations in physics. Suppose we wanted to rotate a vector t into a vector g. Given two vectors r and o which represent normals of two reflection planes we could simply reflect twice with the operation.

g = -rotor

The result is a rotation around an axis orthogonal to both vectors o and r. Here the geometric product of the o and r produces a bivector called a rotor instance.

Lie Algebra

A Lie algebra is a set of elements closed under commutation. For example, the set of operators {x,p,1} over a complex field are closed under commutation. Take any two and see that they either commute to zero [x,1]=0 or to an element of the algebra, i.e. [x,p]=i. Lie algebras make quantum mechanics easier to calculate in that you work with no wave equations or differential operators. The Hamiltonian is simply formed out of Lie algebra elements, such as the ladder operator version of the simple harmonic oscillator.

Applications

Abstract algebras have nontrivial applications in many areas. One area that seems to be a research frontier is molecular dynamics. To see more go to Dynamics.