Jordan algebra
Introduction
A Jordan algebra is a nonassociative algebra over a field where the multiplication satisfies the commutative law and the Jordan identity. Named after the physicist Pascual Jordan, these algebras were first introduced in the context of quantum mechanics in the early 20th century.
Definition
Formally, a Jordan algebra is a set A equipped with a bilinear operation, usually denoted as multiplication, that satisfies the following conditions:
1. Commutativity: For all a, b in A, a * b = b * a. 2. Jordan identity: For all a, b in A, (a * a) * (b * a) = a * ((a * b) * a).
History
The concept of a Jordan algebra was first introduced by Pascual Jordan, John von Neumann, and Eugene Wigner in their study of quantum mechanics. The algebraic structure they discovered was later named after Jordan in recognition of his contributions to the field.
Types of Jordan Algebras
There are several types of Jordan algebras, including:
1. Special Jordan algebra: This is the most common type of Jordan algebra, defined over a field with characteristic not equal to 2. 2. Exceptional Jordan algebra: This is a 27-dimensional algebra over the real numbers, also known as the Albert algebra. 3. Jordan-Banach algebra: This is a Jordan algebra equipped with a compatible norm, making it a Banach space.
Applications
Jordan algebras have found applications in various areas of mathematics and physics. In particular, they play a crucial role in the mathematical formulation of quantum mechanics. They are also used in the study of symmetric spaces and in the classification of simple Lie algebras.