Non-associative algebra
Introduction
Non-associative algebra is a branch of algebra that studies algebraic structures which do not necessarily satisfy the associative property. This field of mathematics is distinct from associative algebra, where the associative law always holds. Non-associative algebras include structures such as loops, quasigroups, Lie algebras, Jordan algebras, and octonions.
Non-Associative Structures
Non-associative algebra primarily deals with algebraic structures that do not follow the associative law. These structures include:
Loops
A loop is a non-associative structure that has a binary operation and an identity element. The binary operation is not necessarily associative, but each element in the loop has an inverse.
Quasigroups
A Quasigroup is a type of non-associative structure that has a binary operation. In a quasigroup, the division operation is always possible, but the binary operation is not necessarily associative.
Lie Algebras
Lie algebras are non-associative structures that are used in the study of differential equations, geometry, and theoretical physics. They are named after the Norwegian mathematician Sophus Lie, who introduced them in the context of his theory of continuous transformation groups.
Jordan Algebras
Jordan algebras are non-associative structures that are commutative and satisfy the Jordan identity. They were introduced by Pascual Jordan in his work on quantum mechanics.
Octonions
Octonions are a type of non-associative algebra that extends the quaternions. They are a normed division algebra over the real numbers, but unlike the real numbers, the complex numbers, and the quaternions, the octonions do not satisfy the associative law.
Properties of Non-Associative Algebras
Non-associative algebras have several properties that distinguish them from associative algebras. These include:
Non-Associativity
The defining property of non-associative algebras is the lack of the associative law. This means that for any elements a, b, and c in the algebra, it is not necessarily true that (a*b)*c = a*(b*c).
Flexibility
Many non-associative algebras satisfy the flexible law, which states that for any elements a and b in the algebra, (a*b)*a = a*(b*a). This is a weaker condition than associativity.
Alternative Law
Some non-associative algebras satisfy the alternative law, which states that for any element a in the algebra, (a*a)*a = a*(a*a). This is also a weaker condition than associativity.
Applications of Non-Associative Algebra
Non-associative algebra has applications in various fields of mathematics and physics. These include:
Differential Equations
Lie algebras are used in the study of differential equations. They provide a method for solving differential equations that arise in the study of continuous transformation groups.
Geometry
Non-associative algebras are used in the study of certain types of geometry, including projective geometry and differential geometry.
Theoretical Physics
Non-associative algebras have applications in theoretical physics, particularly in the study of quantum mechanics and string theory. Jordan algebras, for example, were introduced in the context of quantum mechanics.