Cayley-Dickson notation
Introduction
The Cayley-Dickson notation is a method used in mathematics to construct a sequence of algebras over a field, starting from the real numbers and progressing through complex numbers, quaternions, octonions, and beyond. This construction is named after Arthur Cayley and Leonard Eugene Dickson, who contributed significantly to the development of algebraic structures. The Cayley-Dickson construction is a recursive process that doubles the dimension of the algebra at each step, leading to the creation of hypercomplex numbers.
Historical Background
The Cayley-Dickson construction has its roots in the work of Hamilton, who introduced quaternions in the 19th century. Hamilton's work was later expanded by Cayley and Dickson, who generalized the construction to higher-dimensional algebras. The construction was initially developed to explore the properties of hypercomplex numbers and their applications in various fields of mathematics and physics.
The Construction Process
The Cayley-Dickson construction begins with a base field, typically the real numbers (\(\mathbb{R}\)). Each step in the construction process involves doubling the dimension of the algebra by introducing a new imaginary unit. The process can be summarized as follows:
Step 1: Complex Numbers
Starting with the real numbers \(\mathbb{R}\), the first step introduces an imaginary unit \(i\) such that \(i^2 = -1\). This results in the complex numbers \(\mathbb{C}\), which can be represented as \(\mathbb{R} \oplus \mathbb{R}i\).
Step 2: Quaternions
The next step introduces two new imaginary units, \(j\) and \(k\), satisfying the relations \(i^2 = j^2 = k^2 = ijk = -1\). This results in the quaternions \(\mathbb{H}\), which can be represented as \(\mathbb{C} \oplus \mathbb{C}j\).
Step 3: Octonions
The third step introduces four more imaginary units, resulting in the octonions \(\mathbb{O}\). The octonions are non-associative but still form a normed division algebra. They can be represented as \(\mathbb{H} \oplus \mathbb{H}l\), where \(l\) is a new imaginary unit.
Higher-Dimensional Algebras
The construction can be continued indefinitely, resulting in higher-dimensional algebras such as the sedenions. However, these higher-dimensional algebras lose some of the desirable properties of lower-dimensional algebras, such as associativity and the normed division property.
Properties of Cayley-Dickson Algebras
The algebras constructed using the Cayley-Dickson process exhibit several interesting properties:
Normed Division Algebras
The algebras up to the octonions are normed division algebras, meaning they have a norm that satisfies the multiplicative property \(\|xy\| = \|x\|\|y\|\). This property is lost in higher-dimensional algebras.
Non-Associativity
Starting with the octonions, the algebras become non-associative. This means that the associative law \((xy)z = x(yz)\) does not hold in general. However, the octonions are still alternative, meaning that any subalgebra generated by two elements is associative.
Non-Commutativity
All algebras constructed using the Cayley-Dickson process, except for the complex numbers, are non-commutative. This means that the commutative law \(xy = yx\) does not hold in general.
Applications
The Cayley-Dickson construction has applications in various fields of mathematics and physics:
Theoretical Physics
In theoretical physics, the algebras constructed using the Cayley-Dickson process are used to describe symmetries and particle interactions. For example, the octonions have been used in the study of string theory and exceptional Lie groups.
Computer Graphics
Quaternions, a result of the Cayley-Dickson construction, are widely used in computer graphics for representing rotations and interpolations. They provide a more efficient and numerically stable way to handle rotations compared to traditional methods such as Euler angles.
Number Theory
The Cayley-Dickson construction has implications in number theory, particularly in the study of quadratic forms and algebraic structures. The construction provides a framework for understanding the properties of hypercomplex numbers and their relationships to other mathematical objects.