nLab dyadic rational algebra

Idea

A dyadic rational algebra or dy-algebra is an associative unital algebra over the dyadic rational numbers

Examples

The decimal rational numbers $\mathbb{Z}[1/10]$, the rational numbers $\mathbb{Q}$, and the real numbers $\mathbb{R}$ are dyadic rational algebras.

Any $\mathbb{R}$-algebra, such as the complex numbers, quaternions, and finite dimensional Clifford algebras over the real numbers, is also a dyadic rational algebra.

Properties

Every dyadic rational algebra is a symmetric midpoint algebra with the midpoint operation given by

$$a \vert b \coloneqq \frac{1}{2} (a + b)$$

Symmetric and commutator products

In any dyadic rational algebra, one can define the symmetric product to be

$$a \cdot b \coloneqq \frac{1}{2} (a b + b a)$$

and the commutator product to be

$$a \times b \coloneqq \frac{1}{2} (a b - b a)$$

The symmetric product in a dyadic rational algebra is commutative:

$$a \cdot b = b \cdot a$$

while the commutator product is anticommutative

$$a \times b = -(b \times a)$$

and satisfiea the Jacobi identity

$$a \times (b \times c) + b \times (c \times a) + c \times (a \times b) = 0$$

The bivector subalgebra of a Clifford algebra over the real numbers with the commutator product provides a model for spin geometry and Lie groups, and the vector submodule of a Clifford algebra over the real numbers with the symmetric product is an inner product space.

Related concepts

• dyadic rational number

• dyadic rational module

• symmetric midpoint algebra

• Clifford algebra

• geometric algebra

• spin geometry

References

The relation to symmetric midpoint algebras could be found in

• Peter Freyd, Algebraic real analysis, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 (tac:20-10)

The symmetric and commutator product is defined in this reference for Clifford algebras over the real numbers, but the definition is valid in any dyadic rational algebra:

• Chris Doran, Anthony Lasenby, Geometric algebra for physicists, Cambridge University Press (2003) (pdf)