nLab dyadic rational algebra




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


The decimal rational numbers [1/10]\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.


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

a|b12(a+b) 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

ab12(ab+ba) a \cdot b \coloneqq \frac{1}{2} (a b + b a)

and the commutator product to be

a×b12(abba) a \times b \coloneqq \frac{1}{2} (a b - b a)

The symmetric product in a dyadic rational algebra is commutative:

ab=ba a \cdot b = b \cdot a

while the commutator product is anticommutative

a×b=(b×a) a \times b = -(b \times a)

and satisfiea the Jacobi identity

a×(b×c)+b×(c×a)+c×(a×b)=0 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.


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)

Created on June 19, 2021 at 00:25:46. See the history of this page for a list of all contributions to it.