symmetric monoidal (∞,1)-category of spectra
A dyadic rational algebra or dy-algebra is an associative unital algebra over the dyadic rational numbers
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.
Every dyadic rational algebra is a symmetric midpoint algebra with the midpoint operation given by
In any dyadic rational algebra, one can define the symmetric product to be
and the commutator product to be
The symmetric product in a dyadic rational algebra is commutative:
while the commutator product is anticommutative
and satisfiea the Jacobi identity
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
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:
Created on June 18, 2021 at 20:25:46. See the history of this page for a list of all contributions to it.