0

I try to find any information on the following algebra defined as: $$(a,b)+(c,d)=(a+c,b+d)$$ $$(a,b)*(c,d)=(ac,ac-bd)$$ It is non-associative, but commutative and distributive. Can it be classified somehow? Do there exist any information on it? How can I find its properties?

oddy
  • 51
  • For what it's worth, the components of $(a, b) \ast (c, d)$ are the first components of the componentwise product and the "complex product" $((a, b), (c, d)) \mapsto (a c - b d, a d + b c)$. (Over $\Bbb R$, the complex product is exactly the usual complex multiplication.) – Travis Willse Mar 08 '23 at 16:01

1 Answers1

0

Sounds like you're maybe talking about general non-associative algebras (also known as distributive algebras). Commutativity is just then a property that non-associative algebras can have or not have. I don't always trust Wikipedia but this page would be a good start. There are some good examples like Jordan Algebras

JJP_SWFC
  • 19