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# "From complex numbers to quaternions and beyond" by Valentin Ovsienko

Algebra? Geometry? Number theory? Let us call this subject simply : mathematics. The main goal of these lectures is to explain why do mathematicians invent "complicated" algebraic structures. Our main character will be the algebra of quaternions. Invented by Sir William Hamilton in 1843, quaternions extend complex numbers. But, unlike usual numbers, the algebra of quaternions is non-commutative that makes it more complicated. Non-commutative? Are we sure? We will see that the algebra of quaternions is, in fact, commutative if we understand what the commutativity really means.

## Papa Hamilton's quaternions

There ARE triplets behind COMMUTATIVE quaternions. Understanding their $Z_2^3$ grading actually lead to the full ASSOCIATIVE octonions. OvsienkotranspIssmys.pdf — PDF document, 2.00 MB (2095890 bytes)