Cayley Numbers (A. CAYLEY, 1845) Let 𝒞:=ℋ ×ℋ Consider the following composition law: 𝒞 ×𝒞 ⟶𝒞

step 1 Yes, hello. So first to show that we're closed under addition, so being closed under addition, w

Cayley Numbers (A. CAYLEY, 1845) Let 𝒞:=ℋ ×ℋ Consider the...

Cayley Numbers (A. CAYLEY, 1845) Let $$ \mathcal{C}:=\mathcal{H} \times \mathcal{H} $$ Consider the following composition law: $$ \begin{gathered} \mathcal{C} \times \mathcal{C} \longrightarrow \mathcal{C} \\ \left(\left(H_{1}, H_{2}\right),\left(K_{1}, K_{2}\right)\right) \mapsto\left(H_{1} K_{1}-\bar{K}_{2}^{\prime} H_{2}, H_{2} \bar{K}_{1}^{\prime}+K_{2} H_{1}\right) \end{gathered} $$ Here $\bar{H}^{\prime}$ denotes the conjugate transposed matrix of $H \in \mathcal{H} \subset M(2 \times 2 ; \mathbb{C})$ Show that this defines on an $\mathbb{R}$-bilinear map, which has no zero divisors, i.e. the "product" of two elements in $\mathcal{C}$ is zero, iff one of the two factors vanishes. This "CAYLEY multiplication" is, in general, neither commutative nor associative. A deep theorem (M. A. KERVAIRE (1958), J. MILNOR (1958), J. BOTT (1958)) says that on an $n$-dimensional $(n<\infty)$ real vector space $V$ a bilinear form free of zero divisors can only exist when $n=1,2,4$ or $8 .$ Examples of such structures are the "real numbers", the "complex numbers", the "HAMILTON quaternions" and the "CAYLEY numbers". Compare with the article of F. Hirzebruch, [Hi].

Cayley Numbers (A. CAYLEY, 1845)
Let
$$
\mathcal{C}:=\mathcal{H} \times \mathcal{H}
$$
Consider the following composition law:
$$
\begin{gathered}
\mathcal{C} \times \mathcal{C} \longrightarrow \mathcal{C} \\
\left(\left(H_{1}, H_{2}\right),\left(K_{1}, K_{2}\right)\right) \mapsto\left(H_{1} K_{1}-\bar{K}_{2}^{\prime} H_{2}, H_{2} \bar{K}_{1}^{\prime}+K_{2} H_{1}\right)
\end{gathered}
$$
Here $\bar{H}^{\prime}$ denotes the conjugate transposed matrix of $H \in \mathcal{H} \subset M(2 \times 2 ; \mathbb{C})$ Show that this defines on an $\mathbb{R}$-bilinear map, which has no zero divisors, i.e. the "product" of two elements in $\mathcal{C}$ is zero, iff one of the two factors vanishes. This "CAYLEY multiplication" is, in general, neither commutative nor associative.
A deep theorem (M. A. KERVAIRE (1958), J. MILNOR (1958), J. BOTT (1958)) says that on an $n$-dimensional $(n<\infty)$ real vector space $V$ a bilinear form free of zero divisors can only exist when $n=1,2,4$ or $8 .$ Examples of such structures are the "real numbers", the "complex numbers", the "HAMILTON quaternions" and the "CAYLEY numbers". Compare with the article of F. Hirzebruch, [Hi].

Key Concepts

R-bilinear Map

An R-bilinear map is a function between real vector spaces that is linear in each argument separately. This means if you fix one argument, the map behaves like a linear transformation with respect to the other argument. In the context of normed division algebras, R-bilinearity ensures that scalar multiplication distributes appropriately over the product, maintaining the structure necessary for defining operations like the Cayley multiplication.

Zero Divisors

Zero divisors in an algebraic structure are nonzero elements whose product is zero. A bilinear map or multiplication is said to be free of zero divisors if the product of two elements is zero only when at least one of the elements is zero. This property is crucial in the study of division algebras because it guarantees the existence of multiplicative inverses for nonzero elements, which is fundamental in defining a division algebra.

Normed Division Algebra

A normed division algebra is an algebra over the real numbers equipped with a norm where every nonzero element has a multiplicative inverse, and the norm is multiplicative. This setting provides a rigorous framework for operations where division is well-defined and often leads to rich geometric and algebraic structures. The classical examples include the real numbers, complex numbers, quaternions, and octonions.

Cayley Numbers (Octonions)

The Cayley Numbers, often referred to as the Octonions, form an 8-dimensional normed division algebra. They extend the idea of complex numbers and quaternions but do not satisfy associativity, meaning that the order in which multiplications are performed matters. Despite losing associativity, octonions retain other important properties such as the absence of zero divisors and a well-defined norm, making them a key example in the study of non-associative algebras.

Hurwitz's Theorem

Hurwitz's Theorem is a foundational result in algebra which states that the only finite-dimensional normed division algebras over the real numbers occur in dimensions 1, 2, 4, or 8. This theorem explains why the classical algebras—the real numbers, complex numbers, quaternions, and octonions—are unique in their structure and why no other normed division algebras exist in other finite dimensions.

Non-Commutativity and Non-Associativity

Non-commutativity and non-associativity are key features of certain algebraic systems, particularly in quaternions and octonions. In a non-commutative algebra, the order of multiplication affects the result, and in a non-associative algebra, the grouping of elements during multiplication can change the outcome. These properties are essential in understanding the structural differences and the complexities that arise in higher-dimensional algebras such as the Cayley Numbers.

Transcript

Please give Ace some feedback