Algebra on Manifolds
Definition 1
Let $\mathbb{F}$ be a field, and let $A$ be a $\mathbb{F}$–vector space. Given a binary operation $\times : A \times A \to A$ called the product defined as follows, $A$ is called an algebra over $\mathbb{F}$. For $x, y, z \in A$ and $a, b \in \mathbb{F}$,
Distributive laws: $$ (x + y) \times z = x \times z + y \times z $$ $$ z \times (x + y) = z \times x + z \times y $$ Here $+$ denotes the addition of the vector space $A$.
Compatibility with scalar multiplication: $$ (ax) \times (by) = (ab)(x \times y) $$
Explanation
Often abbreviated as a $\mathbb{F}$–algebra. Since a vector space has addition and scalar multiplication defined, a $\mathbb{F}$–algebra is a structure equipped with the three operations: addition, scalar multiplication, and the product.
Saying that conditions 1 and 2 are satisfied simultaneously simply means that the product $\times$ is a (bi)linear map.
Types
Associative algebras
A $\mathbb{F}$–algebra whose product $\times$ satisfies the associative law is called an associative algebra.
- The space of matrices $M_{n \times n}(\mathbb{R})$ or $M_{n \times n}(\mathbb{C})$. In this case the product is matrix multiplication.
- Group algebra
Non-associative algebras
If the product $\times$ of a $\mathbb{F}$–algebra does not satisfy the associative law, it is called a non-associative algebra.
- The 3-dimensional space with the cross product.
- Lie algebra: In general, Lie algebras are non-associative. Since the Lie bracket is bilinear, a Lie algebra is an algebra over a field.