associative algebra

Definition

An associative algebra (over a field) is a vector space with a “multiplication” operation

(where we generally suppress the notation and put them next to each other) such that

1. Associativity (what makes the difference from a non-associative algebra)
2. Distributive

4. We call commutative if for all .

If you want to rule out the trivial algebra, you can include the requirement that there exists and element such that and

Note: There is an equivalent definition using a ring that has an extra operation.

Diagram of associativity

The associativity condition can also be summed up using the following diagram (where denotes vector multiplication).

Subalgebras

A subset is a subalgebra if

1. is a vector subspace of
2. for all

Using this definition, we can prove that for a subalgebra , the algebra structure of is the unique one for which the inclusion map is an algebra homomorphism.

In other words, inherits its algebra structure from .

Generating subalgebras

For a subset , the subalgebra of generated by is

We can think of it as the smallest subalgebra that contains all of . If is not an ideal, it will have other stuff in it, but this is the “best” subalgebra with as little other stuff as possible.

Explicit construction

is the set of all finite -linear combos of finite products of elements of .

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References