associative algebra

Definition

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

$$\bullet :A\times A⟶A$$

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

1. Associativity (what makes the difference from a non-associative algebra) $a(bc)=(ab)c$
2. Distributive $a(b+c)=ab+ac$

$(a+b)c=ac+bc$ 4. $(\lambda a)(\mu b)=(\lambda \mu )(ab)\forall \lambda ,\mu \in \mathbb{F},a,b\in A$ We call $A$ commutative if $ab=ba$ for all $a,b\in A$.

If you want to rule out the trivial algebra, you can include the requirement that there exists and element $1\in A$ such that $1\ne 0$ and

$$1\cdot a=a\cdot 1=a\forall a\in A$$

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

subalgebras

A subset $B\subset A$ is a subalgebra if

1. $B$ is a vector subspace of $A$
2. $bc\in B$ for all $b,c\in B$
3. $1\in B$

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

In other words, $B$ inherits its algebra structure from $A$.

Generating subalgebras

For a subset $S\subset A$, the subalgebra of $A$ generated by $S$ is

$$\left(S\right):=\bigcap _{\begin{array}{rl}B\subset A& \text{subalgebra}\\ \text{s.t.}& S\subset B\end{array}}B$$

We can think of it as the smallest subalgebra that contains all of $S$. If $S$ 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

$\left(S\right)$ is the set of all finite $\mathbb{F}$-linear combos of finite products of elements of $S$.

$$(S)=\left\{\sum _{j=1}^n{\lambda }_j{a}_{j_1}{a}_{j_2}\dots a_{j_{k_j}}|n,k_j\in {\mathbb{Z}}^{+},{\lambda }_j\in \mathbb{F},a_{j_l}\in S\right\}$$

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References