Question Regarding the Commutativity of F-Algebras when the Algebra is finite dimensional over F.

Question

Let $A$ be some $F$-Algebra, for some field $F$, with the property that $A$ is finite dimensional over $F$. Is $A$ always commutative?

Answer 1

Do you know of any noncommutative algebras?

• Matrix algebras: $M_n(F)$ over fields $F$ (when $n>1$). Dimension $n^2$.
• Group algebras $F[G]$ (when $G$ is nonabelian). Dimension is $|G|$.
• Quaternions $\mathbb{H}$ are noncommutative $\mathbb{R}$-algebras. Dimension is $4$.

These are all associative algebras. (Dietrich gives two canonical examples of nonassociative.)

Answer 2

The definition in wikipedia allows non-commutative and non-associative algebras over $F$, such as Lie algebras or Jordan algebras. A "motivating" example are also the quaternions, which are anti-commutative.