Associative algebra

An associative algebra over a field $K$ is a vector space $A$ equipped with a bilinear product

\cdot : A \times A \to A

that satisfies the associativity condition:

(a \cdot b) \cdot c = a \cdot (b \cdot c), \forall a, b, c \in A .

From any associative algebra, one can define a Lie bracket by

[a, b] := a \cdot b - b \cdot a .

Thus, any associative algebra $A$ gives rise to a Lie algebra $(A, [\cdot, \cdot])$ via the commutator bracket.