Current algebra

Remarks

The bracket is a Lie bracket. Antisymmetry: $[y \otimes b, x \otimes a] = [y, x] \otimes ba = -[x, y] \otimes ab$, using $ba = ab$. The Jacobi identity for $x \otimes a$, $y \otimes b$, $z \otimes c$ is the Jacobi identity of $\mathfrak{g}$ tensored with the single element $abc$.

Example: for $A = k[t]$ one gets the algebra $\mathfrak{g}[t] = \mathfrak{g} \otimes k[t]$ of polynomial maps into $\mathfrak{g}$, often called the current algebra of $\mathfrak{g}$. For $A = k[t, t^{-1}]$ one gets the loop algebra. Let $X$ be a smooth manifold and $\mathfrak{g}$ a real Lie algebra. The smooth maps $X \to \mathfrak{g}$ with pointwise bracket form a current algebra of the same kind, with $A$ the smooth real functions on $X$.

Non-example: if $A$ is not commutative, the same formula does not give a Lie algebra. For $\mathfrak{g} = \mathfrak{sl}_2$ with $[h, e] = 2e$ and $A$ the $2 \times 2$ matrices, take $a$ and $b$ with $ab \neq ba$. Then $[h \otimes a, e \otimes b] = 2e \otimes ab$, while $-[e \otimes b, h \otimes a] = 2e \otimes ba$, so antisymmetry fails.