Affine Lie algebra

Remarks

Example: $A = \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}$ is indecomposable, and $u = (1, 1)^T$ has positive entries with $Au = 0$. So $\mathfrak{g}(A)$ is an affine Lie algebra, usually written $A_1^{(1)}$ or $\widehat{\mathfrak{sl}}_2$.

Non-example: $A = \begin{pmatrix} 2 & -3 \\ -3 & 2 \end{pmatrix}$ is indecomposable, but $Au = 0$ means $2u_1 = 3u_2$ and $3u_1 = 2u_2$, which forces $u = 0$. So clause 2 fails, and $\mathfrak{g}(A)$ is not affine.

Untwisted realization: let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\C$ with a nonzero invariant symmetric bilinear form $(\cdot \mid \cdot)$. On $\widehat{\mathfrak{g}} = (\mathfrak{g} \otimes \C[t, t^{-1}]) \oplus \C c \oplus \C d$ define the bracket by $$[x \otimes t^m, y \otimes t^n] = [x, y] \otimes t^{m+n} + m\, \delta_{m+n, 0}\, (x \mid y)\, c,$$ $$[d, x \otimes t^m] = m\, x \otimes t^m, \qquad [c, \widehat{\mathfrak{g}}] = 0,$$ for $x, y \in \mathfrak{g}$ and $m, n \in \Z$. This Lie algebra is the affine Lie algebra attached to the extended Cartan matrix of $\mathfrak{g}$. Its subspace $\mathfrak{g} \otimes \C[t, t^{-1}]$ is the loop algebra of $\mathfrak{g}$, but with the bracket changed by the central term. The other affine Lie algebras, called twisted, arise in the same way from loops twisted by a diagram automorphism of $\mathfrak{g}$.