Tubular algebras and affine Kac-Moody algebras. (Q995682)

The authors realize each of the affine Kac-Moody algebras of types \(D_4^{(1)}\), \(E_6^{(1)}\), \(E_7^{(1)}\) and \(E_8^{(1)}\) as a quotient Lie algebra of the complex degenerate composition Lie algebra \(L(A)_1^{\mathbb{C}}\) of a certain tubular algebra \(A\). It is also proved that the Lie bracket in these Lie algebras is given by Hall multiplication. Finally, it is shown that the algebra \(L(A)_1^{\mathbb{C}}\), generated by simple \(A\)-modules, coincides with the Lie algebra, generated by those indecomposable \(A\)-modules, whose vector-dimensions are real roots.