A note on \(K_{2}\) of Kac-Moody groups over a field (Q2406409)

Let \(F\) be a field. \textit{H. Matsumoto} [Ann. Sci. Éc. Norm. Supér. (4) 2, 1--62 (1969; Zbl 0261.20025)] gave a presentation of the Milnor \(K\) group \(K_2(F)\) in terms of Steinberg symbols. He also gave a presentation for the symplectic counterpart \(K_2Sp(F)\). It seems that \(K_2Sp(F)\) is now known as the Milnor-Witt \(K\)-group \(K_2^{MW}(F)\). Consider the \(K\)-group \(K_2(A,F)\) associated to any Kac-Moody group \(G(A,F)\) where \(A\) is a generalized Cartan matrix. The author explains how to express \(K_2(A,F)\) in terms of \(K_2(F)\) and \(K_2^{MW}(F)\), using a theorem of \textit{J. Morita} and \textit{U. Rehmann} [Tohoku Math. J. (2) 42, No. 4, 537--560 (1990; Zbl 0701.19001)]. One starts with marking the columns of \(A\). Those that contain only even numbers are marked even and the other ones are marked odd. A matrix with columns marked odd or even is a `matrix with parity'. The author describes a variation on the usual Smith normal form algorithm, resulting in a Smith normal form `with parity' of \(A\). This then leads to a description of the sought \(K_2(A,F)\) as \(K_2(F)\otimes X\oplus K_2^{MW}(F)\otimes Y\) with explicit finitely generated abelian groups \(X\), \(Y\).