Universal central extensions of twisted and untwisted Lie algebras extended over commutative rings
Let \(\Gamma\) be an abelian group, \({\mathcal L} = \oplus_{\alpha \in \Gamma} {\mathcal L}_\alpha\) a \(\Gamma\)-graded Lie algebra over a field \(k\) of characteristic \(p \neq 2\), and \(A = \oplus_{\alpha \in \Gamma} A_\alpha\) a \(\Gamma\)-graded commutative, associative \(k\)-algebra. The universal central extensions of the Lie algebra \(({\mathcal L}, A, \Gamma) = \oplus_{\alpha \in \Gamma} ({\mathcal L}_\alpha \otimes A_\alpha)\) under the assumptions \([{\mathcal L}_0, {\mathcal L}_\alpha] = {\mathcal L}_\alpha\), \(H^1 ({\mathcal L}_0, {\mathcal L}^*_\alpha) = 0\), \(\alpha \in \Gamma\) are studied. The situation considered generalizes most of the cases which have been dealt with by many authors. In the main theorem under the above assumptions the construction of the universal central extension is given. Then the authors consider the untwisted case when \(\Gamma = (0)\) to obtain the structure of universal central extensions in terms of Kähler differentials covering the number of well known results. Under extra assumptions the suggested general construction of universal central extensions is studied in the twisted case when \(\Gamma\) is a cyclic group of order 2 or 3. Various applications of the general construction are presented including classical finite-dimensional simple Lie algebras \({\mathcal L}\) and Lie algebras of Cartan type of characteristic \(p \geq 0\). The results on universal central extensions are used to answer a very interesting question on whether or not the character of the Lie algebra \(L^+_p (A)\) given by generators \(X_0, \dots, X_l\) and defining relations \((\text{ad} X_i)^{-a_{ij} + 1} X_j = 0\), where \(A\) is an affine Cartan matrix, depends on the characteristic \(p\) of the ground field \(k\). It seems to be surprising that it does.
scientific article; zbMATH DE number 763947