Classification of group gradings on simple Lie algebras of types \(\mathcal A\), \(\mathcal B\), \(\mathcal C\) and \(\mathcal D\) (Q626697)

The aim of this paper is the classification, up to isomorphism, of the \(G\)-gradings on simple classical Lie algebras (different from \(\mathcal{D}_4\)), for \(G\) a fixed abelian group, over an algebraically closed field \(\mathbb{F}\) of characteristic different from 2. Such gradings were described in [the authors, J. Pure Appl. Algebra 213, No. 9, 1739--1749 (2009; Zbl 1185.17023), and with \textit{S. Montgomery}, Proc. Am. Math. Soc. 137, No. 4, 1245--1254 (2009; Zbl 1168.17016), \textit{Yu. A. Bahturin, I. P. Shestakov} and \textit{M. V. Zaicev}, J. Algebra 283, No. 2, 849--868 (2005; Zbl 1066.17018), \textit{Yu. A. Bahturin} and \textit{M. V. Zaicev}, J. Lie Theory 16, No. 4, 719--742 (2006; Zbl 1141.17006)], but it is not clear in these references when two of those gradings are isomorphic. Recently a classification in characteristic zero (together with a different description) of the gradings on these algebras has appeared in [\textit{A. Elduque}, J. Algebra 324, No. 12, 3532--3571 (2010; Zbl 1213.17030)], but only fine gradings are classified, and this is done up to equivalence. There is no direct way of getting the classification up to equivalence from the classification up to isomorphism, nor conversely. The difference between both concepts is the following. A grading on an algebra \(U\) is a vector space decomposition \(U=\bigoplus_{g\in G}U_g\) such that \(U_gU_h\subset U_{g h}\) for all \(g,h\in G\). Two \(G\)-gradings: \(U=\bigoplus_{g\in G}U_g\) and \(U=\oplus_{g\in G}U'_g\), are isomorphic if there is an automorphism \(\varphi\in\Aut\, U\) such that \(\varphi(U_g)=U'_g\) for any \(g\in G\). Two gradings over the groups \(G\) and \(H\): \(U=\bigoplus_{g\in G}U_g\) and \(U=\oplus_{h\in H}U'_h\), are equivalent if there is an automorphism \(\varphi\in\Aut\, U\) such that, for each \(g\in G\) with \(U_g\neq0\), there is \(h\in H\) such that \(\varphi(U_g)=U'_h\). In order to deal with fields of arbitrary characteristic, the authors use affine group schemes to reduce the classification of \(G\)-gradings on classical Lie algebras to the classification of \(G\)-gradings on matrix algebras \(R\) and on pairs \((R,\varphi)\), where \(\varphi\) is a suitable anti-automorphism. The key is that the gradings on a finite-dimensional algebra \(U\) over a finitely generated abelian group \(G\) are in correspondence with morphisms of algebraic group schemes \(G^D\to\mathbf{Aut}\, U\), where \(G^D\) is the affine group scheme represented by the group algebra \(\mathbb{F}G\) (this is a Hopf algebra) and \(\mathbf{Aut}\, U\) is the automorphism group scheme of \(U\). Two \(G\)-gradings are isomorphic just when the corresponding morphisms are conjugate by an automorphism of \(U\). The paper begins by giving a classification of the \(G\)-gradings on a matrix algebra, which are necessarily the tensor product of a division grading and an elementary grading, by means of some invariants, and provides also canonical forms for the anti-automorphisms compatible with the grading such that the restriction to the identity component is an involution. The classification of the \(G\)-gradings on the simple Lie algebras of types \(\mathcal{B}\), \(\mathcal{C}\) and \(\mathcal{D}\) follows easily, because these algebras coincide with the space of skew-symmetric elements in a matrix algebra with respect to a symplectic or orthogonal involution. The classification, obtained in terms of numerical and group-theoretical invariants, of the \(G\)-gradings on simple Lie algebras of type \(\mathcal{A}\) is more difficult. The authors obtain a different description from the one in [Yu. A. Bahturin and M. V. Zaicev (loc. cit.)], while at the same time correct a mistake in that paper. They divide the gradings on types I and II according to whether they are produced only by inner automorphisms or not. If \(\overline{\Aut}(R)\) denotes the group of automorphisms and anti-automorphisms of a matrix algebra \(R=M_n(\mathbb{F})\), there is a bijection between the non-isomorphic classes of type I gradings and the \(\overline{\Aut}(R)\)-orbits on the set of \(G\)-gradings on \(R\), and between the non-isomorphic classes of type II gradings and the \(\overline{\Aut}(R)\)-orbits on the set of pairs \((R,\varphi)\) where \(R\) is \( G/\langle h\rangle\)-graded for certain \(h\) and \(\varphi\) is an anti-automorphism of \(R\) preserving the grading and satisfying certain property.