A characterization of 3-graded Lie algebras generated by a pair
A \(3\)-graded Lie algebra is a \(\mathbb{Z}\)-graded Lie algebra with non-zero elements only in degrees \(-1\), \(0\), \(1\). The authors investigate such algebras generated by a pair of elements of respective degrees \(-1\) and \(1\). The free \(3\)-graded Lie algebra \(\mathfrak{g}(x,y)\) generated by \(x\), \(y\) is defined by the natural universal property. Continuing a previous article [New York J. Math. 7, 71--86 (2001; Zbl 0985.17020)], the authors show that, if the ground field \(K\) is of characteristic zero, the free \(3\)-graded Lie algebra \(\mathfrak{g}(x,y)\) is canonically embedded in \(\mathfrak{sl}_{2}(K[t])\). They describe the image of this embedding and the ideals of \(\mathfrak{g}(x,y)\). They also prove that any \(3\)- graded Lie algebra with a pair of generators over a field \(K\) of characteristic \(0\) is isomorphic to a \(3\)-graded Lie subalgebra of \(\mathfrak{sl}_{2}(K[t]/(p))\) for some \(p\in K[t]\). The authors express the hope that \(3\)-graded Lie subalgebras generated by a pair may be a useful tool for a better understanding of more general \(3\)-graded Lie algebras, like those arising from the Kantor-Koecher-Tits construction and their unitary representations.
scientific article; zbMATH DE number 1878439