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Lie algebras graded by the root system $$BC_1$$. (English) Zbl 1015.17028
Lie algebras graded by a root system $$\Delta$$ were introduced by S. Berman and R. V. Moody [Invent. Math. 108, 323-347 (1992; Zbl 0778.17018)], motivated by the study of the intersection matrix algebras which arose in Slodowy’s work on singularities. Essentially, a $$\Delta$$-graded Lie algebra $$L$$ contains a split simple finite-dimensional Lie algebra $$\mathfrak{g}$$ with root system $$\Delta$$, so that a Cartan subalgebra $$\mathfrak{h}$$ of $$\mathfrak{g}$$ acts diagonally on the whole $$L$$ with $$\Delta\cup \{ 0\}$$ as eigenvalues (plus a technical restriction).
Through the work of several authors (Berman, Moody, Benkart, Zelmanov, Neher, Allison, Gao) these algebras have been described in terms of the Lie algebra $$\mathfrak{g}$$, some of its irreducible submodules and some “coordinate algebras”. Depending on $$\Delta$$, associative, alternative and Jordan algebras play a role in the description.
Some interesting Lie algebras, including some finite-dimensional simple Lie algebras over nonalgebraically closed fields of characteristic $$0$$, the “odd symplectic” Lie algebras of Gelfand and Zelevinsky and of Proctor, and certain intersection matrix algebras provide examples of Lie algebras graded by the nonreduced root systems $$BC_r$$, where the grading subalgebra $$\mathfrak{g}$$ may be of type $$B$$, $$C$$ or $$D$$. B. N. Allison, G. Benkart and Y. Gao [Mem. Am. Math. Soc. 751 (2002; Zbl 0998.17031)] described the $$BC_r$$-graded Lie algebras for $$r\geq 2$$, where the case $$r=2$$ is especially involved. The paper under review is devoted to completing this program by describing the $$BC_1$$-graded Lie algebras.
The most important feature of the $$BC_1$$-graded Lie algebras is that they are $$5$$-graded: $$L=L_{-2}\oplus L_{-1}\oplus L_0\oplus L_1\oplus L_2$$, so the authors first study the “Jordan systems” formed by the $$L_i$$’s ($$i\neq 0$$), which they call “Jordan-Kantor pairs”. These are the coordinate algebras that appear in this setting. In case the grading subalgebra $$\mathfrak{g}$$ is of type $$B_1$$ this Jordan-Kantor pair can be described in terms of a structurable algebra [see B. N. Allison, Math. Ann. 237, 133-156 (1978; Zbl 0368.17001)], while for $$\mathfrak{g}$$ of type $$C_1$$ this can be done in terms of $$J$$-ternary algebras [B. N. Allison, Am. J. Math. 98, 285-294 (1976; Zbl 0342.17008)].
The full description of the $$BC_1$$-graded Lie algebras is given in a very nice way, in spite of the many technical difficulties. Moreover, the central extensions, derivations and invariant forms of these algebras are determined, too.