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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.

MSC:
17B70 Graded Lie (super)algebras
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