Dvorsky, Alexander Index of parabolic and seaweed subalgebras of \(\mathfrak{sn}_n\). (English) Zbl 1056.17009 Linear Algebra Appl. 374, 127-142 (2003). Summary: The index of a Lie algebra \(\mathfrak r\) is defined as \(\min_{w\in \mathfrak r^*}\) dim \(\mathfrak r_w\), where \(\mathfrak r_w\) is the stabilizer of \(w\) under the coadjoint action of \(\mathfrak r\) on \(\mathfrak r^*\). We derive simple formulas for the index of parabolic subalgebras of the even orthogonal algebra \(\mathfrak g=\mathfrak{so}(sn,\mathbf k)\), \({\mathbf k}\) denotes an algebraically closed field. We use these formulas to prove the conjecture that ind \(\mathfrak q < n\) for any parabolic subalgebra or seaweed subalgebra (intersection of two weakly opposite parabolics) \(\mathfrak q\) of \(\mathfrak{so}(2n,\mathbf k)\). Some partial results for subalgebras of \(\mathfrak g=\mathfrak{so}(2n=1,\mathbf k)\) are also obtained. Cited in 6 Documents MSC: 17B20 Simple, semisimple, reductive (super)algebras 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) Keywords:Parabolic subalgebras; Seaweed subalgebras; Lie algebra index; Frobenius Lie algebra × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Dergachev, V.; Kirillov, A., Index of Lie algebras of seaweed type, J. Lie Theory, 10, 331-343 (2000) · Zbl 0980.17001 [2] Dixmier, J., Algebres enveloppantes (1974), Gauthier-Villars: Gauthier-Villars Paris-Montreal · Zbl 0308.17007 [3] Dvorsky, A., Generic representations of parabolic subgroups. Case \(I-- SO(2n, C)\), Comm. Alg., 23, 1369-1402 (1995) · Zbl 0833.20052 [4] Elashvili, A., On the index of horospherical subalgebras of semisimple Lie algebras, Tr. Tbiliss. Mat. Inst., 77, 116-126 (1985), (in Russian) · Zbl 0626.17004 [5] Ooms, A., On Frobenius Lie algebras, Comm. Alg., 8, 13-52 (1980) · Zbl 0421.17004 [6] Panyushev, D., Inductive formulas for the index of seaweed Lie algebras, Mosc. Math. J., 1, 221-241 (2001) · Zbl 0998.17008 [7] Raı̈s, M., L’indice des produits semi-directs \(E×ρg\), C. R. Acad. Sci. Paris, 287, 195-197 (1978) · Zbl 0387.17002 [8] Wolf, J. A., Unitary representations of maximal parabolic subgroups of the classical groups, Mem. Amer. Math. Soc., 8 (180), 1-193 (1976) · Zbl 0344.22016 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.