Irreducible module decompositions of rank 2 symmetric hyperbolic Kac-Moody Lie algebras by $\mathfrak{sl}_2$ subalgebras which are generalizations of principal $\mathfrak{sl}_2$ subalgebras
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説明
There exist principal $\mathfrak{sl}_2$ subalgebras for hyperbolic Kac-Moody Lie algebras. In the case of rank 2 symmetric hyperbolic Kac-Moody Lie algebras, certain $\mathfrak{sl}_2$ subalgebras are constructed. These subalgebras are generalizations of principal $\mathfrak{sl}_2$ subalgebras. We show that the rank 2 symmetric hyperbolic Kac-Moody Lie algebras themselves are irreducibly decomposed under the action of this $\mathfrak{sl}_2$ subalgebras. Furthermore, we classify irreducible components of the decomposition. In particular, we obtain multiplicities of unitary principal series and complementary series.
28 pages