2014/02/19 by Anthony Joseph, Joseph, Anthony
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.1402.4681
Let \mathfrak a be an algebraic Lie algebra. An adapted pair for \mathfrak a is pair (h,η) consisting of an ad-semisimple element of h ∈ \mathfrak a and a regular element of η∈ \mathfrak a^* satisfying (ad h)η=-η. An adapted pair (h,η) is said to satisfy integrality if ad h has integer eigenvalues on \mathfrak a. Integrality is shown to hold for any Frobenius Lie algebra which is a biparabolic subalgebra of a semisimple Lie algebra; but may fail in general. Call \mathfrak a regular if there are no proper semi-invariant polynomial functions on \mathfrak a^* and if the subalgebra of invariant functions is polynomial. In this case there are no known counter-examples to integrality. It is shown that if \mathfrak a is the canonical truncation of a biparabolic subalgebra of a simple Lie algebra \mathfrak g which is regular and admits an adapted pair (h,η), then the eigenvalues of ad h on \mathfrak a lie in (1)/(m)\mathbb Z, where m is a coefficient of a simple root in the highest root of \mathfrak g. Let \mathfrak a be a regular Lie algebra admitting an adapted pair (h,η). Let \mathfrak a_\mathbb Z be the subalgebra spanned by the eigensubspaces of ad h with integer eigenvalue. It is shown that the canonical truncation of \mathfrak a_\mathbb Z is regular. Sufficient knowledge of the relation between the generators for the invariant polynomial functions on \mathfrak a^* and on \mathfrak a^*_\mathbb Z can then lead to establishing the integrality of (h,η). This method is used to show the integrality of an adapted pair for a truncated parabolic subalgebra of a simple Lie algebra of type C.