The characters of involutive automorphisms of simple Lie algebras

The paper deals with the arbitrary finite-dimensional irreducible representations of simple complex Lie algebras g of types B2 and G2 and real
forms so(1,4), so(3,2) and G of these algebras. Involutive automorphisms
θ on Cartan subalgebras of these algebras are considered. Formulae for
characters value χ(θ) are obtained. This allows us to find the number of
linearly independent space-like and time-like vectors in the representation
space

Consider a simple complex Lie algebra g and an irreducible finite-dimensional
representation φ : g → sl(V). Denote by gr the real form of inner type for
algebra g. Then φ(gr) ⊆ su(p,q), where p − q = δ(gr) is the signature of
the invariant Hermitian form on V. It is possible to find δ(gr) using Weyl
character formula for the representation φ. In [1], F.I.Karpelevich
derived formulae for δ in the case of classical Lie algebras. In [2], Lie algebras
su(p,q) are considered convenient tables for δ in the case a small rank of
g is found. In [3, 4], formulae for |δ| in the case of gr = G,FI,FII,so(p,q)
were presented. Nevertheless, in applications, the exact tables including the
sign of δ are necessary. We
obtain in this paper the tables of δ in terms of the marks of the highest weight
of representation φ. In addition, a similar table for any representation of exceptional
Lie algebras of type G2 is derived. And it is not necessary to know the system
of all weights of the representation φ

References

\bibitem{1}F.I.Karpelevich:"Simple subalgebras of real Lie algebras", Trudy

Mosk.Mat.Obshch., Vol.4,(1955), pp.3-112.

\bibitem{2}J.Patera and R.T.Sharp:"Signatures of finite $\mathfrak{su}(p,q)$ representations",

J.Math.Phys., Vol.25,(1984),pp.2128-2131.MR0748387(85j:22042).

\bibitem{3}A.N.Rudy:"Signatures of finite representation of real, simple Lie algebras", J.Phys.A:Math.Gen., Vol.26,(1993),

pp.5873-5880.MR1252794(94i:17014).

\bibitem{4}A.N.Rudy:"Signatures of finite classical Lie algebra representations", J.Phys.A:Math.Gen., Vol.28,(1995),

pp.1641-1653.MR1338050(96e:17017).