## H. Tachikawa, Claus M. Ringel's Quasi-Frobenius Rings and Generalizations QF-3 and QF-1 PDF

By H. Tachikawa, Claus M. Ringel

ISBN-10: 3540065016

ISBN-13: 9783540065012

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Extra info for Quasi-Frobenius Rings and Generalizations QF-3 and QF-1 Rings

Example text

I) If g is a non-commutative simple complex Lie algebra, then ν is non-trivial precisely in the cases g = Al , l ≥ 2; D2m+1 , m ≥ 1; E6 . In these cases, ν is the only non-trivial symmetry of the Dynkin diagram. s (ii) If g = i=1 gi , where gi are simple, then ν ∈ Aut Π induces the corresponds ing involution νi on each component of the decomposition Π = i=1 Πi , where Πi is the system of simple roots of gi , i = 1, . . 23)). (iii) ν lies in the centre Z(Aut Π). Proof. (i) We have only to investigate the cases g = Al , l ≥ 2, g = Dl , l ≥ 3, and g = E6 , since for other simple g the Dynkin diagram has no symmetries (see Table 1).

Choosing h0 ∈ t in such a way that αi (h0 ) = log ci , i = 1, . . , l, we see that θ and exp(ad h0 ) coincide on the generators ei , fi . Hence θ = exp(ad h0 ) ∈ T . We are now going to extend this semidirect decomposition to the entire group Aut g. Theorem 1. Let g be a complex semisimple Lie algebra. Then Aut g = Int g Ψ(Aut Π) , (3) and the corresponding projection Aut g → Aut Π coincides with Φ on Aut(g, t, Π). Proof. First we prove that Aut g = (Int g) Aut(g, t, Π). 1), there exists ϕ ∈ Int g such that ϕθ(t) = t.

Z ∈ p. Since the mapping µ(id × exp ad) : K × p → Aut g0 is a homeomorphism, it should map the connected components of K × p onto those of Aut g0 . It follows that K ◦ × p is mapped onto Int g0 . Hence µ(K ◦ × P ) = Int g0 . The decomposition Int g0 = K ◦ P implies the last assertion of the theorem. 40 §5. Cartan decompositions and maximal compact subgroups Corollary. The subgroups K and K ◦ are maximal compact subgroups of the Lie groups Aut g0 and Int g0 , respectively. Proof. Suppose that we have a linear group L such that K L ⊂ Aut g0 .