2-Generator conditions in linear groups by Wehrfritz B.A.F.

By Wehrfritz B.A.F.

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Funct. Anal.

We have to distinguish two cases. First, let us suppose that for any simple root α ∈ , the roots α and σc (α) are not connected in the Dynkin diagram of . In this case, one can easily show that σc (eα ) = eσc (α) , so that s(α) = 1 for all real roots α ∈ re . For any root α ∈ let us denote by ασc its restriction to the subspace hσc ⊂ h. Then the set {mα ασc | α ∈ re } is the set of real roots of an affine root system which we will denote by σc . Now suppose that there exists some α ∈ such that α = σc (α) are not orthogonal.

J. Reine Angew. Math. : On unstable bundles over elliptic curves. Publ. Res. Inst. Math. Sci. : Infinite-dimensional Lie Algebras. : Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. of Math. :Root Systems and Elliptic Curves. Invent. Math. : Conjugacy classes of non-connected semisimple algebraic groups. PhD thesis, Univ. : Stable principal bundles on a compact Riemann surface. Math. Ann. : On moduli spaces of flat connections with non-simply connected structure group. Nucl. Phys.

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