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Maximal indexes of Tits algebras. (English) Zbl 0854.20060

Summary: Let \(G\) be a split simply connected semisimple algebraic group over a field \(F\) and let \(C\) be the center of \(G\). It is proved that the maximal index of the Tits algebras of all inner forms of \(G_L\) over all field extensions \(L/F\) corresponding to a given character \(\chi\) of \(C\) equals the greatest common divisor of the dimensions of all representations of \(G\) which are given by the multiplication by \(\chi\) being restricted to \(C\). An application to the discriminant algebra of an algebra with an involution of the second kind is given.

MSC:

20G15 Linear algebraic groups over arbitrary fields
14F22 Brauer groups of schemes
20G05 Representation theory for linear algebraic groups
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
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