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Isotropic trialitarian algebraic groups. (English) Zbl 0920.20037

The four infinite families of absolutely simple affine algebraic groups, namely, groups of type \(A_n\), \(B_n\), \(C_n\) and \(D_n\), over a field \(F\) of characteristic \(\neq 2\) are more or less well understood except for the groups of type \(D_4\). Also, the groups of type \({^1D_4}\) and \({^2D_4}\) may be considered to be well understood. The author of this paper focuses his attention on the so-called trialitarian groups, i.e., groups of type \({^3D_4}\) and \({^6D_4}\). For any trialitarian group \(G\) defined over \(F\), there is a separable cubic extension \(L\) of \(F\), determined up to \(F\)-algebra isomorphism, such that \(G\) is of type \({^1D_4}\) over the Galois closure of \(L\) over \(F\). There is a central simple algebra over \(L\) of degree 8, also determined up to \(F\)-algebra isomorphism, called Allen invariant of \(G\). By modifying a construction from M.-A. Knus et al. [The Book of Involutions (Colloq. Publ. 44, Am. Math. Soc., Providence, RI) (1998)], the author constructs all isotropic algebraic groups of type \({^3D_4}\) and \({^6D_4}\) over any field of characteristic \(\neq 2\). Also, the author provides an isomorphism criterion for such groups. Here, the Allen invariants play an important role.
Reviewer: Li Fuan (Beijing)

MSC:

20G15 Linear algebraic groups over arbitrary fields
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References:

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