Anisotropic group

2020 Mathematics Subject Classification: Primary: 20.27 [MSN][ZBL]

An anisotropic algebraic group over a field $k$ is a linear algebraic group $G$ defined over $k$ [1] and of $k$-rank zero, i.e. not containing non-trivial $k$-split tori (cf. Splittable group). Classical examples of anisotropic groups include the orthogonal groups of quadratic forms that do not vanish over $k$; and algebraic groups of elements of reduced norm one in division algebras over $k$. If $G$ is semi-simple, and if the characteristic of $k$ is zero, then $G$ is anisotropic over $k$ if and only if $G_k$ contains non-trivial unipotent elements. (For the field of real numbers or the field of $p$-adic numbers this is equivalent to saying that $G_k$ is compact.) The classification of arbitrary semi-simple groups over the field $k$ reduces essentially to the classification of anisotropic groups over $k$ [2].

References

[1]	A. Borel, Linear algebraic groups, Benjamin (1969) | MR0251042 | Zbl 0186.33201
[2]	J. Tits, Classification of algebraic semisimple groups, in Algebraic Groups and Discontinuous Subgroups, Proc. Symp. Pure Math., 9, Amer. Math. Soc. (1966) pp. 33–62 | MR0224710 | Zbl 0238.20052