Adjoint group

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of a group

The linear group that is the image of the Lie group or algebraic group under the adjoint representation (cf. Adjoint representation of a Lie group). The adjoint group is contained in the group of automorphisms of the Lie algebra of , and its Lie algebra coincides with the adjoint algebra of . A connected semi-simple group is a group of adjoint type (i.e. is isomorphic to its adjoint group) if and only if its roots generate the lattice of rational characters of the maximal torus; the centre of such a group is trivial. If the ground field has characteristic zero and is connected, then is uniquely determined by the Lie algebra and is either called the adjoint group or the group of inner automorphisms of . In particular, if is semi-simple, coincides with the connected component of the identity in .


[1] L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) MR0201557 Zbl 0882.01025 Zbl 0534.22001 Zbl 0079.03903 Zbl 0058.26003 Zbl 0022.17104 Zbl 0016.20305 Zbl 0016.20304 Zbl 0015.24901
[2] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) MR0218496 Zbl 0132.27803
[3] J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039



[a1] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) pp. Chapt. 2; 3 (Translated from French) MR0682756 Zbl 0319.17002
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Adjoint group. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article