View source for Adjoint group

''of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a0108301.png" />''

The linear group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a0108302.png" /> that is the image of the Lie group or algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a0108303.png" /> under the adjoint representation (cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]). The adjoint group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a0108304.png" /> is contained in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a0108305.png" /> of automorphisms of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a0108306.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a0108307.png" />, and its Lie algebra coincides with the adjoint algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a0108308.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a0108309.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a01083010.png" /> is connected, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a01083011.png" /> is uniquely determined by the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a01083012.png" /> and is either called the adjoint group or the group of inner automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a01083013.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a01083014.png" /> is semi-simple, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a01083015.png" /> coincides with the connected component of the identity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a01083016.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,   "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)  {{MR|0201557}} {{ZBL|0882.01025}} {{ZBL|0534.22001}} {{ZBL|0079.03903}} {{ZBL|0058.26003}} {{ZBL|0022.17104}} {{ZBL|0016.20305}} {{ZBL|0016.20304}} {{ZBL|0015.24901}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.-P. Serre,   "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)  {{MR|0218496}} {{ZBL|0132.27803}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.E. Humphreys,   "Linear algebraic groups" , Springer  (1975)  {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  pp. Chapt. 2; 3  (Translated from French)  {{MR|0682756}} {{ZBL|0319.17002}} </TD></TR></table>