View source for Lie algebra, exceptional

A simple Lie algebra (see [[Lie algebra, semi-simple|Lie algebra, semi-simple]]) that is not classical. Over an algebraically closed field of characteristic zero there are 5 exceptional Lie algebras: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058390/l0583901.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058390/l0583902.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058390/l0583903.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058390/l0583904.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058390/l0583905.png" />, of dimension 78, 133, 248, 52, and 14, respectively. The indices are the ranks of these Lie algebras. The simplest linear representations of these exceptional Lie algebras have dimensions 27, 56, 248, 26, and 7, respectively. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058390/l0583906.png" /> is the algebra of derivations (cf. [[Derivation in a ring|Derivation in a ring]]) of the [[Cayley–Dickson algebra|Cayley–Dickson algebra]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058390/l0583907.png" /> is the algebra of derivations of the unique exceptional [[Jordan algebra|Jordan algebra]], which can be represented as the algebra of Hermitian matrices of order three over the Cayley–Dickson algebra. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058390/l0583908.png" /> is the linear envelope of the derivations and the multiplications by elements of the exceptional Jordan algebra. The algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058390/l0583909.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058390/l05839010.png" /> and the forms of all the exceptional Lie algebras over algebraically non-closed fields are also connected with the Cayley–Dickson algebra. The various models of exceptional Lie algebras are obtained by considering their gradings by means of cyclic groups (cf. also [[Graded algebra|Graded algebra]]). The connected Lie groups corresponding to the exceptional Lie algebras, called exceptional groups, are often denoted by the same letters. For example, the complex group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058390/l05839011.png" /> is the group of automorphisms of the Cayley–Dickson algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058390/l05839012.png" />; the complex group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058390/l05839013.png" /> is the group of automorphisms of the exceptional Jordan algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058390/l05839014.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,   "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Jacobson,   "Exceptional Lie algebras" , M. Dekker  (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Freudenthal,   "Oktaven, Ausnahmengruppen und Oktavengeometrie" , Math. Inst. Univ. Utrecht  (1960)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B.A. Rozenfel'd,   "Einfache Lie-gruppen und nichteuklidische Geometrie" , ''Algebraical and topological foundations of geometry'' , Pergamon  (1962)  pp. 135–155</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.B. Vinberg,   "A construction of the exponential simple Lie algebras"  ''Trudy Sem. Vektor. Tenzor. Anal.'' , '''13'''  (1966)  pp. 7–9  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J. Tits,   "Algèbres alternatives, algèbres de Jordan et algèbres de Lie exceptionelles I. Construction"  ''Indag. Mat.'' , '''28'''  (1966)  pp. 233–237</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  J. Tits,   "Tabellen zu den einfachen Lie Grupppen und ihren Darstellungen" , ''Lect. notes in math.'' , '''40''' , Springer  (1967)</TD></TR></table>



====Comments====
See [[Lie algebra, semi-simple|Lie algebra, semi-simple]] for the Cartan matrices and Dynkin diagrams of the exceptional Lie algebras. The same article also contains information on the classification of simple Lie algebras over arbitrary fields, in particular over the real field.