In the mathematical theory of linear algebraic groups, a Tits index (or index) is an object used to classify semisimple algebraic groups defined over a base field k, not assumed to be algebraically closed.
The possible irreducible indices were classified by Jacques Tits,[1] and this classification is reproduced below.
This representation captures the full information of the index except when the underlying Dynkin diagram is D4, in which case one must distinguish between an action by the cyclic group C3 or the permutation group S3.
Alternatively, an index can be represented using the name of the underlying Dynkin diagram together with additional superscripts and subscripts, to be explained momentarily.
This representation, together with the labeled Dynkin diagram described in the previous paragraph, captures the full information of the index.
The notation for an index is of the form gXtn,r, where In the description, there are given (only for classical groups), a representative of the isogeny class of the group of the given Tits index.
Description: Algebraic group: The special unitary group SU(n+1)/d(D,h), where D is a central division algebra of degree d over a separable quadratic extension k' of k, and where h is a nondegenerate hermitian form of index r relative to the unique non-trivial k-automorphism of k' .
Full name: Bn,r Image: Conditions: None.
Distinguished vertices: 1, 2,...,r. Description: Algebraic group: The special orthogonal group SO2n+1(k,q), where q is a quadratic form of index r, and defect 1 if k has characteristic 2.
Description: Algebraic group: The special unitary group SU2n/d(D,h), where D is a division algebra of degree d over k and h is a nondegenerate antihermitian form relative to a k-linear involution σ of D (also called an "involution of the first kind") such that the fixed-point subring Dσ has dimension d(d + 1)/2; or equivalently, when d > 1 and char k ≠ 2, the group SU2n/d where D and h are as above except that h is hermitian and D has dimension d(d − 1)/2.
Description: Algebraic group: If k has characteristic ≠ 2, the special unitary group SU2n/d(D,h), where D is a division algebra of degree d over k and h is a hermitian form of discriminant 1 and index r, relative to a k-linear involution σ of D, an "involution of the first kind such that the subring Dσ has dimension d(d + 1)/2; or equivalently, when d > 1 and char k ≠ 2, the group SU2n/d where D and h are as above except that h is anti-hermitian form of discriminant 1 and index r, and D has dimension d(d − 1)/2.
Description: Algebraic group: The same as for 1D(d)n,r, except that all forms in question have now discriminant ≠ 1.
Image: Description: Algebraic Group: The automorphism group of an exceptional simple Jordan algebra J that does not contain nonzero nilpotent elements.
Image: Description: Algebraic Group: The automorphism group of an exceptional simple Jordan algebra J containing nonzero nilpotent elements, no two of which are nonproportional and orthogonal.
Image: Description: Algebraic Group: The automorphism group of an exceptional simple Jordan algebra J containing nonproportional orthogonal nilpotent elements.