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SL(2, R) is the group of all linear transformations of R 2 that preserve oriented area. It is isomorphic to the symplectic group Sp(2, R) and the special unitary group SU(1, 1). It is also isomorphic to the group of unit-length coquaternions. The group SL ± (2, R) preserves unoriented area: it may reverse orientation.
In mathematics, the special linear group SL(n, R) of degree n over a commutative ring R is the set of n × n matrices with determinant 1, ...
In mathematics, the special linear Lie algebra of order over a field, denoted or (,), is the Lie algebra of all the matrices (with entries in ) with trace zero and with the Lie bracket [,]:= given by the commutator. This algebra is well studied and understood, and is often used as a model for the study of other Lie algebras.
Considering their abstract group theory, many linear groups have a "special" subgroup, usually consisting of the elements of determinant 1 over the ground field, and most of them have associated "projective" quotients, which are the quotients by the center of the group. For orthogonal groups in characteristic 2 "S" has a different meaning.
In the theory of Lie algebras, an sl 2-triple is a triple of elements of a Lie algebra that satisfy the commutation relations between the standard generators of the special linear Lie algebra sl 2. This notion plays an important role in the theory of semisimple Lie algebras , especially in regard to their nilpotent orbits .
This article gives a table of some common Lie groups and their associated Lie algebras.. The following are noted: the topological properties of the group (dimension; connectedness; compactness; the nature of the fundamental group; and whether or not they are simply connected) as well as on their algebraic properties (abelian; simple; semisimple).
In mathematics, the projective special linear group PSL(2, 7), isomorphic to GL(3, 2), is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane .
The families are in turn indexed by special representations of the Weyl group, or equivalently by 2-sided cells of the Weyl group. For example, the group E 8 (F q) has 46 families of unipotent characters corresponding to the 46 special representations of the Weyl group of E 8. There are 23 families with 1 character, 18 families with 4 ...