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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, with the group operations of ordinary matrix multiplication and matrix inversion.
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.
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.
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 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 .
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 .
For example, a linear algebraic group is solvable if it has a composition series of linear algebraic subgroups such that the quotient groups are commutative. Also, the normalizer , the center , and the centralizer of a closed subgroup H of a linear algebraic group G are naturally viewed as closed subgroup schemes of G .
As the adjoint representation is injective, a semisimple Lie algebra is a linear Lie algebra under the adjoint representation. This may lead to some ambiguity, as every Lie algebra is already linear with respect to some other vector space (Ado's theorem), although not necessarily via the adjoint representation. But in practice, such ambiguity ...