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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.
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 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.
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 .
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 classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special [1] automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. [2]
Conditions (1)–(3) are linear and can all be understood in terms of ordinary Lie algebras. Condition (4) is nonlinear, and is the most difficult one to verify when constructing a Lie superalgebra starting from an ordinary Lie algebra ( g 0 {\displaystyle {\mathfrak {g}}_{0}} ) and a representation ( g 1 {\displaystyle {\mathfrak {g}}_{1}} ).
These two basic examples of commutative linear algebraic groups, the multiplicative and additive groups, behave very differently in terms of their linear representations (as algebraic groups). Every representation of the multiplicative group G m {\displaystyle \mathbf {G} _{\mathrm {m} }} is a direct sum of irreducible representations .