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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. This is the normal subgroup of the general linear group given by the kernel of the determinant
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.
special orthogonal group: real orthogonal matrices with determinant 1 Y 0 Z n=2 Z 2 n>2 Spin(n) n>2 SO(1) is a single point and SO(2) is isomorphic to the circle group, SO(3) is the rotation group of the sphere. so(n) n(n−1)/2 SE(n) special euclidean group: group of rigid body motions in n-dimensional space. N 0 se(n) n + n(n−1)/2 Spin(n)
Then C × is the internal direct product of the circle group T of unit complex numbers and the group R + of positive real numbers under multiplication. If n is odd, then the general linear group GL(n, R) is the internal direct product of the special linear group SL(n, R) and the subgroup consisting of all scalar matrices.
The generator of any continuous symmetry implied by Noether's theorem, the generators of a Lie group being a special case. In this case, a generator is sometimes called a charge or Noether charge, examples include: angular momentum as the generator of rotations, [3] linear momentum as the generator of translations, [3]
The group GL n (K) itself; The special linear group SL n (K) (the subgroup of matrices with determinant 1); The group of invertible upper (or lower) triangular matrices; If g i is a collection of elements in GL n (K) indexed by a set I, then the subgroup generated by the g i is a linear group.
The projective special linear group, PSL, is defined analogously, as the induced action of the special linear group on the associated projective space. Explicitly: PSL(V) = SL(V) / SZ(V) where SL(V) is the special linear group over V and SZ(V) is the subgroup of scalar transformations with unit determinant.