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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
The group GL(n, F) and its subgroups are often called linear groups or matrix groups (the automorphism group GL(V) is a linear group but not a matrix group). These groups are important in the theory of group representations , and also arise in the study of spatial symmetries and symmetries of vector spaces in general, as well as the study of ...
The group SL(2, R) acts on its Lie algebra sl(2, R) by conjugation (remember that the Lie algebra elements are also 2 × 2 matrices), yielding a faithful 3-dimensional linear representation of PSL(2, R). This can alternatively be described as the action of PSL(2, R) on the space of quadratic forms on R 2. The result is the following representation:
For example, the group PSL 2 (R) is not a group of 2 × 2 matrices, but it has a faithful representation as 3 × 3 matrices (the adjoint representation), which can be used in the general case. Many Lie groups are linear, but not all of them. The universal cover of SL 2 (R) is not linear, as are many solvable groups, for instance the quotient of ...
Special groups include the general linear group, the special linear group, and the symplectic group. Special groups are necessarily connected. Products of special groups are special. The projective linear group is not special because there exist Azumaya algebras, which are trivial over a finite separable extension, but not over the base field.
The rotation group SO(3) is a symmetry of Euclidean space and all fundamental laws of physics, the Lorentz group O(3,1) is a symmetry group of spacetime of special relativity. The special unitary group SU(3) is the symmetry group of quantum chromodynamics and the symplectic group Sp( m ) finds application in Hamiltonian mechanics and quantum ...
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