Special linear group

Matrix Groups

Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006

The special linear group SL(n, F) is the subgroup of GL(n, F) consisting of matrices of determinant 1. If V is a vector space of dimension n over F, the group of invertible linear operators on V having determinant 1 is denoted by SL(V).

Symmetries and Group Theory

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Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018

Mattias Blennow

Problem 4.29. The special linear group of complex 2 × 2 matrices SL(2, ℂ) is defined as the subset of complex 2 × 2 matrices that have determinant one. Verify that this set forms a subgroup of GL(2, ℂ) and determine the constraints on the matrices in its Lie algebra.

Estimates for the volume variation of compact submanifolds driven by a stochastic flow

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Published in Dynamical Systems, 2022

Diego Sebastian Ledesma, Robert Andres Galeano Anaya, Fabiano Borges da Silva

Let be the special linear group of matrices with determinant 1 and its compact subgroup of orthogonal matrices. If and denote their respective Lie algebras, then by the Cartan decomposition we have Note that is a basis for such that For some we can write , where , and therefore and Thus, in the representation as corollary above, we get Then the flow associated with Y preserves the volume of .

Special linear group

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Matrix Groups

Symmetries and Group Theory

Estimates for the volume variation of compact submanifolds driven by a stochastic flow