Semisimple elements of a lie algebra - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T13:16:16Z http://mathoverflow.net/feeds/question/58063 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58063/semisimple-elements-of-a-lie-algebra Semisimple elements of a lie algebra Michele Torielli 2011-03-10T12:07:24Z 2011-03-10T13:18:59Z <p>Let \$G\subset GL_n(\mathbb{C})\$ be an algebraic group of dimension n, and let \$\mathfrak{g}\$ its Lie algebra.Is there a relations between the maximal number of independent semisimple elements of \$G\$ and the maximal number of independent semisimple elements \$\mathfrak{g}\$?</p> http://mathoverflow.net/questions/58063/semisimple-elements-of-a-lie-algebra/58070#58070 Answer by Jim Humphreys for Semisimple elements of a lie algebra Jim Humphreys 2011-03-10T13:18:59Z 2011-03-10T13:18:59Z <p>Assuming you mean "linearly independent" here, it's important to understand that the maximal numbers in question depend on the chosen faithful linear representation of the group rather than on its intrinsic properties as an algebraic group. For example, there are many conjugate maximal tori (of the same dimension) living inside the full upper triangular matrix group, but such a torus may or may not consist of diagonal matrices depending on the choice you make.</p> <p>Also, the maximal number of independent matrices in the Lie algebra may well differ, as seen in the case of the trivial algebraic group whose Lie algebra is zero. </p> <p>I'm not at all sure what the question has to do with Lie theory (?)</p>