how many injective homomorphism between two lie algebra sl2 and sp6 up to conjugate by Sp6? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T17:20:40Z http://mathoverflow.net/feeds/question/36762 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/36762/how-many-injective-homomorphism-between-two-lie-algebra-sl2-and-sp6-up-to-conjuga how many injective homomorphism between two lie algebra sl2 and sp6 up to conjugate by Sp6? TOM 2010-08-26T13:31:22Z 2010-08-26T17:40:34Z <p>how many injective homomorphism between two lie algebra \$sl_2 \$and \$sp_6\$ up to conjugate by\$Sp_6\$ ?</p> http://mathoverflow.net/questions/36762/how-many-injective-homomorphism-between-two-lie-algebra-sl2-and-sp6-up-to-conjuga/36772#36772 Answer by Jim Humphreys for how many injective homomorphism between two lie algebra sl2 and sp6 up to conjugate by Sp6? Jim Humphreys 2010-08-26T15:48:34Z 2010-08-26T15:48:34Z <p>To follow up Skip's comment, assuming the underlying field is algebraically closed of characteristic 0, the number of possible embeddings (= injective homomorphisms) of <code>\$\mathfrak{s}\mathfrak{l}_2\$</code> into <code>\$\mathfrak{s}\mathfrak{p}_6\$</code> up to conjugacy by the adjoint group should be 7. This is the number of nonzero nilpotent conjugacy classes in the simple Lie algebra <code>\$C_3\$</code>, which in turn are in natural bijection (using Jacobson-Morozov theory) with the <code>\$\mathfrak{s}\mathfrak{l}_2\$</code>-triples. A good source for the basic theory is Section 11 of Chapter 8 in the Bourbaki treatise <em>Groupes et algebres de Lie</em> (published in English by Springer), supplemented by data on hilpotent orbits in books like those by Collingwood-McGovern and Carter. As Skip points out, general questions of this sort were studied systematically by Dynkin and later refined or generalized by others. </p> http://mathoverflow.net/questions/36762/how-many-injective-homomorphism-between-two-lie-algebra-sl2-and-sp6-up-to-conjuga/36774#36774 Answer by José Figueroa-O'Farrill for how many injective homomorphism between two lie algebra sl2 and sp6 up to conjugate by Sp6? José Figueroa-O'Farrill 2010-08-26T15:53:23Z 2010-08-26T17:40:34Z <p>As a follow-up to Jim's answer (which came in as I was typing an inferior answer), let me add that the 7 possible embeddings are given in the \$C_3\$ entry of Table VI in the paper: <em><a href="http://www.ams.org/mathscinet-getitem?mr=310139" rel="nofollow">Classification of semisimple subalgebras of simple Lie algebras</a></em> by Lorente and Gruber. It's of course based on Dynkin, but they work out the details up to rank 6.</p> <hr> <p><strong>Added</strong></p> <p>The defining vectors for the 7 embeddings are given by: (1,0,0), (1,1,0), (1,1,1), (2,2,0), (3,1,0), (3,1,1) and (5,3,1). Recall that the embedding with defining vector \$(a,b,c)\$ is one for which the Cartan generator \$H\$ of the \$\mathfrak{sl}(2)\$ subalgebra is given by \$H = a H_1 + b H_2 + c H_3\$, where \$(H_i)\$ is an orthonormal basis of a Cartan subalgebra of \$C_3\$ containing \$H\$.</p>