The L sub 2(11)-subalgebra of the Monster algebra - Decelle, Sophie (Ars mathematica contemporanea, 2014)
We study a subalgebra ?$V$? of the Monster algebra, ?$V_\mathbb{M}$?, generated by three Majorana axes ?$a_x$?, ?$a_y$? and ?$a_z$ ?indexed by the ?$2A$?-involutions ?$x$?, ?$y$? and ?$z$? of ?$\mathbb{M}$?, the Monster simple group. We use the notation ?$V = \langle \langle a_x, a_y, a_z \rangle \rangle$?. We assume that ?$xy$? is another ?$2A$?-involution and that each of ?$xz$?, ?$yz$? and ?$xyz$? has order 5. Thus a subgroup ?$G$? of ?$\mathbb{M}$? generated by ?$\{x, y, z\}$? is a non-trivial quotient of the group ?$G^{(5, 5, 5)} = \langle x, y, z | x^2, y^2, (xy)^2, z^2, (xz)^5, (yz)^5, (xyz)^5 \rangle$?. It is known that ?$G^{(5, 5, 5)}$? is isomorphic to the projective special linear group ?$L_2(11)$? which is simple, so that ?$G$? is isomorphic to ?$L_2(11)$?. It was proved by S. Norton that (up to conjugacy) ?$G$? is the unique ?$2A$?-generated ?$L_2(11)$?-subgroup of? $V_\mathbb{M}$? and that? $K = C_\mathbb{M}(G)$? is isomorphic to the Mathieu group ?$M_{12}$?. For any pair ?$\{t, s\}$? of ?$2A$?-involutions, the pair of Majorana axes ?$\{a_t, a_s\}$? generates the dihedral subalgebra ?$\langle \langle a_t, a_s \rangle \rangle$? of ?$V_\mathbb{M}$?, whose structure has been described in S. P. Norton, The Monster algebra, some new formulae, Contemp. Math. 193 (1996), 297306. In particular, the subalgebra ?$\langle \langle a_t, a_s \rangle \rangle$? contains the Majorana axis ?$a_{tst}$? by the conjugacy property of dihedral subalgebras. Hence from the structure of its dihedral subalgebras, ?$V$? coincides with the subalgebra of? $V_\mathbb{M}$? generated by the set of Majorana axes ?$\{a_t | t \in T\}$?, indexed by the 55 elements of the unique conjugacy class ?$T$? of involutions of ?$G \cong L_2(11)$?. We prove that ?$V$? is 101-dimensional, linearly spanned by the set? $\{a_t · a_s | s, t \in T\}$?, and with ?$C_{V_\mathbb{M}}(K) = V \oplus ι_\mathbb{M}$?, where ?$ι_\mathbb{M}$? is the identity of ?$V_\mathbb{M}$?. Lastly we present a recent result of Á. Seress proving that ?$V$? is equal to the algebra of the unique Majorana representation of ?$L_2(11)$?.
Keywords: algebra; Conway-Griess-Norton algebra; grupe; Majorana representation; Monster group