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  • The ▫$L_2(11)$▫-subalgebra of the Monster algebra
    Decelle, Sophie
    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▫ $\{a_t \cdot a_s | s, t \in T\}$▫, and with ▫$C_{V_\mathbb{M}}(K) = V \oplus \iota_\mathbb{M}$▫, where ▫$\iota_\mathbb{M}$▫ 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 ▫$L_2(11)$▫.
    Vir: Ars mathematica contemporanea : special issue Bled'11 (Vol. 7, no. 1, 2014, str. 83-103)
    Vrsta gradiva - prispevek na konferenci ; neleposlovje za odrasle
    Leto - 2014
    Jezik - angleški
    COBISS.SI-ID - 16793433

    Povezava(-e):

    http://amc-journal.eu/index.php/amc/article/view/259/240
    Digitalna knjižnica Slovenije - dLib.si

vir: Ars mathematica contemporanea : special issue Bled'11 (Vol. 7, no. 1, 2014, str. 83-103)

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