Matsuo代数的双轴与Monster型子代数
Double axes and subalgebras of Monster type in Matsuo algebras
摘要：
轴代数是一类由幂等元生成的交换非结合代数代数，称为轴代数，伴随作用是半单的，满足一个指定的融合律。轴代数是由Hall、Frederen和Shpectorov \cite{hrs,hrs1}作为Ivanov的Majorana代数的广泛推广而引入的，其公理是从Monster散在单群的Griess代数的性质导出的。Monster型的轴代数类包括Monster和许多其它散在单群的Majorana代数，经典单群和一些例外单群的Jordan代数，以及对应于$3$ -转置群的Matsuo代数。因此，Monster型的轴代数统一了有限单群理论中的几个分支。这里示出了双轴，即，Matsuo代数中两正交轴的和满足Monster型融合律$(2η,η)$ .对由两个单轴或双轴生成的本原子代数进行了完全分类，并将$3$生成的本原子代数分类到这三种情况之一.这些分类进一步导致了一般的翻转构造输出了丰富多样的Monster型轴代数。将翻转构造应用于与对称群相关的Matsuo代数的情形，得到了此类代数的三个新的显式无穷级数。
英文摘要：
Axial algebras are a class of commutative non-associative algebras generated by idempotents, called axes, with adjoint action semi-simple and satisfying a prescribed fusion law. Axial algebras were introduced by Hall, Rehren and Shpectorov \cite{hrs,hrs1} as a broad generalisation of Majorana algebras of Ivanov, whose axioms were derived from the properties of the Griess algebra for the Monster sporadic simple group. The class of axial algebras of Monster type includes Majorana algebras for the Monster and many other sporadic simple groups, Jordan algebras for classical and some exceptional simple groups, and Matsuo algebras corresponding to $3$-transposition groups. Thus, axial algebras of Monster type unify several strands in the theory of finite simple groups. It is shown here that double axes, i.e., sums of two orthogonal axes in a Matsuo algebra, satisfy the fusion law of Monster type $(2η,η)$. Primitive subalgebras generated by two single or double axes are completely classified and $3$-generated primitive subalgebras are classified in one of the three cases. These classifications further lead to the general flip construction outputting a rich variety of axial algebras of Monster type. An application of the flip construction to the case of Matsuo algebras related to the symmetric groups results in three new explicit infinite series of such algebras.
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