论文标题
假想的$ \ mathfrak {gl} _2 $ subalgebras in Monster Lie代数代数
Vertex operators for imaginary $\mathfrak{gl}_2$ subalgebras in the Monster Lie Algebra
论文作者
论文摘要
怪物谎言代数$ \ mathfrak m $是顶点代数的物理空间的商对应于等级2的代数甚至单模型晶格$ \ textrm {ii} _ {1,1} $。我们构建了$ \ mathfrak m $ isomorphic of $ \ mathfrak {gl} _ {2} $的$ \ mathfrak m $同构的子代数的顶点代数元素，对应于每个想象的简单根，表示为$（1，j）$ $ j> 0 $。我们的方法要求在满足某些自然条件的$ v^{\ natural} $中存在成对的主要向量，我们证明了这一点。我们表明，在$ v^\ natural $中，怪物有限的简单组$ \ mathbb {m mathbb {m} $在$ v^\ natural $中的子空间中诱导了$ \ mathbb {m} $ - 在$ \ mathfrak {gl} _2 _2 _2 _2 $ subgras的集合上的动作，对应于固定的固定的想象简单的根源。我们将生成函数用于$ v^\ Natural $的主要向量子空间的尺寸，以证明此操作对于$ j $的小值而言是不平凡的。
The Monster Lie algebra $\mathfrak m$ is a quotient of the physical space of the vertex algebra $V=V^\natural\otimes V_{1,1}$, where $V^\natural$ is the Moonshine module vertex operator algebra of Frenkel, Lepowsky, and Meurman, and $V_{1,1}$ is the vertex algebra corresponding to the rank 2 even unimodular lattice $\textrm{II}_{1,1}$. We construct vertex algebra elements that project to bases for subalgebras of $\mathfrak m$ isomorphic to $\mathfrak{gl}_{2}$, corresponding to each imaginary simple root, denoted $(1,j)$ for $j>0$. Our method requires the existence of pairs of primary vectors in $V^{\natural}$ satisfying some natural conditions, which we prove. We show that the action of the Monster finite simple group $\mathbb{M}$ on the subspace of primary vectors in $V^\natural$ induces an $\mathbb{M}$-action on the set of $\mathfrak{gl}_2$ subalgebras corresponding to a fixed imaginary simple root. We use the generating function for dimensions of subspaces of primary vectors of $V^\natural$ to prove that this action is non-trivial for small values of $j$.
