Constructing a Lie group analog for the Monster Lie algebra

魔群李代数的李群类似物

A Lie group analog for the Monster Lie algebra

本文围绕魔群李代数$\frak m$展开，通过引入新的可和性概念，构造了相关李群类似物$G(\frak m)$及其完备幂幺群$\Uhp$等，还构造了指数映射和伴随表示类似物，证明了魔群$\mathbb{M}$对相关结构的作用及兼容性。

Vertex operators and sporadic groups

In the 1980's, the work of Frenkel, Lepowsky and Meurman, along with that of Borcherds, culminated in the notion of vertex operator algebra, and an example whose full symmetry group is the largest sporadic simple group: the Monster. Thus it was shown that the vertex operators of mathematical physics play a role in finite group theory. In this article we describe an extension of this phenomenon by introducing the notion of enhanced vertex operator algebra, and constructing examples that realize other sporadic simple groups, including one that is not involved in the Monster.

Conformal Field Theories with Sporadic Group Symmetry

The monster sporadic group is the automorphism group of a central charge $c=24$ vertex operator algebra (VOA) or meromorphic conformal field theory (CFT). In addition to its $c=24$ stress tensor $T(z)$, this theory contains many other conformal vectors of smaller central charge; for example, it admits $48$ commuting $c=\frac12$ conformal vectors whose sum is $T(z)$. Such decompositions of the stress tensor allow one to construct new CFTs from the monster CFT in a manner analogous to the Goddard-Kent-Olive (GKO) coset method for affine Lie algebras. We use this procedure to produce evidence for…

Moonshine for Rudvalis's sporadic group I

We introduce the notion of vertex operator superalgebra with enhanced conformal structure, which is a refinement of the notion of vertex operator superalgebra. We exhibit several examples, including a particular one which is self-dual, and whose full symmetry group is a direct product of a cyclic group of order seven with the sporadic simple group of Rudvalis. We thus obtain an analogue of Monstrous Moonshine for a sporadic group not involved in the Monster. Two variable analogues of the usual McKay--Thompson series are naturally associated to the action of the Rudvalis group on this object, a…

On analytic analogues of quantum groups

In this paper we present a new construction of analytic analogues of quantum groups over non-Archimedean fields and construct braided monoidal categories of their representations. We do this by constructing analytic Nichols algebras and use Majid's double-bosonisation construction to glue them together. We then go on to study the rigidity of these analytic quantum groups as algebra deformations of completed enveloping algebras through bounded cohomology. This provides the first steps towards a $p$-adic Drinfel'd-Kohno Theorem, which should relate this work to Furusho's $p$-adic Drinfel'd assoc…

A global analogue of the Springer resolution for $SL_2$

The global nilpotent cone N is a singular stack associated to the choice of an algebraic group G, a smooth projective curve X, and a line bundle L on X, which is of fundamental importance to the Geometric Langlands Program, and which is of emerging importance to the Classical Langlands Program. In analogy with the ordinary Springer resolution, we construct and study a resolution of singularities of N in the special case where G=SL_2. As an immediate application, we prove that N is equidimensional and also provide an enumeration of its irreducible components. We hope this is the first step in c…