Lin Xianzu
It is well known that to each infinite class of classical groups over a commutative ring $R$, we can associate an infinite loop space by Quillen's plus construction. In this paper we generalize this fact to the case of affine Kac-Moody groups.
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In this paper we determine the rational homotopy type of the classifying space of a generic Kac-Moody group by computing its rational cohomology ring. As an application we determine the rational homology Hopf algebra of the generic Kac-Moody group.
In this paper, we study the rational cohomology rings of indefinite Kac-Moody groups and their flag manifolds. By extracting the information of cohomology from the Poincaré series, we are able to determine the rational cohomology rings of Kac-Moody groups and their flag manifolds. Since Kac-Moody groups and their flag manifolds are rational formal, we also determine their rational homotopy groups and rational homotopy types.
We compute the mod $p$ cohomology algebra of a family of infinite discrete Kac-Moody groups of rank two defined over finite fields of characteristic different from $p$.
In this paper, we compute the rational cohomology groups of the classifying space of a simply connected Kac-Moody group of infinite type. The fundamental principle is "from finite to infinite". That is, for a Kac-Moody group G(A) of infinite type, the input data for computation are the rational cohomology of classifying spaces of parabolic subgroups of G(A)(which are of finite type), and the homomorphisms induced by inclusions of these subgroups. In some special cases, we can further determine the cohomology rings. Our method also applies to study the mod p cohomology of the classifying spaces…
Cohomology of classifying spaces of rank 3 Kac-Moody groups
本文用Weyl群不变量及其商群的直和刻画秩3 Kac-Moody群分类空间的有理数和模p上同调，证明每个素数p都对应p-挠元，并基本确定了有理数上同调的环结构。
We give a topological interpretation of the highest weight representations of Kac-Moody groups. Given the unitary form G of a Kac-Moody group (over C), we define a version of equivariant K-theory, K_G on the category of proper G-CW complexes. We then study Kac-Moody groups of compact type in detail (see Section 2 for definitions). In particular, we show that the Grothendieck group of integrable hightest weight representations of a Kac-Moody group G of compact type, maps isomorphically onto K_G^*(EG), where $EG$ is the classifying space of proper G-actions. For the affine case, this agrees very…
We construct a map from the classifying space of a discrete Kac-Moody group over the algebraic closure of the field with p elements to the classifying space of a complex topological Kac-Moody group and prove that it is a homology equivalence at primes q different from p. This generalises a classical result of Quillen-Friedlander-Mislin for Lie groups. As an application, we construct unstable Adams operations for general Kac-Moody groups compatible with the Frobenius homomorphism. In contrast to the Lie case, the homotopy fixed points of these unstable Adams operations cannot be realized at q a…
In this paper we discuss the isomorphism types of parabolic subgroups in Kac-Moody groups. The results have applications in the study of topology of Kac-Moody groups and their classifying spaces.
In this paper, we prove that the ring of polynomial invariants of the Weyl group for an indecomposable and indefinite Kac-Moody Lie algebra is generated by invariant symmetric bilinear form or is trivial depending on $A$ is symmetrizable or not. The result was conjectured by Moody and assumed by Kac. As applications we discuss the rational homotopy types of Kac-Moody groups and their flag manifolds.
Higher homotopy groups of topological Kac-Moody groups, Whitehead tower, and string groups
本文证明拓扑分裂实Kac-Moody群的同伦群可数，从而推出其Whitehead塔存在，并给出$E_n$系列弦群的构造方法。
We study spin structures on affine Kac-Moody symmetric spaces and obtain sufficient conditions for their existence.\ As a by product of this, we obtain a spin-c representation of certain Kac-Moody quadratic subgroups of type E.
In this paper, we discuss the Poincaré series of Kac-Moody Lie algebras, especially for indefinite type. Firstly, we compute the Poincaré series of certain indefinite Kac-Moody Lie algebras whose Cartan matrices have the same type of $2\times 2$ principal sub-matrices. Secondly, we show that the Poincaré series of Kac-Moody Lie algebras satisfy certain interesting properties. Lastly we give some applications of the Poincaré series to other fields. Particularly we construct some counter examples to a conjecture of Victor Kac\cite{Kac_85} and a conjecture of Chapavalov, Leites and Stekolshchik\c…
Stability phenomena for Kac-Moody groups
本文研究了通过扩展广义Dynkin图得到的Kac-Moody群族的稳定性现象，并利用同调分解技术展示了其在弦理论中的应用。
We prove sign-alternation of the structure constants in the basis of structure sheaves of opposite Schubert varieties in the torus-equivariant Grothendieck group of coherent sheaves on the flag varieties $G/P$ associated to an arbitrary symmetrizable Kac-Moody group $G$, where $P$ is any parabolic subgroup. This generalizes the work of Anderson-Griffeth-Miller from the finite case to the general Kac-Moody case, and affirmatively answers a conjecture of Lam-Schilling-Shimozono regarding the signs of the structure constants in the case of the affine Grassmannian.
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