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Zhao Xu-an

Publications and source records attributed to Zhao Xu-an.

9 recordsLinked to original sources
arXiv2025

Cohomology of classifying spaces of rank 3 Kac-Moody groups

We represent the rational and mod $p$ cohomology groups of classifying spaces of rank 3 Kac-Moody groups by a direct sum of the invariants of Weyl groups and their quotients. As an application, the authors conclude that there is a $p$-torsion for each prime $p$ in the integral cohomology groups of classifying spaces of rank 3 Kac-Moody groups. We also determine the ring structure of the rational cohomology with one exception case.

math.AT↗
arXiv2021

The rational cohomology groups of the classifying spaces of Kac-Moody groups

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 of Kac-Moody groups.

math.AT↗
arXiv2016

Differential geometry of general affine plane curves

In this paper we study the general affine geometry of curves in affine space $A^2$. For a regular plane curves we define two kinds of moving frames. The first is of minimal order in all moving frames.The second is the Frenet moving frame. We get the moving equations of these moving frames. And we prove that curvature and signature are the complete invariants of regular curves. As application we give a complete classification of constant curvature curves in $A^2$.

math.DG↗
arXiv2015

A Geometric Definition Of Schubert Polynomials and Dual Schubert Polynomials For Classical Lie Groups

In this paper, we first discuss the topological properties of projective Stiefel manifolds, we compute their cohomology rings and classify their cohomology endomorphisms; Then by embedding the flag manifold of a classical Lie group into its corresponding infinite dimensional projective Stiefel manifold(which is homotopic to the product of infinite dimensional complex projective space $\mathbb{C}P^{\infty}$), we define the Schubert polynomials and dual Schubert polynomials. Finally we discuss the property and the computation of these polynomials.

math.AT↗
arXiv2015

The ranks of homotopy groups of Kac-Moody groups

Let $A$ be a Cartan matrix and $G(A)$ be the Kac-Moody group associated to Cartan matrix $A$. In this paper, we discuss the computation of the rank $i_k$ of homotopy group $π_k(G(A))$. For a large class of Kac-Moody groups, we construct Lie algebras with grade from the Poincaré series of their flag manifolds. And we interpret $i_{2k}$ as the dimension of the degree $2k$ homogeneous component of the Lie algebra we constructed. Since the computation of $i_{2k-1}$ is trivial, this gives a combinatorics interpretation of $i_k$ for all $k>0$.

math.AT↗
arXiv2013

Poincaré series and rational cohomology rings of Kac-Moody groups and their flag manifolds

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.

math.AT↗
arXiv2012

Polynomial invariants of Weyl groups for Kac-Moody groups

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.

math.AC↗
arXiv2012

On the Poincaré series of Kac-Moody Lie algebras

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\cite{CDR_10}.

math.RT↗