Canonical bases and quantum coordinate ring

对称空间上的坐标环

Coordinate rings on symmetric spaces

本文利用量子对称对的典范基理论，为对称空间构建了整模型、典范基与好滤过，并给出了其上双不变函数空间的典范基与整形式。

On dual canonical bases

The dual basis of the canonical basis of the modified quantized enveloping algebra is studied, in particular for type $A$. The construction of a basis for the coordinate algebra of the $n\times n$ quantum matrices is appropriate for the study the multiplicative property. It is shown that this basis is invariant under multiplication by certain quantum minors including the quantum determinant. Then a basis of quantum SL(n) is obtained by setting the quantum determinant to one. This basis turns out to be equivalent to the dual canonical basis.

Vertex $F$-algebras and their $ϕ$-coordinated modules

In this paper, for every one-dimensional formal group $F$ we formulate and study a notion of vertex $F$-algebra and a notion of $ϕ$-coordinated module for a vertex $F$-algebra where $ϕ$ is what we call an associate of $F$. In the case that $F$ is the additive formal group, vertex $F$-algebras are exactly ordinary vertex algebras. We give a canonical isomorphism between the category of vertex $F$-algebras and the category of ordinary vertex algebras. Meanwhile, for every formal group we completely determine its associates. We also study $ϕ$-coordinated modules for a general vertex $\Z$-graded a…

Remarks on $ϕ$-coordinated modules for quantum vertex algebras

This paper is about $ϕ$-coordinated modules for weak quantum vertex algebras. Among the main results, several canonical connections among $ϕ$-coordinated modules for different $ϕ$ are established. For vertex operator algebras, a reinterpretation of Frenkel-Huang-Lepowsky's theorem on contragredient module is given in terms of $ϕ$-coordinated modules.

The homogeneous coordinate ring of the quantum projective plane

We define holomorphic structures on canonical line bundles on the quantum projective plane. The space of holomorphic sections of these line bundles will determine the quantum homogeneous coordinate ring of $\qp^2_q$. We also show that the holomorphic structure of $\qp^2_q$ is naturally represented by a twisted positive Hochschild 4-cocycle.

Coordinate Calculi on Associative Algebras

A new notion of an optimal algebra for a first order coordinate differential was introduced in \cite{BKO}. Some relevant examples are indicated. Quadratic identities in the optimal algebras and calculi on quadratic algebras are studied. Canonical construction of a quantum de Rham complex for the coordinate differential is proposed. The relations between calculi and various generalizations of the Yang--Baxter equation are established.