本项目围绕典型群与量子群不变量理论的两个基本定理展开，主要研究与不变量理论有关的各种代数的表示理论，特别是与辛型和正交型量子不变量理论相关的Brauer代数、Birman-Murakami-Wenzl代数的结构理论。我们已经知道辛型量子张量空间在带相应参数的Birman-Murakami-Wenzl代数中的零化子是由某个反对称子生成的主理想，因此猜测：正交型量子张量空间在带相应参数的Birman-Murakami-Wenzl代数中的零化子也是由某个元素生成的主理想。这是本课题的核心研究目标。更进一步，我们将试图运用不变量理论揭示Brauer代数、Birman-Murakami-Wenzl代数的一些结构信息，例如Jacobson根、块、分解数等。

This project is based on the two fundamental theorems of the invariant theory for the classical and quantum groups. We mainly study the representation theory of various algebras related to the invariant theory. Especially, the structures of Brauer algebras and Birman-Murakami-Wenzl algebras are studied in the symplectic and orthogonal cases. It is well-known that the annihilator of symplectic quantized tensor space in Birman-Murakami-Wenzl algebra with special parameters is the principal ideal generated by some anti-symmetrizer. We hence conjecture that, in the orthogonal case, the annihilator of quantized tensor space in the Birman-Murakami-Wenzl algebra with special parameters is also a principal ideal generated by some quasi-idempotent. This is the core objective of this project. Furthermore, we will try to understand the structures of Brauer algebras and Birman-Murakami-Wenzl algebras using the corresponding invariant theory. For example, the Jacobson radicals, blocks and decomposition numbers etc are studied.