表示论与量子可积性
Representation theory and quantum integrability
摘要：
我们描述了$U(\mathfrak{g})$和$U_q(\mathfrak{g})$的无限维表示的新构造，其中$\mathfrak{g}$是$\mathfrak{gl}(N)$和$\mathfrak{sl}(N)$。这些结构的应用户田型量子可积理论进行了讨论。借助这些无限维表象，我们成功地建立了量子可积性的群论方法与基于Yangian及其推广的表示论的量子逆散射方法之间的直接联系。在$U_q(\mathfrak{g})$的情况下，所考虑的表示自然具有$U_q(\mathfrak{g})\otimes U_{\tilde q}(\check{\mathfrak{g}})$-双模的结构，其中$\check {\mathfrak{g}}$是$\mathfrak{g}$和$\log q/2πi=- (\log{\tilde q}/2πi)^{-1}$的朗兰兹对偶。这种双模结构是代数及其对偶的Morita等价的一种表现。
英文摘要：
We describe new constructions of the infinite-dimensional representations of $U(\mathfrak{g})$ and $U_q(\mathfrak{g})$ for $\mathfrak{g}$ being $\mathfrak{gl}(N)$ and $\mathfrak{sl}(N)$. The application of these constructions to the quantum integrable theories of Toda type is discussed. With the help of these infinite-dimensional representations we manage to establish direct connection between group theoretical approach to the quantum integrability and Quantum Inverse Scattering Method based on the representation theory of Yangian and its generalizations. In the case of $U_q(\mathfrak{g})$ the considered representation is naturally supplied with the structure of $U_q(\mathfrak{g})\otimes U_{\tilde q}(\check{\mathfrak{g}})$-bimodule where $\check {\mathfrak{g}}$ is Langlands dual to $\mathfrak{g}$ and $\log q/2πi=- (\log{\tilde q}/2πi)^{-1}$. This bimodule structure is a manifestation of the Morita equivalence of the algebra and its dual.
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