Geometric realizations of Lusztig's symmetries of symmetrizable quantum groups

Minghui Zhao Beijing Forestry University

Quantum Algebra Representation Theory mathscidoc:1702.29003

Algebras and Representation Theory, 2017
Let $\mathbf{U}$ be the quantum group and $\mathbf{f}$ be the Lusztig's algebra associated with a symmetrizable generalized Cartan matrix. The algebra $\mathbf{f}$ can be viewed as the positive part of $\mathbf{U}$. Lusztig introduced some symmetries $T_i$ on $\mathbf{U}$ for all $i\in I$. Since $T_i(\mathbf{f})$ is not contained in $\mathbf{f}$, Lusztig considered two subalgebras ${_i\mathbf{f}}$ and ${^i\mathbf{f}}$ of $\mathbf{f}$ for any $i\in I$, where ${_i\mathbf{f}}=\{x\in\mathbf{f}\,\,|\,\,T_i(x)\in\mathbf{f}\}$ and ${^i\mathbf{f}}=\{x\in\mathbf{f}\,\,|\,\,T^{-1}_i(x)\in\mathbf{f}\}$. The restriction of $T_i$ on ${_i\mathbf{f}}$ is also denoted by $T_i:{_i\mathbf{f}}\rightarrow{^i\mathbf{f}}$. The geometric realization of $\mathbf{f}$ and its canonical basis are introduced by Lusztig via some semisimple complexes on the variety consisting of representations of the corresponding quiver. When the generalized Cartan matrix is symmetric, Xiao and Zhao gave geometric realizations of Lusztig's symmetries in the sense of Lusztig. In this paper, we shall generalize this result and give geometric realizations of ${_i\mathbf{f}}$, ${^i\mathbf{f}}$ and $T_i:{_i\mathbf{f}}\rightarrow{^i\mathbf{f}}$ by using the language 'quiver with automorphism' introduced by Lusztig.
Quantum groups, Lusztig's symmetries, Geometric realizations, Quivers with automorphism
[ Download ] [ 2017-02-19 19:06:34 uploaded by zhaomh ] [ 534 downloads ] [ 0 comments ]
@inproceedings{minghui2017geometric,
  title={Geometric realizations of Lusztig's symmetries of symmetrizable quantum groups},
  author={Minghui Zhao},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170219190634941975474},
  booktitle={Algebras and Representation Theory},
  year={2017},
}
Minghui Zhao. Geometric realizations of Lusztig's symmetries of symmetrizable quantum groups. 2017. In Algebras and Representation Theory. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170219190634941975474.
Please log in for comment!
 
 
Contact us: office-iccm@tsinghua.edu.cn | Copyright Reserved