A category for the adjoint representation. (English) Zbl 1026.17015
Given a finite graph \(\Gamma\), one may have a quiver with the given underlying graph and a quotient algebra of the corresponding path algebra (over the complex field) of the quiver modulo the paths of length greater than one. Let \(\Lambda (\Gamma)\) be the trivial extension of this quotient algebra. In the paper under review the authors construct an Abelian category \(\mathcal C\) based on both the category of all finite-dimensional graded left \(\Lambda (\Gamma)\)-modules and the category of all vector spaces over the complex field. On this category \(\mathcal C\) the authors define some functors and then use these functors to give a realization of the quantum group \(U_q(\mathfrak g)\) of the simple Lie algebra \(\mathfrak g\) if \(\Gamma\) is the Dynkin diagram determining \(\mathfrak g\). Some other properties related to the action of the braid group (defined by the graph \(\Gamma\)) on the derived category of the \(\Lambda(\Gamma)\)-modules are discussed. There is a related paper on \(\lambda(\Gamma)\), as mentioned in the paper, namely, the work of [R. Martínez-Villa, CMS Conf. Proc. 18, 487-504 (1996; Zbl 0860.16010)].
Reviewer: Xi Changchang (Beijing)
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 16G10 | Representations of associative Artinian rings |
Citations:
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