Extremal projectors of two-parameter Kashiwara algebras (Q2924875)

Let \(\mathfrak g\) be a finite-dimensional semisimple Lie algebra over \(\mathbb C\) and \(q\in \mathbb C^{\times}\) trascendent over \(\mathbb Q\). M. Kashiwara introduced in his study of crystal basis the so called Kashiwara algebra \(B_q(\mathfrak g)\), related to the quantized enveloping algebra \(U_q(\mathfrak g)\); see \textit{M. Kashiwara} [Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005)]. This algebra was studied further in [\textit{T. Nakashima}, Commun. Math. Phys. 164, No. 2, 239--258 (1994; Zbl 0806.17011), Commun. Math. Phys. 244, No. 2, 285--296 (2004; Zbl 1075.17009)]; in particular, the category \(\mathcal O( B_q(\mathfrak g))\) of upper bounded \(B_q(\mathfrak g)\)-modules was shown to be semisimple, and a parametrization of its simple objects was given. In the present paper, the definition of the Kashiwara algebra is extended to the 2-parameter context; the category of its upper bounded modules is shown to be semisimple, and a parametrization of its simple objects is given.