Irreducible modules for the quantum affine algebra \(U_{q}(\mathfrak {\widehat{sl}_2})\) and its Borel subalgebra
Let \(\mathbb{F}\) be an algebraically closed field of arbitrary characteristic and let \(q\) be an element of \(\mathbb{F}\) that is not a root of unity. The aim of the paper under review is to relate the finite-dimensional irreducible modules of the quantum affine algebra \(U_q(\widehat{\mathfrak{sl}}_2)\) over \(\mathbb{F}\) to the finite-dimensional irreducible modules of its standard Borel subalgebra \(U_q(\mathfrak{b})\). \textit{V. Chari} and \textit{A. Pressley} [Commun. Math. Phys. 142, No. 2, 261--283 (1991; Zbl 0739.17004)] classified the finite-dimensional irreducible modules of \(U_q(\widehat{ \mathfrak{sl}}_2)\) up to isomorphism. Moreover, they showed that every finite-dimensional irreducible modules of \(U_q(\widehat{\mathfrak{sl}}_2)\) is a tensor product of evaluation modules (possibly twisted by an automorphism). The latter are defined via the quantum evaluation homomorphism \(U_q(\widehat{\mathfrak{sl}}_2)\to U_q(\mathfrak{sl}_2)\) introduced by \textit{M. Jimbo} [Lett. Math. Phys. 11, 247--252 (1986; Zbl 0602.17005)]. It should be remarked that the proofs of the paper under review do not use this realization but are quite elementary and require only linear algebra. The main result of the paper establishes a bijection between the isomorphism classes of finite-dimensional irreducible \(U_q(\widehat{\mathfrak{sl}}_2)\)-modules of type \((\varepsilon_0, \varepsilon_1)\) and the isomorphism classes of finite-dimen\-sional irreducible \(\mathcal{B}\)-modules of type \(\alpha\), where \(\mathcal{B}\) is a certain unital associative \(\mathbb{F} \)-algebra, \(\varepsilon_0,\varepsilon_1\in\{\pm 1\}\) such that \(\varepsilon_0\varepsilon_1\) is the eigenvalue of the central element \(K_0K_1\) in \(U_q(\widehat{\mathfrak{sl}}_2)\), and \(\alpha\) is a non-zero element in \(\mathbb{F}\). Most of the paper is devoted to the proof of this result which is an adaptation of a construction of \textit{T. Ito} and the second author in [Tridiagonal pairs and the quantum affine algebra \(U_q(\widehat{\mathfrak{sl}}_2)\), Ramanujan J. 13, No. 1--3, 39--62 (2007; Zbl 1128.16028)] for obtaining \(U_q(\widehat{\mathfrak{sl}}_2)\)-actions from a certain type of tridiagonal pair. As a consequence of the above result the authors show that the restriction of every finite-dimensional irreducible \(U_q(\widehat{\mathfrak{sl}}_2)\)-module to its standard Borel subalgebra \(U_q(\mathfrak{b})\) remains irreducible and that the action of any finite-dimensional irreducible \(U_q(\mathfrak{b})\)-module extends uniquely to an irreducible \(U_q(\widehat{ \mathfrak{sl}}_2)\)-action. By combining this with the results of {Chari} and {Pressley} mentioned above it follows that every finite-dimensional irreducible \(U_q(\mathfrak{b})\)-module can be obtained from a (possibly twisted) tensor product of evaluation modules.
scientific article; zbMATH DE number 2119560
scientific article