A class of infinite-dimensional weight \(sl(3)\)-modules
The paper deals with \(sl(3)\) modules from the category \(E_\lambda\). This category consists of \(sl(3)\) modules \(V\) such that (i) the set of all weights of \(V\) is \(\lambda +\Pi\), where \(\Pi\) is the lattice generated by the roots of algebra, (ii) the elements \(e_2\) and \(f_1\) from the standard set of algebraic generators \(\{e_i, f_i, h_i\}\), \(i=1,2\), act on \(V\) as invertible operators. The author considers the family of \(d\)-dimensional \(Q(sl(3))\) modules \(\{Y_{d,\nu}\}\) of special type and the corresponding family \(\{V_{d,\nu, \lambda}\}\) of \(sl(3)\) modules from the category \(E_\lambda\). Here \(d\in \mathbb{N}\), \(v\in \mathbb{C}\) and \(Q(sl(3))\) is an associative algebra with the identity element, four generators \(x_0,x_1,y_0,y_1\), and the following relations: \[ x_0y_0-y_0 x_0 =x_1y_0+ y_1x_0, \quad x_1y_1- y_1x_1= x_1+y_1, \quad y_0x_1 -x_1y_0= -y_1x_1-x_0, \] \[ x_0y_1-y_1x_0=x_1y_1+y_0, \quad x_0x_1-x_1x_0= -x_1^2+x_0, \quad y_0y_1 - y_1y_0= y^2_1- y_0. \] From the results of the paper it follows that any irreducible torsion-free \(sl(3)\) module with finite-dimensional weight subspaces is isomorphic to one of the modules \(V_{d,\nu,\lambda}\).
scientific article; zbMATH DE number 1179588