Irreducible non-dense \(A_ 1^{(1)}\)-modules. (English) Zbl 0863.17021
From the introduction: For \(A=\left (\begin{smallmatrix} 2 & -2 \\ -2 & 2 \end{smallmatrix} \right)\) let \({\mathcal G}= {\mathcal G} (A)\) be the associated Kac-Moody algebra over the complex numbers \(\mathbb{C}\) with Cartan subalgebra \(H\subset {\mathcal G}\), 1-dimensional center \(\mathbb{C} c\subset H\) and root system \(\Delta\).
A \({\mathcal G}\)-module \(V\) is called a weight if \(V= \bigoplus_{\lambda \in H^*} V_\lambda\), \(V_\lambda= \{v\in V\mid hv= \lambda(h)v\) for all \(h \in H\}\). If \(V\) is an irreducible weight \({\mathcal G}\)-module then \(c\) acts on \(V\) as a scalar. We will call this scalar the level of \(V\). For a weight \({\mathcal G}\)-module \(V\), set \(P(V)= \{\lambda \in H^* \mid V_\lambda \neq 0\}\).
Let \(Q= \sum_{\varphi\in\Delta} \mathbb{Z} \varphi\). If a weight \({\mathcal G}\)-module \(V\) is irreducible then clearly \(P(V) \subset \lambda+ Q\) for some \(\lambda \in H^*\). An irreducible weight \({\mathcal G}\)-module \(V\) is called dense if \(P(V)= \lambda + Q\) for some \(\lambda\in H^*\), and non-dense otherwise.
Irreducible dense modules whose weight spaces are all one-dimensional were classified by S. Spirin [Funkts. Anal. Prilozh. 21, 84-85 (1987; Zbl 0628.17012)] for the algebra \(A^{(1)}_1\) and by D. Britten, F. Lemire and F. Zorzitto [Commun. Algebra 18, 3307-3321 (1990; Zbl 0713.17016)] in the general case. It follows from their paper that such modules exist only for algebras \(A_n^{(1)}\), \(C_n^{(1)}\). V. Chari and A. Pressley constructed a family of irreducible integrable dense modules with all infinite-dimensional weight spaces. These modules can be realized as a tensor product of standard highest weight modules with so-called loop modules [Math. Ann. 277, 543-562 (1987; Zbl 0608.17009)].
In the present paper the author studies irreducible non-dense weight \({\mathcal G}\)-modules. The main result is the classification of all irreducible non-dense \({\mathcal G}\)-modules having at least one finite-dimensional weight subspace. This includes, in particular, all irreducible highest weight modules. Moreover, he shows that this classification includes all irreducible modules of non-zero level whose weight spaces are all finite-dimensional.
A \({\mathcal G}\)-module \(V\) is called a weight if \(V= \bigoplus_{\lambda \in H^*} V_\lambda\), \(V_\lambda= \{v\in V\mid hv= \lambda(h)v\) for all \(h \in H\}\). If \(V\) is an irreducible weight \({\mathcal G}\)-module then \(c\) acts on \(V\) as a scalar. We will call this scalar the level of \(V\). For a weight \({\mathcal G}\)-module \(V\), set \(P(V)= \{\lambda \in H^* \mid V_\lambda \neq 0\}\).
Let \(Q= \sum_{\varphi\in\Delta} \mathbb{Z} \varphi\). If a weight \({\mathcal G}\)-module \(V\) is irreducible then clearly \(P(V) \subset \lambda+ Q\) for some \(\lambda \in H^*\). An irreducible weight \({\mathcal G}\)-module \(V\) is called dense if \(P(V)= \lambda + Q\) for some \(\lambda\in H^*\), and non-dense otherwise.
Irreducible dense modules whose weight spaces are all one-dimensional were classified by S. Spirin [Funkts. Anal. Prilozh. 21, 84-85 (1987; Zbl 0628.17012)] for the algebra \(A^{(1)}_1\) and by D. Britten, F. Lemire and F. Zorzitto [Commun. Algebra 18, 3307-3321 (1990; Zbl 0713.17016)] in the general case. It follows from their paper that such modules exist only for algebras \(A_n^{(1)}\), \(C_n^{(1)}\). V. Chari and A. Pressley constructed a family of irreducible integrable dense modules with all infinite-dimensional weight spaces. These modules can be realized as a tensor product of standard highest weight modules with so-called loop modules [Math. Ann. 277, 543-562 (1987; Zbl 0608.17009)].
In the present paper the author studies irreducible non-dense weight \({\mathcal G}\)-modules. The main result is the classification of all irreducible non-dense \({\mathcal G}\)-modules having at least one finite-dimensional weight subspace. This includes, in particular, all irreducible highest weight modules. Moreover, he shows that this classification includes all irreducible modules of non-zero level whose weight spaces are all finite-dimensional.
MSC:
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
