Integrable highest weight modules over affine superalgebras and Appell's function
Let \(\mathfrak g=\mathfrak g_0\oplus \mathfrak g_1\) be a finite dimensional simple or abelian complex Lie superalgebra with a symmetric invariant bilinear form \((x |y)\). The associated affine Lie superalgebra \(\widehat{\mathfrak g}=\left(\mathbb C[t^{\pm 1}]\otimes \mathfrak g\right)\oplus \mathbb C k\oplus \mathbb C d \) with the central element \(K\) is defined by multiplication \[ [t^m\otimes a , t^n \otimes b]= t^{n+m} \otimes [a , b]+m\delta_{m,-n}(a |b)K; \] \[ [d , t^m\otimes a] =-mt^m\otimes a, \] for all \(m,n\in \mathbb Z\) and for all \(a,b\in \mathfrak g\). Moreover \[ (t^n\otimes a |t^n\otimes b)=\delta_{m,-n}(a |b),\quad (K |d)=-1 \] \[ (t^m\otimes a |\mathbb C K +\mathbb C d)=(K |K)=(d |d)=0. \] It is assumed that \(\mathfrak g_0\) is reductive and \(\mathfrak g_0=\oplus_{0j}^N\mathfrak g_{0j}\) where \(\mathfrak g_{00}\) is abelian and \(\mathfrak g_{0j}, j\geq 1,\) are simple Lie algebras. A \(\mathfrak g\)-module \(V\) is integrable if \(V\) is integrable as \(\widehat{\mathfrak g}_{0j}\)-module for some subset \(J\) in \(\{0,1,\ldots,N\}\). It means that the affine Lie algebra associated to the Cartan subalgebra of \( \mathfrak g_{0j}\) is diagonalizable on \(V\) and \(T^m\otimes \mathfrak g_{\alpha}\) is locally finite on \(V\) for all \(m\in \mathbb Z\) and any root \(\alpha\) of \(\widehat{\mathfrak g}_{0j}\). Given a Cartan subalgebra \(\mathfrak h\) in \(\mathfrak g_0\) and \(\Lambda\in\widehat{h^*}\) one defines an irreducible highest weight \(\widehat{\mathfrak g}\)-module \(L(\Lambda)\) as the unique irreducible \(\widehat{\mathfrak g}\)-module such that \(hv=\Lambda(h)v\) for some nonzero vector \(v\) and for all \(h\in \widehat{ \mathfrak h}\). Moreover \(\mathfrak n_+v=(T^m\otimes \mathfrak g)v=0\) for all \(m\in \mathbb Z\). The level of \(L(\Lambda)\) is equal to \(\Lambda(K)\). There is given a construction of level 1 irreducible highest weight module \(L(\Lambda)\) over \(\widehat{\mathfrak gl}(m |n)\) and a character formula for \(\widehat{\mathfrak sl}(m |1)\) which is a product of a theta function, a power of an eta function and an Appell function. These results are applied to a classification of irreducible integrable highest weight modules over affine superalgebras of types \(\widehat A(m,n)\), \(\widehat B(m,n)\), \(\widehat C(n)\), \(\widehat D(m,n)\), \(\widehat D(m,n, a)\), \(\widehat F(n)\), \(\widehat G(3)\).
scientific article; zbMATH DE number 1631893