Regular Kac-moody Superalgebras and Integrable Highest Weight Modules

We define regular Kac-Moody superalgebras and classify them using integrable modules. We give conditions for irreducible highest weight modules of regular Kac-Moody superalgebras to be integrable. This paper is a major part of the proof for the classification of finite-growth contragredient Lie superalgebras. The results of this paper are a crucial part of the proof for the classification of contragredient Lie superalgebras with finite growth, and in particular, for the classification of finite-growth Kac-Moody superalgebras [2, 3]. Previously, such a classification was known only for contragredient Lie superalgebras with either symmetrizable Cartan matrices [11, 12], or Cartan matrices with no zeros on the main diagonal, i.e. contragredient Lie superalgebras without simple isotropic roots [5]. Several of the results of this paper are surveyed in [10]. A contragredient Lie superalgebra g(A) is a Lie superalgebra defined by a Cartan matrix A [4, 6]. A Lie superalgebra usually has more than one Cartan matrix. However, an odd reflection at a regular simple isotropic root allows one to move from one base to another [9]. An odd reflection yields a new Cartan matrix A ′ such that g(A ′) and g(A) are isomorphic as Lie superalgebras. A matrix which satisfies certain numerical conditions is called a generalized Cartan matrix (see Definition 1.7). If A is a generalized Cartan matrix then all simple isotropic roots are regular. A contragredient Lie superalgebra g(A) is said to be regular Kac-Moody if A and any matrix A ′ , obtained by a sequence of odd reflections of A, are generalized Cartan matrices. If A is a generalized Cartan matrix and g(A) has no simple isotropic roots, then g(A) is regular Kac-Moody by definition. Hence, we restrict our attention to regular Kac-Moody 1 superalgebras which have a simple isotropic root. Remarkably, there are only a finite number of such families. It is shown in [3] that if g(A) is a finite-growth contragredient Lie superal-gebra and the defining matrix A has no zero rows, then simple root vectors of g(A) act locally nilpotently on the adjoint module. This implies certain conditions on A which are only slightly weaker than the conditions for the matrix to be a generalized Cartan matrix. For a finite-growth Lie superalgebra, these matrix conditions should still hold after odd reflections, which leads to the definition of a regular Kac-Moody superalgebras. Remarkably, these superalgebras almost always have finite growth. The exception is the family: Q ± …