Representations of the Exceptional Lie Superalgebra E(3,6) Iii: Classification of Singular Vectors

Abstract

Abstract. We continue the study of irreducible representations of the exceptional Lie superalgebra E(3, 6). This is one of the two simple infinite-dimensional Lie superalgebras of vector fields which have a Lie algebra sℓ(3) × sℓ(2) × gℓ(1) as the zero degree component of its consistent Z-grading. We provide the classification of the singular vectors in the degenerate Verma modules over E(3, 6), completing thereby the classification and construction of all irreducible E(3, 6)-modules that are L0-locally finite. Introduction.There are only two infinite-dimensional simple linearly compact Lie superalgebras, E(3, 6) and E(3, 8), that have the Lie algebra sℓ(3) × sℓ(2) × gℓ(1) as the zero degree component g 0 in their consistent Z-grading [2]. In the present paper we continue the study of irreducible representations of E(3, 6), the Lie superalgebra which has apparent relations to the Standard Model (see [4]).The article is a sequel to our papers [3, 4] and together they provide the classification and description of all irreducible L 0 -locally finite representations of E(3, 6) (see [3] for the definition).As was shown in [3] the problem can be solved in two steps. First one obtains the classification of the so called degenerate (generalized) Verma modules (the Verma modules that have non-trivial singular vectors), and then one describes the irreducible factors of these degenerate Verma modules.In [4], both probelms were solved modulo the classification of singular vectors. In this article we provide the proof of this classification (the result was announced in [4]). Thus we complete the classification of the irreducible E(3, 6)-modules that are L 0 -locally finite.As before all vector spaces, linear maps and tensor products are considered over the field C of complex numbers.

Low Degree Morphisms of E(5, 10)-Generalized Verma Modules

“…This completes the proof of Equations (4), (6) and(7). In order to deduce Equation(5)we consider the coefficient of ∂ a in the previous equation (the coefficients of ∂ b and ∂ c provide…”

Classification of Degenerate Verma Modules for E(5, 10)

“…Then g ≥0 := ⊕ j≥0 g j is a maximal open subalgebra of g of minimal codimension. In the case of g = E (3,6) and E (3,8) the subalgebra g 0 is isomorphic to sl 3 ⊕ sl 2 ⊕ C, and for g = E (5,10), g 0 is isomorphic to sl 5 , which hints to connections to particle physics [8].…”

“…The infinite-dimensional simple linearly compact Lie superalgebras were classified in [6] and explicitely described in [5]; all their maximal open subalgebras were classified in [4]. The complete list consists of ten "classical" series (which include the Lie-Cartan series), and five exceptional examples, denoted by E (1,6), E (3,6), E (3,8), E (4,4), and E (5,10). With the exception of E (4,4), these Lie superalgebras carry a Z-gradation, compatible with the parity:…”

“…where d = 2 for E (1,6), E (3,6) and E (5,10), and d = 3 for E (3,8). Then g ≥0 := ⊕ j≥0 g j is a maximal open subalgebra of g of minimal codimension.…”

“…In a series of papers [7,8,10] this problem was solved for the exceptional linearly compact Lie superalgebra E (3,6). It turned out that all the morphisms between the degenerate finite Verma modules over E (3,6) can be arranged in an infinite number of complexes, and cohomology of these complexes was computed in [8] as well.…”