Lie Theory and Its Applications in Physics

Abstract

ABSTRACT. Using deformation theory, Braverman and Joseph obtained an alternative characterisation of the Joseph ideal for simple Lie algebras, which included even type A. In this note we extend that characterisation to define a remarkable quadratic ideal for sl(m|n). When m − n > 2 we prove the ideal is primitive and can also be characterised similarly to the construction of the Joseph ideal by Garfinkle. PRELIMINARIESWe use the notation g = sl(m|n). See [CW] for the definition and more information on sl(m|n) and Lie superalgebras. We take the Borel subalgebra b to be the space of upper triangular matrices and the Cartan subalgebra h diagonal matrices, both with zero supertrace. With slight abuse of notation we will write elements of h * as elements of C m+n , using bases {ε j , i = 1, . . . , m} of C m and {δ j , i = 1, . . . , n} of C n , with the restriction that the coefficients add up to zero. With this choice and convention, the system of positive roots is given by ∆ + = ∆ + 0 ∪ ∆ + 1 , whereThe Borel subalgebra leads to a triangular decomposition of g given by n -⊕ h ⊕ n + where b = h ⊕ n + . A highest weight vector v λ of a weight module M satisfies n + · v λ = 0 and h · v λ = λ (h) · v λ . The corresponding weight λ ∈ h * will be called a highest weight. We use the notation L(λ ) for the simple module with highest weight λ ∈ h * . We also set ρ 0 = 1 2 ∑ α∈∆ + 0 α and ρ = ρ 0 − 1 2 ∑ γ∈∆ + 1 γ, so concretelyWe choose the form (·, ·) on C m+n , and on h * by restriction, by settingFrom now on we consider only weights λ which are integral, that is (λ + ρ, α ∨ ) ∈ Z for all α ∈ ∆ 0 , with α ∨ := 2α/(α, α). If (λ + ρ, α ∨ ) > 0, for all α ∈ ∆ + 0 , we say that the integral weight λ is dominant regular. Denote by C the quadratic Casimir operator. It is an element of the center of U(g) and it acts on a highest weight vector of weight λ by the scalarWe denote by M ∨ the dual module of M in category O, see e.g. [Hu, chapter 3]. The functor ∨ is exact and contravariant, we have that L(λ ) ∨ ∼ = L(λ ) and for finite dimensional modules (M ⊗ N) ∨ ∼ = M ∨ ⊗ N ∨ .We set V = C m|n the natural representation of g. We will use the notation A i j for an element in V ⊗V * and we have the identification V ⊗V * ∼ = V * ⊗V given by 1