An Analog of the Classical Invariant Theory for Lie Superlagebras

Let V be a finite-dimensional superspace over C and g a simple (or a “close” to simple) matrix Lie superalgebra, i.e., a Lie subsuperalgebra in gl(V ). Under the classical invariant theory for g we mean the description of g-invariant elements of the algebra A p,q k,l = S . (V k ⊕Π(V ) ⊕ V ∗p ⊕Π(V )). We give such description for gl(V ), sl(V ) and osp(V ) and their “odd” analogs: q(V ), sq(V ); pe(V ) and spe(V ). This is a detailed exposition of my short announcement “An Analog of the Classical Invariant theory for Lie Superalgebras” published in Funktsional’nyj Analiz i ego Prilozheniya, 26, no. 3, 1992, 88–90. For prerequisites on Lie superalgebras see Appendix borrowed from [L]. §1. Setting of the problem. Formulation of the results 1.0. Let V be a finite-dimensional superspace over C and g an arbitrary matrix Lie superalgebra, i.e., a Lie subsuperalgebra in gl(V ). Under the classical invariant theory for g we mean the description of g-invariant elements of the algebra A p,q k,l = S . (V k ⊕ Π(V ) ⊕ V ∗p ⊕Π(V )). Clearly, A k,l = S . (U ⊗ V ⊕ V ∗ ⊗W ), where dimU = (k, l) and dimW = (p, q). Therefore, on A k,l there also act Lie superalgebras gl(U) and gl(W ); hence, the universal enveloping algebra U(gl(U⊗W )) also acts on A k,l . The elements of the enveloping algebra will be called polarization operators. These operators commute with the natural gl(V )-action. A set M of g-invariants will be called a basic one if the algebra of invariants coincides with the least subalgebra containing M and invariant with respect to polarization operators. For every series of the classical Lie superalgebras (i.e., simple ones and their central extensions) we describe such a set M. Introduce Z2-graded sets: T = {1, . . . , k, 1̄, . . . , ē}, S = {1, . . . , p, 1̄, . . . , q̄}, I = {1, . . . , n, 1̄, . . . , m} here the odd elements are bared and even elements not bared and select bases in the spaces U , W , V : {ut}t∈T , {ws}s∈S, {li}i∈I 1991 Mathematics Subject Classification. 17A70, 13A50.