Surjectivity of the invariant projection from T(g) to Z(g)

Determine whether the restriction of the natural projection from the tensor algebra T(g) to the universal enveloping algebra U(g) is surjective when restricted to the g-invariant subalgebra T(g)^g, i.e., whether the map η′: T(g)^g → Z(g) is surjective for classical Lie superalgebras g = gl(m|n), osp(m|2n), q(n), and p(n). This question arises from the non-commutative diagram relating the g-invariant subalgebras T(g)^g, S(g)^g, and U(g) via the supersymmetrization map ψ.

Background

The paper studies explicit invariants of classical Lie superalgebras and their relationships across three algebras: the tensor algebra T(g), the supersymmetric algebra S(g), and the universal enveloping algebra U(g). A central concern is understanding how g-invariants in T(g) map to central elements in U(g).

The authors present a diagram connecting T(g)g and S(g)g via the canonical map η and S(g)g to U(g) via the supersymmetrization map ψ. While η is surjective and ψ is bijective on invariants, the overall diagram does not commute, raising the question of whether the natural projection from T(g) to U(g), restricted to g-invariants (denoted η′), is surjective onto Z(g). This uncertainty motivates much of the subsequent development using Schur–Weyl-type dualities to construct central elements.

References

In the following diagram, the map η is surjective and ψ is bijective. However, it is unknown whether η is surjective or not due to the lack of commutativity.

— The Schur-Weyl duality and Invariants for classical Lie superalgebras  (2411.17093 - Luo et al., 2024) in Introduction (non-commutative diagram of g-invariants)