Primitive Ideals of U ( 𝖘 𝖑 ( ∞ ) ) $${\mathrm {U}}(\mathfrak {sl}(\infty ))$$ and the Robinson–Schensted Algorithm at Infinity

Annihilators of simple integrable weight $\mathfrak{sl}(\infty)$-modules

“…Now we have the following result. In [PP18] a parametrization of all primitive ideals of U(sl(∞)) is given in terms of quadruples (r, g, X, Y ), where r, g ∈ Z ≥0 and X, Y are Young diagrams. The primitive ideal associated to (r, g, X, Y ) is denoted by I(r, g, X, Y ).…”

“…Recently the primitive ideals of the enveloping algebra of the Lie algebra sl(∞) has been classified by A. Petukhov and I. Penkov in [PP18]. As a next step, one should like to compute the annihilators of natural classes of simple sl(∞)-modules.…”

“…However, in the last year several classes of simple weight modules have been discussed in the literature; see [Nam18,CP18,GP18]. Furthermore, in [PP18], an algorithm for computing the primitive ideals of any simple highest weight module is presented, and the problem of computing primitive ideals is open only for simple non highest weight modules. For the class of simple bounded weight modules, this problem was solved by D. Grantcharov and I. Penkov in [GP18].…”

“…Besides simple highest weight modules, Dimitrov's class contains simple integrable modules without a highest weight. Modulo the results of [PP18], the annihilator problem reduces to the computation of the annihilators of the latter modules. Our main result, Theorem 4.1, shows that the annihilators in question equal the annihilators of certain highest weight modules, which we construct explicitly.…”

“…Our main result, Theorem 4.1, shows that the annihilators in question equal the annihilators of certain highest weight modules, which we construct explicitly. Then our claim follows by applying the algorithm given in [PP18].…”