On ideals in U(sl()),U(o()),U(sp())\operatorname{U}(\mathfrak {sl} (\infty)), \operatorname{U}(\mathfrak {o} (\infty)), \operatorname{U}(\mathfrak {sp} (\infty))

  • Ivan Penkov

    Jacobs-Universität Bremen, Germany
  • Alexey Petukhov

    The University of Manchester, UK
On ideals in $\operatorname{U}(\mathfrak {sl} (\infty)), \operatorname{U}(\mathfrak {o} (\infty)), \operatorname{U}(\mathfrak {sp} (\infty))$ cover
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We provide a review of results on two-sided ideals in the enveloping algebra U(g())\operatorname{U}(\mathfrak g(\infty)) of a locally simple Lie algebra g()\mathfrak g(\infty). We pay special attention to the case when g()\mathfrak g(\infty) is one of the finitary Lie algebras sl(),o(),sp()\mathfrak{sl}(\infty), \mathfrak o(\infty), \mathfrak{sp}(\infty). The main results include a description of all integrable ideals in U(g())\operatorname{U}(\mathfrak g(\infty)), as well as a criterion for the annihilator of an arbitrary (not necessarily integrable) simple highest weight module to be nonzero. This criterion is new for g()=o(),sp()\mathfrak g(\infty)=\mathfrak o(\infty), \mathfrak{sp}(\infty). All annihilators of simple highest weight modules are integrable ideals for g()=sl(),\mathfrak g(\infty)=\mathfrak{sl}(\infty), o()\mathfrak o(\infty). Finally, we prove that the lattices of ideals in U(o())\operatorname{U}(\mathfrak o(\infty)) and U(sp())\operatorname{U}(\mathfrak{sp}(\infty)) are isomorphic.