[Paper Review] On some representations of nilpotent Lie algebras and superalgebras

Shantala Mukherjee|ArXiv.org|Sep 24, 2004

Algebraic structures and combinatorial models11 references3 citations

TL;DR

This paper refines Benoist's orbit method results for nilpotent Lie algebras by determining the exact number of simple modules annihilated by a primitive ideal and sharing a common eigenvector for a subalgebra when the coadjoint orbit is two-dimensional. It further extends Bell and Musson's work on superalgebras by characterizing cases where quotient algebras of universal enveloping superalgebras by graded-primitive ideals are purely Weyl algebras, and explicitly links the size of these Weyl algebras to the structure of the ideals.

ABSTRACT

Let $G$ be a simply connected, nilpotent Lie group with Lie algebra $\gee$. The group $G$ acts on the dual space $\gee^*$ by the coadjoint action. %% which partitions $\gee^*$ into coadjoint orbits. By the orbit method of Kirillov, the simple unitary representations of $G$ are in bijective correspondence with the coadjoint orbits in $\gee^*$, which in turn are in bijective correspondence with the primitive ideals of the universal enveloping algebra of $\gee$. The number of simple $\gee$-modules which have a common eigenvector for a particular subalgebra of $\gee$ and are annihilated by a particular primitive ideal $I$ is shown by Benoist to depend on geometric properties of a certain subvariety of the coadjoint orbit corresponding to $I$. We determine the exact number of such modules when the coadjoint orbit is two-dimensional. Bell and Musson showed that the algebras obtained by factoring the universal enveloping superalgebra of a Lie superalgebra by graded-primitive ideals are isomorphic to tensor products of Weyl algebras and Clifford algebras. We describe certain cases where the factors are purely Weyl algebras and determine how the sizes of these Weyl algebras depend on the graded-primitive ideals.

Motivation & Objective

• To refine Benoist's geometric characterization of the number of simple modules annihilated by a primitive ideal and sharing a common eigenvector for a subalgebra in nilpotent Lie algebras.
• To determine when the quotient of a universal enveloping superalgebra by a graded-primitive ideal is isomorphic to a purely Weyl algebra.
• To establish how the size of the Weyl algebra factor in such quotients depends on the structure of the graded-primitive ideal.
• To extend the orbit method and primitive ideal correspondence to Lie superalgebras, particularly in the context of nilpotent superalgebras.
• To explore the implications of these results for the classification of simple modules and the structure of primitive ideals in nilpotent Lie and superalgebras.

Proposed method

• Uses the orbit method and coadjoint action to relate primitive ideals in the universal enveloping algebra to coadjoint orbits in the dual space 𝔤*.
• Applies Dixmier's correspondence between primitive ideals, coadjoint orbits, and Weyl algebra factors in quotient algebras.
• Employs polarizations and induced representations to construct irreducible modules and identify their annihilators.
• Applies results from Bell and Musson on graded-primitive ideals in superalgebras, showing quotients are tensor products of Weyl and Clifford algebras.
• Analyzes specific examples, including the Heisenberg Lie superalgebra, to demonstrate cases where the quotient is purely a Weyl algebra.
• Uses the structure of the Lie superalgebra and the annihilator of a highest weight module to determine the isomorphism type of the quotient algebra.

Experimental results

Research questions

• RQ1 For a finite-dimensional nilpotent Lie algebra, when the coadjoint orbit is two-dimensional, what is the exact number of simple modules annihilated by a primitive ideal and sharing a common eigenvector for a fixed subalgebra?
• RQ2 Under what conditions is the quotient of the universal enveloping superalgebra of a nilpotent Lie superalgebra by a graded-primitive ideal isomorphic to a purely Weyl algebra?
• RQ3 How does the dimension of the Weyl algebra factor in such quotients depend on the structure of the graded-primitive ideal?
• RQ4 When is the variety Ω_f ∩ (f + 𝔪⊥) Lagrangian for a coadjoint orbit Ω_f of dimension greater than two?
• RQ5 Can a theory of σ-spherical simple modules be developed for finite-dimensional nilpotent Lie superalgebras with an involution, and is there a one-to-one correspondence with a subset of graded-primitive ideals?

Key findings

• For two-dimensional coadjoint orbits, the number of simple modules annihilated by a primitive ideal and sharing a common eigenvector for a subalgebra is determined exactly by geometric properties of a subvariety of the orbit.
• In the case of the Heisenberg Lie superalgebra with λ(z) = 1 and λ(𝔤₁) = 0, the quotient 𝔘/P_λ is isomorphic to M₂(ℂ) ⊗ 𝒜₁, showing a non-pure Weyl algebra factor.
• For the Lie superalgebra 𝔤 = 𝔤𝔩(m,n)+, the quotient 𝔘/P is isomorphic to 𝒜_{r_m + r_n}, where r_m and r_n are integers bounded by s_m and s_n, respectively, indicating a purely Weyl algebra structure.
• The size of the Weyl algebra factor in the quotient depends on the choice of polarisation and the structure of the annihilating ideal, with r_m and r_n uniquely determined by the graded-primitive ideal.
• The results confirm that the quotient by a graded-primitive ideal can be purely a Weyl algebra in specific cases, such as when the superalgebra has no nontrivial Clifford algebra components.
• The paper provides explicit constructions of irreducible modules via induced representations from polarizations, and identifies their annihilators as graded-primitive ideals.