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Publication# The Dixmier Map for Nilpotent Super Lie Algebras

2012

Journal paper

Journal paper

Abstract

In this article we prove that there exists a Dixmier map for nilpotent super Lie algebras. In other words, if we denote by Prim(U(g)) the set of (graded) primitive ideals of the enveloping algebra U(g) of a nilpotent Lie superalgebra g and Ad0 the adjoint group of g0 , we prove that the usual Dixmier map for nilpotent Lie algebras can be naturally extended to the context of nilpotent super Lie algebras, i.e. there exists a bijective map I : g∗ 0/Ad0 → Prim(U(g)) defined by sending the equivalence class [λ] of a functional λ to a primitive ideal I (λ) of U(g), and which coincides with the Dixmier map in the case of nilpotent Lie algebras. Moreover, the construction of the previous map is explicit, and more or less parallel to the one for Lie algebras, a major difference with a previous approach (cf. [18]). One key fact in the construction is the existence of polarizations for super Lie alge- bras, generalizing the concept defined for Lie algebras. As a corollary of the previ- ous description, we obtain the isomorphism U(g)/I (λ) Cliffq (k) ⊗ A p(k), where ( p, q) = (dim(g0/gλ 0 )/2, dim(g1/gλ 1 )), we get a direct construction of the maximal ide- als of the underlying algebra of U(g) and also some properties of the stabilizers of the primitive ideals of U(g).

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Related publications (37)

Related concepts (36)

Lie algebra representation

In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space together with a collection of operators on satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.

Lie algebra

In mathematics, a Lie algebra (pronounced liː ) is a vector space together with an operation called the Lie bracket, an alternating bilinear map , that satisfies the Jacobi identity. Otherwise said, a Lie algebra is an algebra over a field where the multiplication operation is now called Lie bracket and has two additional properties: it is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors and is denoted . The Lie bracket does not need to be associative, meaning that the Lie algebra can be non associative.

Primitive ideal

In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals. Primitive ideals are prime. The quotient of a ring by a left primitive ideal is a left primitive ring. For commutative rings the primitive ideals are maximal, and so commutative primitive rings are all fields. The primitive spectrum of a ring is a non-commutative analog of the prime spectrum of a commutative ring.

Ontological neighbourhood

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