Dixmier mapping
Dixmier map
A mapping first defined for nilpotent Lie algebras by J. Dixmier in 1963 [a6], based on the orbit method of A.A. Kirillov [a12]. In 1966, Dixmier extended his definition to solvable Lie algebras [a7] (here and below, all Lie algebras are of finite dimension over an algebraically closed field of characteristic zero, cf. also Lie algebra; Lie algebra, solvable).
The Dixmier mapping is an equivariant mapping with respect to the adjoint algebraic group from the dual space
of a solvable Lie algebra
into the space of primitive ideals of the enveloping algebra
of
(cf. also Universal enveloping algebra; the adjoint algebraic group
of
is the smallest algebraic subgroup of the group of automorphisms of the Lie algebra
whose Lie algebra in the algebra of endomorphisms of
contains the adjoint Lie algebra of
). All ideals of
are stable under the action of
.
The properties of the Dixmier mapping have been studied in detail. In particular, the Dixmier mapping allows one to describe (the) primitive ideals of the enveloping algebra and to describe the centre of
.
The dual of
is equipped with the Zariski topology and the space of primitive ideals of
with the Jacobson topology.
The Dixmier mapping (for solvable) is surjective [a9], injective (modulo the action of the adjoint algebraic group
) [a17], continuous [a5], and even open [a13].
Hence the Dixmier mapping induces a homeomorphism between and the space
of primitive ideals of
and allows a complete classification of the primitive ideals of
. The openness was an open question for quite a long while.
The Dixmier construction goes as follows: If is a linear form on the Lie algebra
, one chooses a subalgebra
of
which is a polarization of
. This means that the subalgebra
is an isotropic subspace of maximal dimension for the skew-symmetric bilinear form
(on
); hence the dimension of
is one half of
, where
is the stabilizer of
in
with respect to the co-adjoint action of
in
.
For solvable Lie algebras such polarizations always exist, whereas for arbitrary Lie algebras this is, in general, not the case. Let denote the linear form on
defined as the trace of the adjoint action of
in
. The linear form
on
defines a one-dimensional representation of the enveloping algebra
. Let
denote its kernel and
the largest two-sided ideal in
contained in
. This is nothing else but the kernel of the so-called twisted induction from
to
of the one-dimensional representation of
given by
. In the case of a nilpotent Lie algebra, the twist
is zero. The twisted induction on the level of enveloping algebras corresponds to the unitary induction on the level of Lie groups.
The ideal obtained (in the solvable case) in this way is independent of the choice of the polarization [a7], hence this ideal may be denoted by
. The ideal
is a (left) primitive ideal, i.e. the annihilator of an irreducible representation (left module) of
. It is known that for enveloping algebras of Lie algebras, left and right primitive ideals coincide (see [a7] in the solvable case and [a14] in the general case). It should be noted that for solvable Lie algebras
all prime ideals (hence especially all primitive ideals) of
are completely prime [a7].
For solvable Lie algebras , the Dixmier mapping associates to a linear form
of
this primitive ideal
. The
-equivariance follows immediately from the fact that this construction commutes with automorphisms of
. For a general description and references, see [a3] and [a8].
The definition has been extended in several directions:
1) To the Dixmier–Duflo mapping [a10], defined for all Lie algebras but only on the set of elements of
having a solvable polarization. In particular, this set contains the open set of linear forms whose orbits under
have maximal dimension. For solvable Lie algebras one gets the usual definition.
2) To the -parameter Duflo mapping [a11]. This mapping is defined for algebraic Lie algebras
(cf. also Lie algebra, algebraic). The first parameter is a so-called linear form on
of unipotent type, the second parameter is a primitive ideal in the reductive part of the stabilizer in
of the first parameter. The mapping goes into the space of primitive ideals of
. This mapping coincides with the Dixmier mapping if
is nilpotent and it can be related to the Dixmier mapping if
is algebraic and solvable. For
semi-simple, the mapping reduces to the identity. This
-parameter Duflo mapping is surjective [a11] and it is injective modulo the operation of
[a16].
3) The Dixmier mapping for . This was done by W. Borho, using the above Dixmier procedure [a1]. The problem is its being well-defined. Because of the twist in the induction, this Dixmier mapping for
(
) is continuous only on sheets of
but not as mapping in the whole. The sheets of
are the maximal irreducible subsets of the space of linear forms whose
-orbits have a fixed dimension. The Dixmier mapping for
is surjective on the space of primitive completely prime ideals of
[a15] and it is injective modulo
[a4].
4) The Dixmier mapping on polarizable sheets (sheets in which every element has a polarization) in the semi-simple case. This was done by W. Borho. This map is well-defined [a2] and continuous, and it is conjectured to be injective modulo (the conjecture is still open, October 1999).
References
[a1] | W. Borho, "Definition einer Dixmier–Abbildung für ![]() |
[a2] | W. Borho, "Extended central characters and Dixmier's map" J. Algebra , 213 (1999) pp. 155–166 |
[a3] | W. Borho, P. Gabriel, R. Rentschler, "Primideale in Einhüllenden auflösbarer Lie–Algebren" , Lecture Notes Math. , 357 , Springer (1973) |
[a4] | W. Borho, J.C. Jantzen, "Über primitive Ideale in der Einhüllenden einer halbeinfachen Lie-Algebra" Invent. Math. , 39 (1977) pp. 1–53 |
[a5] | N. Conze, M. Duflo, "Sur l'algèbre enveloppante d'une algèbre de Lie résoluble" Bull. Sci. Math. , 94 (1970) pp. 201–208 |
[a6] | J. Dixmier, "Représentations irréductibles des algèbres de Lie nilpotents" An. Acad. Brasil. Ci. , 35 (1963) pp. 491–519 |
[a7] | J. Dixmier, "Representations irreductibles des algebres de Lie résolubles" J. Math. Pures Appl. , 45 (1966) pp. 1–66 |
[a8] | J. Dixmier, "Enveloping algebras" , Amer. Math. Soc. (1996) (Translated from French) |
[a9] | M. Duflo, "Sur les extensions des representations irreductibles des algèbres de Lie contenant un ideal nilpotent" C.R. Acad. Sci. Paris Ser. A , 270 (1970) pp. 504–506 |
[a10] | M. Duflo, "Construction of primitive ideals in enveloping algebras" I.M. Gelfand (ed.) , Lie Groups and their representations: Summer School of the Bolyai Janos Math. Soc. (1971) , Akad. Kiado (1975) |
[a11] | M. Duflo, "Théorie de Mackey pour les groupes de Lie algébriques" Acta Math. , 149 (1982) pp. 153–213 |
[a12] | A.A. Kirillov, "Unitary representations of nilpotent Lie groups" Uspekhi Mat. Nauk , 17 (1962) pp. 57–110 (In Russian) |
[a13] | O. Mathieu, "Bicontinuity of the Dixmier map" J. Amer. Math. Soc. , 4 (1991) pp. 837–863 |
[a14] | C. Moeglin, "Ideaux primitifs des algèbres enveloppantes" J. Math. Pures Appl. , 59 (1980) pp. 265–336 |
[a15] | C. Moeglin, "Ideaux primitifs completement premiers de l'algèbre enveloppante de ![]() |
[a16] | C. Moeglin, R. Rentschler, "Sur la classification des ideaux primitifs des algèbres enveloppantes" Bull. Soc. Math. France , 112 (1984) pp. 3–40 |
[a17] | R. Rentschler, "L'injectivite de l'application de Dixmier pour les algèebres de Lie résolubles" Invent. Math. , 23 (1974) pp. 49–71 |
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