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 $G$ from the dual space $\mathfrak{g} ^ { * }$ of a solvable Lie algebra $\frak g$ into the space of primitive ideals of the enveloping algebra $U ( \mathfrak { g } )$ of $\frak g$ (cf. also Universal enveloping algebra; the adjoint algebraic group $G$ of $\frak g$ is the smallest algebraic subgroup of the group of automorphisms of the Lie algebra $\frak g$ whose Lie algebra in the algebra of endomorphisms of $\frak g$ contains the adjoint Lie algebra of $\frak g$). All ideals of $U ( \mathfrak { g } )$ are stable under the action of $G$.

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 $U ( \mathfrak { g } )$ and to describe the centre of $U ( \mathfrak { g } )$.

The dual $\mathfrak{g} ^ { * }$ of $\frak g$ is equipped with the Zariski topology and the space of primitive ideals of $U ( \mathfrak { g } )$ with the Jacobson topology.

The Dixmier mapping (for $\frak g$ solvable) is surjective [a9], injective (modulo the action of the adjoint algebraic group $G$) [a17], continuous [a5], and even open [a13].

Hence the Dixmier mapping induces a homeomorphism between $\mathfrak { g } ^ { * } / G$ and the space $\operatorname{Prim}( U ( \mathfrak{g} ) )$ of primitive ideals of $U ( \mathfrak { g } )$ and allows a complete classification of the primitive ideals of $U ( \mathfrak { g } )$. The openness was an open question for quite a long while.

The Dixmier construction goes as follows: If $f$ is a linear form on the Lie algebra $\frak g$, one chooses a subalgebra $\mathfrak h $ of $\frak g$ which is a polarization of $f$. This means that the subalgebra $\mathfrak h $ is an isotropic subspace of maximal dimension for the skew-symmetric bilinear form $f ( [ \cdot , \cdot ] )$ (on $\frak g$); hence the dimension of $\mathfrak h $ is one half of $\operatorname { dim } \mathfrak { g } - \operatorname { dim } \mathfrak { g } ( f )$, where $\mathfrak { g } ( f )$ is the stabilizer of $f$ in $\frak g$ with respect to the co-adjoint action of $\frak g$ in $\mathfrak{g} ^ { * }$.

For solvable Lie algebras such polarizations always exist, whereas for arbitrary Lie algebras this is, in general, not the case. Let $\operatorname{tr}$ denote the linear form on $\mathfrak h $ defined as the trace of the adjoint action of $\mathfrak h $ in $\mathfrak{g}/\mathfrak{h}$. The linear form $f + 1 / 2 \operatorname{tr}$ on $\mathfrak h $ defines a one-dimensional representation of the enveloping algebra $U ( \mathfrak{h} )$. Let $J$ denote its kernel and $I ( f , \mathfrak{h} )$ the largest two-sided ideal in $U ( \mathfrak { g } )$ contained in $U ( {\frak g} ) J$. This is nothing else but the kernel of the so-called twisted induction from $U ( \mathfrak{h} )$ to $U ( \mathfrak { g } )$ of the one-dimensional representation of $U ( \mathfrak{h} )$ given by $f$. In the case of a nilpotent Lie algebra, the twist $1 / 2 \operatorname{tr}$ is zero. The twisted induction on the level of enveloping algebras corresponds to the unitary induction on the level of Lie groups.

The ideal $I ( f , \mathfrak{h} )$ obtained (in the solvable case) in this way is independent of the choice of the polarization [a7], hence this ideal may be denoted by $I ( f )$. The ideal $I ( f )$ is a (left) primitive ideal, i.e. the annihilator of an irreducible representation (left module) of $U ( \mathfrak { g } )$. 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 $\frak g$ all prime ideals (hence especially all primitive ideals) of $U ( \mathfrak { g } )$ are completely prime [a7].

For solvable Lie algebras $\frak g$, the Dixmier mapping associates to a linear form $f$ of $\frak g$ this primitive ideal $I ( f )$. The $G$-equivariance follows immediately from the fact that this construction commutes with automorphisms of $\frak g$. 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 $\frak g$ but only on the set of elements of $\mathfrak{g} ^ { * }$ having a solvable polarization. In particular, this set contains the open set of linear forms whose orbits under $G$ have maximal dimension. For solvable Lie algebras one gets the usual definition.

2) To the $2$-parameter Duflo mapping [a11]. This mapping is defined for algebraic Lie algebras $\frak g$ (cf. also Lie algebra, algebraic). The first parameter is a so-called linear form on $\frak g$ of unipotent type, the second parameter is a primitive ideal in the reductive part of the stabilizer in $\frak g$ of the first parameter. The mapping goes into the space of primitive ideals of $U ( \mathfrak { g } )$. This mapping coincides with the Dixmier mapping if $\frak g$ is nilpotent and it can be related to the Dixmier mapping if $\frak g$ is algebraic and solvable. For $\frak g$ semi-simple, the mapping reduces to the identity. This $2$-parameter Duflo mapping is surjective [a11] and it is injective modulo the operation of $G$ [a16].

3) The Dixmier mapping for ${\frak sl} ( n )$. 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 ${\frak sl} ( n )$ ($= \mathfrak { g }$) is continuous only on sheets of $\mathfrak{g} ^ { * }$ but not as mapping in the whole. The sheets of $\mathfrak{g} ^ { * }$ are the maximal irreducible subsets of the space of linear forms whose $G$-orbits have a fixed dimension. The Dixmier mapping for ${\frak sl} ( n )$ is surjective on the space of primitive completely prime ideals of $U ( \mathfrak { sl } ( n ) )$ [a15] and it is injective modulo $G$ [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 $G$ (the conjecture is still open, October 1999).

References

[a1]	W. Borho, "Definition einer Dixmier–Abbildung für ${\frak sl} ( n , {\bf C} )$" Invent. Math. , 40 (1977) pp. 143–169 MR442046 Zbl 0346.17014
[a2]	W. Borho, "Extended central characters and Dixmier's map" J. Algebra , 213 (1999) pp. 155–166 MR1674672
[a3]	W. Borho, P. Gabriel, R. Rentschler, "Primideale in Einhüllenden auflösbarer Lie–Algebren" , Lecture Notes Math. , 357 , Springer (1973) Zbl 0293.17005
[a4]	W. Borho, J.C. Jantzen, "Über primitive Ideale in der Einhüllenden einer halbeinfachen Lie-Algebra" Invent. Math. , 39 (1977) pp. 1–53 MR0453826 Zbl 0327.17002
[a5]	N. Conze, M. Duflo, "Sur l'algèbre enveloppante d'une algèbre de Lie résoluble" Bull. Sci. Math. , 94 (1970) pp. 201–208 MR0283037
[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 MR0200393
[a8]	J. Dixmier, "Enveloping algebras" , Amer. Math. Soc. (1996) (Translated from French) MR1451138 MR1393197 Zbl 0867.17001
[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 MR257160
[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) MR399194
[a11]	M. Duflo, "Théorie de Mackey pour les groupes de Lie algébriques" Acta Math. , 149 (1982) pp. 153–213 MR0688348 Zbl 0529.22011
[a12]	A.A. Kirillov, "Unitary representations of nilpotent Lie groups" Uspekhi Mat. Nauk , 17 (1962) pp. 57–110 (In Russian) MR0142001 Zbl 0106.25001
[a13]	O. Mathieu, "Bicontinuity of the Dixmier map" J. Amer. Math. Soc. , 4 (1991) pp. 837–863 MR1126380 MR1115787 Zbl 0743.17013 Zbl 0762.17008
[a14]	C. Moeglin, "Ideaux primitifs des algèbres enveloppantes" J. Math. Pures Appl. , 59 (1980) pp. 265–336 MR0604473 Zbl 0454.17006
[a15]	C. Moeglin, "Ideaux primitifs completement premiers de l'algèbre enveloppante de ${\frak gl} ( n , {\bf C} )$" J. Algebra , 106 (1987) pp. 287–366
[a16]	C. Moeglin, R. Rentschler, "Sur la classification des ideaux primitifs des algèbres enveloppantes" Bull. Soc. Math. France , 112 (1984) pp. 3–40 MR0771917 Zbl 0549.17007
[a17]	R. Rentschler, "L'injectivite de l'application de Dixmier pour les algèebres de Lie résolubles" Invent. Math. , 23 (1974) pp. 49–71