Difference between revisions of "Dixmier mapping"
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''Dixmier map'' | ''Dixmier map'' | ||
A mapping first defined for nilpotent Lie algebras by J. Dixmier in 1963 [[#References|[a6]]], based on the [[Orbit method|orbit method]] of A.A. Kirillov [[#References|[a12]]]. In 1966, Dixmier extended his definition to solvable Lie algebras [[#References|[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]]; [[Lie algebra, solvable|Lie algebra, solvable]]). | A mapping first defined for nilpotent Lie algebras by J. Dixmier in 1963 [[#References|[a6]]], based on the [[Orbit method|orbit method]] of A.A. Kirillov [[#References|[a12]]]. In 1966, Dixmier extended his definition to solvable Lie algebras [[#References|[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]]; [[Lie algebra, solvable|Lie algebra, solvable]]). | ||
− | The Dixmier mapping is an equivariant mapping with respect to the adjoint [[Algebraic group|algebraic group]] | + | The Dixmier mapping is an equivariant mapping with respect to the adjoint [[Algebraic group|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|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 | + | 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 | + | The dual $\mathfrak{g} ^ { * }$ of $\frak g$ is equipped with the [[Zariski topology|Zariski topology]] and the space of primitive ideals of $U ( \mathfrak { g } )$ with the Jacobson topology. |
− | The Dixmier mapping (for | + | The Dixmier mapping (for $\frak g$ solvable) is surjective [[#References|[a9]]], injective (modulo the action of the adjoint algebraic group $G$) [[#References|[a17]]], continuous [[#References|[a5]]], and even open [[#References|[a13]]]. |
− | Hence the Dixmier mapping induces a homeomorphism between | + | 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 | + | 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|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 | + | 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 | + | The ideal $I ( f , \mathfrak{h} )$ obtained (in the solvable case) in this way is independent of the choice of the polarization [[#References|[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 [[#References|[a7]]] in the solvable case and [[#References|[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 [[#References|[a7]]]. |
− | For solvable Lie algebras | + | 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 [[#References|[a3]]] and [[#References|[a8]]]. |
The definition has been extended in several directions: | The definition has been extended in several directions: | ||
− | 1) To the Dixmier–Duflo mapping [[#References|[a10]]], defined for all Lie algebras | + | 1) To the Dixmier–Duflo mapping [[#References|[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) To the $2$-parameter Duflo mapping [[#References|[a11]]]. This mapping is defined for algebraic Lie algebras $\frak g$ (cf. also [[Lie algebra, algebraic|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 [[#References|[a11]]] and it is injective modulo the operation of $G$ [[#References|[a16]]]. |
− | 3) The Dixmier mapping for | + | 3) The Dixmier mapping for ${\frak sl} ( n )$. This was done by W. Borho, using the above Dixmier procedure [[#References|[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 $U ( \mathfrak { sl } ( n ) )$ [[#References|[a15]]] and it is injective modulo $G$ [[#References|[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 [[#References|[a2]]] and continuous, and it is conjectured to be injective modulo | + | 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 [[#References|[a2]]] and continuous, and it is conjectured to be injective modulo $G$ (the conjecture is still open, October 1999). |
====References==== | ====References==== | ||
− | <table>< | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> W. Borho, "Definition einer Dixmier–Abbildung für ${\frak sl} ( n , {\bf C} )$" ''Invent. Math.'' , '''40''' (1977) pp. 143–169 {{MR|442046}} {{ZBL|0346.17014}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> W. Borho, "Extended central characters and Dixmier's map" ''J. Algebra'' , '''213''' (1999) pp. 155–166 {{MR|1674672}} {{ZBL|}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> W. Borho, P. Gabriel, R. Rentschler, "Primideale in Einhüllenden auflösbarer Lie–Algebren" , ''Lecture Notes Math.'' , '''357''' , Springer (1973) {{MR|}} {{ZBL|0293.17005}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> W. Borho, J.C. Jantzen, "Über primitive Ideale in der Einhüllenden einer halbeinfachen Lie-Algebra" ''Invent. Math.'' , '''39''' (1977) pp. 1–53 {{MR|0453826}} {{ZBL|0327.17002}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> N. Conze, M. Duflo, "Sur l'algèbre enveloppante d'une algèbre de Lie résoluble" ''Bull. Sci. Math.'' , '''94''' (1970) pp. 201–208 {{MR|0283037}} {{ZBL|}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> J. Dixmier, "Représentations irréductibles des algèbres de Lie nilpotents" ''An. Acad. Brasil. Ci.'' , '''35''' (1963) pp. 491–519 {{MR|}} {{ZBL|}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> J. Dixmier, "Representations irreductibles des algebres de Lie résolubles" ''J. Math. Pures Appl.'' , '''45''' (1966) pp. 1–66 {{MR|0200393}} {{ZBL|}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> J. Dixmier, "Enveloping algebras" , Amer. Math. Soc. (1996) (Translated from French) {{MR|1451138}} {{MR|1393197}} {{ZBL|0867.17001}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> 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 {{MR|257160}} {{ZBL|}} </td></tr><tr><td valign="top">[a10]</td> <td valign="top"> 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) {{MR|399194}} {{ZBL|}} </td></tr><tr><td valign="top">[a11]</td> <td valign="top"> M. Duflo, "Théorie de Mackey pour les groupes de Lie algébriques" ''Acta Math.'' , '''149''' (1982) pp. 153–213 {{MR|0688348}} {{ZBL|0529.22011}} </td></tr><tr><td valign="top">[a12]</td> <td valign="top"> A.A. Kirillov, "Unitary representations of nilpotent Lie groups" ''Uspekhi Mat. Nauk'' , '''17''' (1962) pp. 57–110 (In Russian) {{MR|0142001}} {{ZBL|0106.25001}} </td></tr><tr><td valign="top">[a13]</td> <td valign="top"> O. Mathieu, "Bicontinuity of the Dixmier map" ''J. Amer. Math. Soc.'' , '''4''' (1991) pp. 837–863 {{MR|1126380}} {{MR|1115787}} {{ZBL|0743.17013}} {{ZBL|0762.17008}} </td></tr><tr><td valign="top">[a14]</td> <td valign="top"> C. Moeglin, "Ideaux primitifs des algèbres enveloppantes" ''J. Math. Pures Appl.'' , '''59''' (1980) pp. 265–336 {{MR|0604473}} {{ZBL|0454.17006}} </td></tr><tr><td valign="top">[a15]</td> <td valign="top"> C. Moeglin, "Ideaux primitifs completement premiers de l'algèbre enveloppante de ${\frak gl} ( n , {\bf C} )$" ''J. Algebra'' , '''106''' (1987) pp. 287–366 {{MR|}} {{ZBL|}} </td></tr><tr><td valign="top">[a16]</td> <td valign="top"> C. Moeglin, R. Rentschler, "Sur la classification des ideaux primitifs des algèbres enveloppantes" ''Bull. Soc. Math. France'' , '''112''' (1984) pp. 3–40 {{MR|0771917}} {{ZBL|0549.17007}} </td></tr><tr><td valign="top">[a17]</td> <td valign="top"> R. Rentschler, "L'injectivite de l'application de Dixmier pour les algèebres de Lie résolubles" ''Invent. Math.'' , '''23''' (1974) pp. 49–71 {{MR|}} {{ZBL|}} </td></tr></table> |
Latest revision as of 17:02, 1 July 2020
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 |
Dixmier mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dixmier_mapping&oldid=50465