In mathematics, the Dixmier mapping describes the space Prim(''U''(''g'')) of

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 ar ...

s of the

universal enveloping algebra In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representati ...

''U''(''g'') of a finite-dimensional

solvable Lie algebra In mathematics, a Lie algebra \mathfrak is solvable if its derived series terminates in the zero subalgebra. The ''derived Lie algebra'' of the Lie algebra \mathfrak is the subalgebra of \mathfrak, denoted : mathfrak,\mathfrak/math> that consist ...

''g'' over an

algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...

of characteristic 0 in terms of

coadjoint orbit In mathematics, the coadjoint representation K of a Lie group G is the dual of the adjoint representation. If \mathfrak denotes the Lie algebra of G, the corresponding action of G on \mathfrak^*, the dual space to \mathfrak, is called the coadjoint ...

s. More precisely, it is a homeomorphism from the space of orbits ''g''^*/''G'' of the dual ''g''^* of ''g'' (with the

Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...

) under the action of the adjoint group ''G'' to Prim(''U''(''g'')) (with the

Jacobson topology In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra ''A'', denoted ''Â'', is the set of unitary equivalence classes of irreducible *-representations of ''A''. A *-representation π of ''A'' on a Hilbert space ''H'' is irreducibl ...

). The Dixmier map is closely related to the

orbit method In mathematics, the orbit method (also known as the Kirillov theory, the method of coadjoint orbits and by a few similar names) establishes a correspondence between irreducible unitary representations of a Lie group and its coadjoint orbits: or ...

, which relates the irreducible representations of a nilpotent Lie group to its coadjoint orbits. introduced the Dixmier map for

nilpotent Lie algebra In mathematics, a Lie algebra \mathfrak is nilpotent if its lower central series terminates in the zero subalgebra. The ''lower central series'' is the sequence of subalgebras : \mathfrak \geq mathfrak,\mathfrak\geq mathfrak,[\mathfrak,\mathfrak ...

s and then in extended it to solvable ones. describes the Dixmier mapping in detail.

Construction

Suppose that ''g'' is a completely solvable Lie algebra, and ''f'' is an element of the dual ''g''^*. A polarization of ''g'' at ''f'' is a subspace ''h'' of maximal dimension subject to the condition that ''f'' vanishes on [''h'',''h''], that is also a subalgebra. The Dixmier map ''I'' is defined by letting ''I''(''f'') be the kernel of the twisted induced representation Ind^~(''f'', ''h'',''g'') for a polarization ''h''.

References

* * * *{{eom, id=Dixmier_mapping Lie algebras