# Irreducible representations of Heisenberg algebra

I need some references for irreducible representations of the Heisenberg algebra with three generators or the category of its finite length modules.

Here, the Heisenberg algebra with three generators $x$, $y$ and $z$ is defined to be the Lie algebra whose underlying vector space is generated by $x$, $y$, and $z$, and the commutator relations are give as follows: $[x,y]=-[y,x]=z$, and all other commuator relations are zero. So it is a nilpotent (and hence solvable) Lie algebra with a one dimensional center $Kz$, where $K$ is the ground field of zero characteristic.

(The Heisenberg algbera in $2n+1$ vectors is defined in same way, for more details I would suggest J. Diximier, "Enveloping algberas".)

• springerlink.com/content/kl6g82r836g23n82 – Steve Huntsman Jun 9 '10 at 23:32
• Which Heisenberg algebra? – S. Carnahan Jun 9 '10 at 23:34
• Do you mean the "canonical commutation relations"? If that is the case, maybe Petz:"An Invitation to the Algebra of Canonical Commutation Relations" could help? Or are you looking for something more advanced/particular? – Tim van Beek Jun 10 '10 at 14:10
• Do you want representations for which $z$ acts by a nonzero scalar, or arbitrary representations? Most of the theory, e.g., Stone-von-Neumann theorem, only applies to central representations. Also, does your ground field have characteristic zero, and are you only considering representations on complex vector spaces? – S. Carnahan Jun 11 '10 at 2:15
• To emphasize what Scott was saying, there are at least two completely different "representation theories" of Heisenberg algebra: (1) algebraic representations (which correspond to representations of the Heisenberg group as an algebraic group) and (2) unitary representations. Details of both theories depend on whether (a) $K$ is a local field of characteristic 0 (b) $K$ is finite (c) $K$ is a local field of positive characteristic. – Victor Protsak Jun 11 '10 at 3:01

If we just consider central representations, i.e., those for which $z$ acts by a nonzero scalar, then up to a certain kind of equivalence (given by conjugation with algebra isomorphisms) there is a unique irreducible representation of the algebra. It is infinite dimensional, given by polynomials in one variable. If the action of $z$ is multiplication by $\lambda \in K^\times$, then we can write the representation as $K[x]$, where the action of $x$ is multiplication by $x$, and the action of $y$ is $\lambda \frac{\partial}{\partial x}$.

If you want to rigidify the equivalence to isomorphism, (i.e., you want to remember the specific actions of the elements $x,y,z$ instead of just the algebra structure), then there is a parameter space of irreducible representations, with one coordinate describing the action of $z$ (i.e., taking values in nonzero elements of $K$), and the rest of the coordinates describing a line in the span of $x,y$. The construction generalizes to the $2n+1$ dimensional setting, where the representation is given by polynomial functions on a Lagrangian subspace of a $2n$ dimensional symplectic vector space, but you lose uniqueness. The parameter space of these representations involves a torus and the Lagrangian Grassmannian, but I don't remember if it has the geometric structure of a product or some kind of fibration.

The finite length modules are in bijection with finitely generated holonomic D-modules on the affine line, since the central condition endows the modules with an action of the Weyl algebra, which is a quotient of the universal enveloping algebra. As zamanjan notes in a comment here, when $n > 1$, this fails rather spectacularly. Stafford (1983) showed that there are irreducible modules with characteristic cycle of dimension $2n-1 > n$, and Bernstein-Lunts showed that in the $n=2$ case, this is in some sense a property of the generic irreducible module.

For non-central representations, things are considerably messier. When $z$ acts trivially, you're asking for pairs (or $2n$-tuples) of commuting matrices, and when $z$ is arbitrary, you can have finite dimensional representations look a lot like representations of arbitrary nilpotent lie algebras.

Regarding references, I guess anything about D-modules should work, but depending on your background, they may be hard to read. Howe has a paper called "A century of Lie theory", and Rosenberg has a paper called "A selective history of the Stone-von-Neumann theorem", both of which could be useful.

• Scott, you got it backwards: Kirillov's orbit method generalizes Stone-von Neumann theorem about irreducible $\textit{unitary}$ representations of the Heisenberg $\textit{group}$ to other nilpotent groups. So it says nothing about fin-dim indecomposable representations of the Heisenberg Lie algebra. There are Lie algebra analogues of the orbit method, but they have to do with primitive ideals, not with irreps (which cannot be classified in infinite dimensions). The most complete theory of "central" (z acts by scalar) representations for $n=1$, i.e. $A_1$-modules, is due to Berest and Wilson. – Victor Protsak Jun 12 '10 at 5:07
• Thanks, that clears up some confusion I had. I have removed the sentence about the orbit method, since it doesn't seem to be germane here. – S. Carnahan Jun 12 '10 at 5:11
• The finite length modules are not in bijection with finitely generated holonomic D-modules when n>1. Even simple modules need not be holonomic as shown by Stafford, Bernstein-Lunts. – zamanjan Nov 14 '11 at 3:29
• You're quite right - it seems I am 23 years out of date. I suppose I can take comfort in the fact that some famous people made the same mistake. I'll edit once I understand the result better. – S. Carnahan Nov 24 '11 at 9:16