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Unitary and Nonunitary Representations of the Heisenberg-Weyl Lie Algebra

Andrew Douglas, Hubert de Guise, Joe Repka

Abstract

We examine unitary and nonunitary representations of the Heisenberg-Weyl Lie algebra $\mathfrak{hw}_n$, with particular emphasis on tensor products of unitary representations and on indecomposable nonunitary representations. In the unitary setting, the irreducible representations with nontrivial central character are the Schrödinger representations, as classified by the Stone-von Neumann theorem. Although tensor products of these representations are considered in the literature, we give a detailed Lie-algebraic analysis and construct explicit unitary intertwining operators, including the case where the central characters sum to zero. In the nonunitary setting, we consider a natural realization of $\mathfrak{hw}_n$ as a subalgebra of the real symplectic Lie algebra $\mathfrak{sp}_{2n+2}(\mathbb R)$ and prove that every finite-dimensional complex irreducible representation of $\mathfrak{sp}_{2n+2}(\mathbb{R})$ remains indecomposable upon restriction to $\mathfrak{hw}_n$. This yields a large natural family of finite-dimensional, nonunitary indecomposable representations of $\mathfrak{hw}_n$.

Unitary and Nonunitary Representations of the Heisenberg-Weyl Lie Algebra

Abstract

We examine unitary and nonunitary representations of the Heisenberg-Weyl Lie algebra , with particular emphasis on tensor products of unitary representations and on indecomposable nonunitary representations. In the unitary setting, the irreducible representations with nontrivial central character are the Schrödinger representations, as classified by the Stone-von Neumann theorem. Although tensor products of these representations are considered in the literature, we give a detailed Lie-algebraic analysis and construct explicit unitary intertwining operators, including the case where the central characters sum to zero. In the nonunitary setting, we consider a natural realization of as a subalgebra of the real symplectic Lie algebra and prove that every finite-dimensional complex irreducible representation of remains indecomposable upon restriction to . This yields a large natural family of finite-dimensional, nonunitary indecomposable representations of .
Paper Structure (13 sections, 9 theorems, 108 equations, 2 tables)

This paper contains 13 sections, 9 theorems, 108 equations, 2 tables.

Table of Contents

  1. Introduction
  2. The Heisenberg--Weyl Group and Lie Algebra
  3. Unitary Representations: Schrödinger Representations
  4. Derived Representations of $\mathfrak{hw}_n$
  5. Tensor Products of Schrödinger Representations
  6. Nonunitary Representations via Symplectic Embedding
  7. Closed Subsets of Root Systems and Regular Subalgebras
  8. The Real and Complex Symplectic Algebras
  9. Indecomposable Representations of $\mathfrak{hw}_n$ via Symplectic Embedding
  10. Conclusion
  11. Appendix: Constructing lowest weight states
  12. Single oscillator
  13. Constructing lowest weight states in the decomposition

Key Result

Theorem 3.1

For each $\lambda \in \mathbb{R}\setminus\{0\}$, there exists, up to unitary equivalence, a unique irreducible unitary representation of the Heisenberg--Weyl group $HW_n$ with central character This representation is unitarily equivalent to the Schrödinger representation $\pi_\lambda$ on $L^2(\mathbb{R}^n)$.

Theorems & Definitions (20)

  • Theorem 3.1: Stone--von Neumann, group version
  • Remark 3.2: Vanishing central character
  • Theorem 3.3: Stone--von Neumann, Lie algebra version
  • Theorem 3.4
  • proof
  • Remark 3.5: Non-irreducibility of the tensor product
  • Theorem 3.6
  • proof
  • Remark 3.7: Outside the scope of the Stone--von Neumann framework
  • Lemma 4.1
  • ...and 10 more