[Paper Review] On quantizing semisimple basic algebras

Mark J. Gotay|ArXiv.org|Dec 18, 2000

Advanced Algebra and Geometry12 references3 citations

TL;DR

This paper establishes that polynomial quantizations of coordinate rings on basic nilpotent coadjoint orbits of semisimple Lie algebras—specifically in sl(2,ℝ)*—exist but are essentially trivial, as they vanish on quadratic and higher-degree polynomials. It further proves that no consistent polynomial quantization exists for basic semisimple orbits in sl(2,ℝ)*, demonstrating a Groenewold-Van Hove obstruction in this case due to the inhomogeneity of the orbit's defining ideal.

ABSTRACT

We show that there is a consistent polynomial quantization of the coordinate ring of a basic nilpotent coadjoint orbit of a semisimple Lie group. We also show, at least in the case of a nilpotent orbit in sl(2,R)*, that any such quantization is essentially trivial. Furthermore, we prove that the coordinate ring of a basic semisimple orbit in sl(2,R)* cannot be consistently polynomially quantized.

Motivation & Objective

• To determine whether consistent polynomial quantizations exist for the coordinate ring of a basic coadjoint orbit in the dual of a semisimple Lie algebra.
• To investigate the structure of polynomial quantizations when the orbit's defining ideal is homogeneous (as in nilpotent orbits) versus inhomogeneous (as in semisimple orbits).
• To clarify the role of the coadjoint orbit's geometry—particularly homogeneity of its ideal—in enabling or obstructing polynomial quantization.
• To extend the understanding of Groenewold-Van Hove obstructions beyond finite-dimensional or compact cases to infinite-dimensional polynomial quantizations on noncompact symplectic manifolds.
• To analyze the case of sl(2,ℝ)* in detail, distinguishing between nilpotent and semisimple orbits in terms of quantization feasibility.

Proposed method

• Construct a polynomial quantization by exploiting the conical structure of nilpotent coadjoint orbits, which ensures the coordinate ring splits as a semidirect product: P(M) = (ℝ ⊕ 𝔟) ⋉ P₂(M), where P₂(M) is the ideal of quadratic and higher-degree polynomials.
• Use the homogeneity of the ideal I(M) for nilpotent orbits to define a quantization that vanishes on P₂(M), resulting in a trivial but consistent quantization map.
• Apply representation-theoretic techniques from sl(2,ℝ) theory, including the use of the Casimir operator and the action of E± on eigenvectors of H, to analyze the spectrum and commutation relations.
• Derive a recursion relation (3.13) for the quantized Casimir operator Q(h²) in terms of the eigenvalues of H, showing that polynomial solutions are quadratic in n, while transcendental solutions exist.
• Use the digamma function to construct general solutions of the recursion, demonstrating that non-polynomial quantizations are possible, but polynomial ones are not.
• Apply the principle that a polynomial quantization must preserve the algebraic structure of P(M), and show that this fails when I(M) is inhomogeneous, as in semisimple orbits.

Experimental results

Research questions

• RQ1 Can a consistent polynomial quantization be constructed for the coordinate ring of a basic nilpotent coadjoint orbit in the dual of a semisimple Lie algebra?
• RQ2 Is any such polynomial quantization necessarily trivial, especially when the ideal I(M) is homogeneous?
• RQ3 Does the inhomogeneity of the ideal I(M) for semisimple orbits in sl(2,ℝ)* lead to an obstruction against polynomial quantization?
• RQ4 Can the existence of non-polynomial (transcendental) quantizations coexist with the non-existence of polynomial ones in the same setting?
• RQ5 To what extent does the homogeneity of the orbit's defining ideal determine the possibility of polynomial quantization?

Key findings

• A consistent polynomial quantization of the coordinate ring P(M) exists for any basic nilpotent coadjoint orbit M in 𝔟*, due to the conical and homogeneous nature of the orbit's ideal I(M).
• Any such polynomial quantization on a nilpotent orbit in sl(2,ℝ)* must be trivial, as it vanishes identically on P₂(M), the ideal of quadratic and higher-degree polynomials.
• No consistent polynomial quantization exists for basic semisimple orbits in sl(2,ℝ)*, as shown by a Groenewold-Van Hove obstruction arising from the inhomogeneity of I(M).
• The recursion relation (3.13) for the quantized Casimir operator Q(h²) admits polynomial solutions of the form ξₙ = γ − αn², but also transcendental solutions involving the digamma function, indicating non-polynomial quantizations are possible.
• The existence of polynomial quantizations is obstructed precisely when I(M) is inhomogeneous, as confirmed in the case of semisimple orbits in sl(2,ℝ)*, supporting a broader conjecture linking ideal homogeneity to quantization feasibility.
• The construction of a polynomial quantization relies crucially on the splitting P(M) = (ℝ ⊕ 𝔟) ⋉ P₂(M), which is only valid when I(M) is homogeneous, and fails otherwise.