[Paper Review] Twisted Lie group $C^*$-algebras as strict quantizations

N.P. Landsman|arXiv (Cornell University)|Jul 24, 1998

Advanced Topics in Algebra1 references4 citations

TL;DR

This paper establishes that twisted group C*-algebras $C^*(G,c)$ associated with a smooth multiplier $c \in Z^2(G,U(1))$ on a compact Lie group $G$ constitute a strict quantization of the twisted Lie-Poisson algebra $C^\infty(\g^*_{(\Gamma)})$, where the Planck constant $\hbar$ takes values in $(\mathbb{Z} \setminus \{0\})^{-1}$. The key result is a continuous field of C*-algebras linking the classical algebra $C_0(\g^*)$ at $\hbar = 0$ to the quantum algebra $C^*(G,c)$ for $\hbar \neq 0$, satisfying Dirac's condition asymptotically.

ABSTRACT

A nonzero 2-cocycle $Γ\in Z^2(\g,\R)$ on the Lie algebra $\g$ of a compact Lie group $G$ defines a twisted version of the Lie-Poisson structure on the dual Lie algebra $\g^*$, leading to a Poisson algebra $C^{\infty}(\g_{(Γ)}^*)$. Similarly, a multiplier $c\in Z^2(G,U(1))$ on $G$ which is smooth near the identity defines a twist in the convolution product on $G$, encoded by the twisted group $C^*$-algebra C^*(G,c)$. Further to some superficial yet enlightening analogies between $C^{\infty}(\g^*_{(Γ)})$ and $C^*(G,c)$, it is shown that the latter is a strict quantization of the former, where Planck's constant $\hbar$ assumes values in $(\Z\backslash\{0\})^{-1}$. This means that there exists a continuous field of $C^*$-algebras, indexed by $\hbar\in 0\cup (\Z\backslash\{0\})^{-1}$, for which $\A^0=C_0(\g^*)$ and $\A_{\hbar}=C^*(G,c)$ for $\hbar eq 0$, along with a cross-section of the field satisfying Dirac's condition asymptotically relating the commutator in $\A_{\hbar}$ to the Poisson bracket on $C^{\infty}(\g^*_{(Γ)})$. Note that the `quantization' of $\hbar$ does not occur for $Γ=0$.

Motivation & Objective

• To establish a rigorous connection between twisted group C*-algebras and classical twisted Poisson structures on dual Lie algebras.
• To demonstrate that the twisted group C*-algebra $C^*(G,c)$ serves as a strict quantization of the Poisson algebra $C^\infty(\g^*_{(\Gamma)})$.
• To clarify the role of Planck's constant $\hbar$ in this quantization, showing it takes discrete inverse-integer values.
• To extend the framework of strict quantization to include nontrivial twists via group 2-cocycles and Lie algebra 2-cocycles.
• To unify geometric quantization with operator algebraic structures in the context of compact Lie groups with central extensions.

Proposed method

• Construct a continuous field of C*-algebras $\mathcal{A}_\hbar$ indexed by $\hbar \in \{0\} \cup (\mathbb{Z} \setminus \{0\})^{-1}$, with $\mathcal{A}^0 = C_0(\g^*)$ and $\mathcal{A}_\hbar = C^*(G,c)$ for $\hbar \neq 0$.
• Define a twisted Lie-Poisson structure on $\g^*$ using a nonzero 2-cocycle $\Gamma \in Z^2(\g,\mathbb{R})$, yielding the Poisson algebra $C^\infty(\g^*_{(\Gamma)})$.
• Use a smooth multiplier $c \in Z^2(G,U(1))$ near the identity to define a twisted convolution product on $G$, giving rise to the twisted group C*-algebra $C^*(G,c)$.
• Construct a cross-section of the continuous field such that the commutator in $\mathcal{A}_\hbar$ asymptotically reproduces the Poisson bracket on $C^\infty(\g^*_{(\Gamma)})$ as $\hbar \to 0$, satisfying Dirac's condition.
• Leverage the correspondence between the Lie algebra 2-cocycle $\Gamma$ and the group 2-cocycle $c$ via the exponential map, ensuring consistency between classical and quantum structures.
• Ensure the quantization is strict by verifying continuity of the field and the asymptotic behavior of the commutator in the $\hbar \to 0$ limit.

Experimental results

Research questions

• RQ1 Can twisted group C*-algebras $C^*(G,c)$ be interpreted as a strict quantization of the twisted Lie-Poisson algebra $C^\infty(\g^*_{(\Gamma)})$?
• RQ2 What is the role of Planck's constant $\hbar$ in this quantization, and why does it take values in $(\mathbb{Z} \setminus \{0\})^{-1}$?
• RQ3 How is the asymptotic relation between the commutator in $C^*(G,c)$ and the Poisson bracket on $\g^*_{(\Gamma)}$ realized in the $\hbar \to 0$ limit?
• RQ4 What is the precise relationship between the Lie algebra 2-cocycle $\Gamma$ and the group 2-cocycle $c$ that ensures consistency between classical and quantum structures?
• RQ5 Is there a continuous field of C*-algebras that interpolates between the classical algebra $C_0(\g^*)$ and the quantum algebra $C^*(G,c)$ for $\hbar \neq 0$?

Key findings

• The twisted group C*-algebra $C^*(G,c)$ is a strict quantization of the twisted Lie-Poisson algebra $C^\infty(\g^*_{(\Gamma)})$ via a continuous field of C*-algebras indexed by $\hbar \in \{0\} \cup (\mathbb{Z} \setminus \{0\})^{-1}$.
• The classical limit $\hbar \to 0$ recovers $C_0(\g^*)$, while for $\hbar \neq 0$, the algebra $\mathcal{A}_\hbar = C^*(G,c)$ captures the quantum structure.
• The commutator in $\mathcal{A}_\hbar$ asymptotically satisfies Dirac's condition, with $\frac{1}{i\hbar}[a,b]_\hbar \to \{a,b\}_{\Gamma}$ as $\hbar \to 0$ for smooth functions $a,b$ on $\g^*$.
• The value of $\hbar$ is restricted to inverse integers, reflecting the topological nature of the twist via the multiplier $c \in Z^2(G,U(1))$.
• The construction establishes a consistent correspondence between the Lie algebra 2-cocycle $\Gamma$ and the group 2-cocycle $c$, ensuring compatibility between classical and quantum levels.
• The quantization is strict, meaning the field of C*-algebras is continuous and the classical limit is well-defined, satisfying the axioms of strict quantization.