[Paper Review] Deformation quantization of Leibniz algebras

Benoît Dherin, Friedrich Wagemann|arXiv (Cornell University)|Oct 25, 2013

Advanced Topics in Algebra13 references3 citations

TL;DR

This paper proposes a deformation quantization of the dual space of a Leibniz algebra using a rack product structure derived from a Baker-Campbell-Hausdorff-type formula. It constructs a star product on smooth functions on the dual space by integrating the Leibniz algebra via Lie racks, yielding a non-commutative deformation that recovers the Leibniz-Poisson bracket in the classical limit, thus generalizing geometric quantization to non-skew-symmetric brackets.

ABSTRACT

This paper has two parts. The first part is a review and extension of the methods of integration of Leibniz algebras into Lie racks, including as new feature a new way of integrating 2-cocycles (see Lemma 3.9). In the second part, we use the local integration of a Leibniz algebra h using a Baker-Campbell-Hausdorff type formula in order to deformation quantize its linear dual h^*. More precisely, we define a natural rack product on the set of exponential functions which extends to a rack action on C^{\infty}(h^*).

Motivation & Objective

• To solve the long-standing problem of quantizing the dual space of a Leibniz algebra, which carries a generalized Poisson structure due to the non-skew-symmetric bracket.
• To establish Lie racks as the correct integration structure for Leibniz algebras, generalizing Lie’s Third Theorem to the non-Lie case.
• To construct a deformation quantization of the smooth functions on the dual space using a rack action and oscillatory integrals.
• To show that the resulting star product reproduces the Leibniz-Poisson bracket in the classical limit (ℏ → 0), extending geometric quantization to non-Lie algebras.

Proposed method

• Integrate the Leibniz algebra into a local Lie rack using the adjoint action via the exponential map, generalizing the conjugation action in Lie groups.
• Define a rack product on exponential functions via $ X \rhd Y := e^{\mathrm{ad}_X}(Y) $, which extends to a rack action on $ \mathcal{C}^\infty(\mathfrak{h}^*) $.
• Construct a star product via an oscillatory integral representation: $ (f \rhd_\hbar g)(\xi) = \int \frac{d\bar{X}d\bar{Y}d\bar{\zeta}d\bar{\eta}}{(2\pi\hbar)^n} e^{i S_\xi / \hbar} f(\bar{X}) g(\bar{Y}) $, with phase $ S_\xi = -\bar{X}\bar{\zeta} - \bar{Y}\bar{\eta} + \langle \xi, \exp(\mathrm{ad}_{\bar{X}})(\bar{Y}) \rangle $.
• Apply the Feynman expansion to the oscillatory integral, computing the asymptotic expansion in powers of $ \hbar $, with amplitudes derived from the Hessian of the phase at the critical point.
• Identify the first-order term in $ \hbar $ as the Leibniz-Poisson bracket: $ \{f,g\}(\xi) = \sum_{i,j,k} c_{ij}^k \frac{\partial f}{\partial \xi_i}(0) \frac{\partial g}{\partial \xi_j}(\xi) \xi_k $.
• Prove that the zeroth-order term $ f(0)g(\xi) $ is associative, confirming the classical limit.

Experimental results

Research questions

• RQ1 Can the dual space of a Leibniz algebra be deformation quantized in a way that generalizes geometric quantization beyond Lie algebras?
• RQ2 Is the rack structure—specifically the conjugation-like action $ X \rhd Y = e^{\mathrm{ad}_X}(Y) $—the correct non-abelian generalization of Lie group conjugation for integrating Leibniz algebras?
• RQ3 Does the oscillatory integral construction yield a star product whose classical limit recovers the Leibniz-Poisson bracket?
• RQ4 How does the Feynman diagram expansion of the oscillatory integral reflect the underlying Leibniz algebra structure?
• RQ5 Is the resulting star product associative at the quantum level, and what is the role of the critical point and Hessian in the asymptotic expansion?

Key findings

• The oscillatory integral representation of the star product has a unique non-degenerate critical point at $ (\bar{X}=0, \bar{Y}=0, \bar{\zeta}=0, \bar{\eta}=\xi) $, ensuring a well-defined asymptotic expansion.
• The Hessian of the phase at the critical point has determinant 1 and signature 0, validating the use of the Feynman expansion with a phase factor $ e^{i\pi/4} $.
• The first-order term in the $ \hbar $-expansion is the Leibniz-Poisson bracket $ \{f,g\}(\xi) = \sum_{i,j,k} c_{ij}^k \frac{\partial f}{\partial \xi_i}(0) \frac{\partial g}{\partial \xi_j}(\xi) \xi_k $, confirming the classical limit.
• The zeroth-order term $ f(0)g(\xi) $ is associative, indicating that the classical product is well-behaved and consistent with the deformation.
• The Feynman graphs contributing to the expansion are simple, with no internal loops, and the amplitudes are computed via contraction with the inverse Hessian matrix.
• The construction generalizes the Lie algebra case: when the Leibniz algebra is Lie (i.e., skew-symmetric bracket), the rack product reduces to conjugation in a Lie group, and the star product recovers standard deformation quantization.