Quantization of Lie-Poisson algebra and Lie algebra solutions of mass-deformed type IIB matrix model
Jumpei Gohara, Akifumi Sako
TL;DR
The work develops a relaxed matrix-regularization framework, termed weak matrix regularization, to quantize Lie-Poisson algebras on varieties defined by Casimir polynomials, connecting classical semisimple Lie algebra solutions of the mass-deformed IKKT matrix model to noncommutative matrix algebras. By embedding into enveloping algebras and employing fixed Gröbner bases, the authors construct quantizations $q_{A/I,\mu}$ that preserve Lie-Poisson brackets to leading order in $\hbar$ and accommodate quotients by Casimir ideals $I(C)$. Explicit realizations for $\mathfrak{su}(2)$ and $\mathfrak{su}(3)$ demonstrate fuzzy spaces (fuzzy sphere and higher-dimensional analogues) and illustrate how quadratic and cubic Casimirs yield distinct geometric varieties such as $\mathbb{R}^8$, $S^7$, and cubic Casimir surfaces. The framework further generalizes to reducible representations, enabling multiple coadjoint orbits to be embedded, thereby enriching the classical-space reflection in the commutative limit. Overall, the paper extends fuzzy-space constructions beyond the quadratic Casimir, providing a scalable path from mass-deformed IKKT classical solutions to Lie-Poisson quantizations via Gröbner-basis techniques and enveloping-algebra quantization.
Abstract
A quantization of Lie-Poisson algebras is studied. Classical solutions of the mass-deformed Ishibashi-Kawai-Kitazawa-Tsuchiya (IKKT) matrix model can be constructed from semisimple Lie algebras whose dimension matches the number of matrices in the model. We consider the geometry described by the classical solutions of the Lie algebras in the limit where the mass vanishes and the matrix size tends to infinity. Lie-Poisson varieties are regarded as such geometric objects. We provide a quantization called ``weak matrix regularization'' of Lie-Poisson algebras (linear Poisson algebras) on the algebraic varieties defined by their Casimir polynomials. This quantization is a generalization of matrix regularization, and neither faithfulness of the map nor the correspondence between integration and trace in the commutative limit is required. Casimir polynomials correspond with Casimir operators of the Lie algebra by the quantization. This quantization is a generalization of the method for constructing the fuzzy sphere. In order to define the weak matrix regularization of the quotient space by the ideal generated by the Casimir polynomials, we take a fixed reduced Gröbner basis of the ideal. The Gröbner basis determines remainders of polynomials. The operation of replacing these remainders with representation matrices of a Lie algebra roughly corresponds to a weak matrix regularization. As concrete examples, we construct weak matrix regularization for $\mathfrak{su}(2)$ and $\mathfrak{su}(3)$. In the case of $\mathfrak{su}(3)$, we not only construct weak matrix regularization for the quadratic Casimir polynomial, but also construct weak matrix regularization for the cubic Casimir polynomial.