Quantization of Lie-Poisson algebra and Lie algebra solutions of mass-deformed type IIB matrix model
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.