Deformation Quantization with Separation of Variables of ₂‚₄(ℂ)

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2025
• Main Authors: Okuda, Taika, Sako, Akifumi
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2025
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/214175
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Deformation Quantization with Separation of Variables of ₂‚₄(ℂ). Taika Okuda and Akifumi Sako. SIGMA 21 (2025), 061, 32 pages

We construct a deformation quantization with separation of variables of the Grassmannian ₂‚₄(ℂ). A star product on ₂‚₄(ℂ) can be explicitly determined as the solution of the recurrence relations for ₂‚₄(ℂ) given by Hara and one of the authors (A. Sako). To provide a solution to the recurrence relations, it is necessary to solve a system of linear equations in each order. However, to give a concrete expression of the general term is not simple because the variables increase with the order of differentiation of the star product. For this reason, there has been no formula to express the general term of the recurrence relations. In this paper, we overcome this problem by transforming the recurrence relations into simpler ones. We solve the recurrence relations using creation and annihilation operators on a Fock space. From this solution, we obtain an explicit formula of a star product with separation of variables on ₂‚₄(ℂ).