gl3 ALGEBRA IN MIXED MATRIX REPRESENTATIONS
DOI:
https://doi.org/10.14311/AP.2025.65.0562Keywords:
mixed representations in Fock space, matrix differential operators, matrix finite-difference (discrete) operators, matrix complex operators in (z, z̄)Abstract
It is demonstrated that the so-called mixed realization of the gl3 algebra generators in terms of matrix differential operators in two variables, as presented by Smirnov-Turbiner (2013), can be “lifted” into the action in the Fock space associated with the five-dimensional Heisenberg algebra h5. A realization of the gl3 generators in terms of matrix finite-difference (translation-invariant) operators, matrix discrete (dilatation-invariant) operators, matrix complex operators in (z, z̄), and their mixtures is presented.
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M. Havlíček. Personal communications, 1988–2010.
Y. F. Smirnov, A. Turbiner. gln+1 algebra of matrix differential operators and matrix quasi-exactly-solvable problems. Acta Polytechnica 53(5):462–469, 2013. https://doi.org/10.14311/AP.2013.53.0462
F. Tremblay, A. V. Turbiner, P. Winternitz. An infinite family of solvable and integrable quantum systems on a plane. Journal of Physics A: Mathematical and Theoretical 42(24):242001, 2009. https://doi.org/10.1088/1751-8113/42/24/242001
A. V. Turbiner, J. C. Lopez Vieyra, M. A. Guadarrama-Ayala. gl(3) polynomial integrable system: different faces of the 3-body/A2 elliptic Calogero model. Symmetry, Integrability and Geometry: Methods and Applications 20:012, 2024. https://doi.org/10.3842/SIGMA.2024.012
Y. Smirnov, A. Turbiner. Lie algebraic discretization of differential equations. Modern Physics Letters A 10(24):1795–1802, 1995. https://doi.org/10.1142/S0217732395001927
C. Chryssomalakos, A. Turbiner. Canonical commutation relation preserving maps. Journal of Physics A: Mathematical and General 34(48):10475, 2001. https://doi.org/10.1088/0305-4470/34/48/312
N. L. Vasilevski. Poly-Fock spaces. In V. M. Adamyan, I. Gohberg, M. Gorbachuk, et al. (eds.), Differential Operators and Related Topics, pp. 371–386. Birkhäuser Basel, Basel, 2000. https://doi.org/10.1007/978-3-0348-8403-7_28
A. V. Turbiner, N. Vasilevski. Poly-analytic functions and representation theory. Complex Analysis and Operator Theory 15(7):110, 2021. https://doi.org/10.1007/s11785-021-01154-y
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Copyright (c) 2025 Alexander V. Turbiner
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