Abstract:
We continue the study of quantum matrix algebras of the GL(m|n) type. We find three alternative forms of the Cayley–Hamilton identity; most importantly, this identity can be represented in a factored form. The factorization allows naturally dividing the spectrum of a quantum supermatrix into subsets of “even” and “odd” eigenvalues. This division leads to a parameterization of the characteristic subalgebra (the subalgebra of spectral invariants) in terms of supersymmetric polynomials in the eigenvalues of the quantum matrix. Our construction is based on two auxiliary results, which are independently interesting. First, we derive the multiplication rule for Schur functions sλ(M), that form a linear basis of the characteristic subalgebra of a Hecke-type quantum matrix algebra; the structure constants in this basis coincide with the Littlewood–Richardson coefficients. Second, we prove a number of bilinear relations in the graded ring Λ of symmetric functions of countably many variables.
Citation:
D. I. Gurevich, P. N. Pyatov, P. A. Saponov, “Quantum matrix algebras of the GL(m|n) type: The structure and spectral parameterization of the characteristic subalgebra”, TMF, 147:1 (2006), 14–46; Theoret. and Math. Phys., 147:1 (2006), 460–485
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\paper Quantum matrix algebras of the $GL(m|n)$ type: The structure and spectral parameterization of the characteristic subalgebra
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\jour Theoret. and Math. Phys.
\yr 2006
\vol 147
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Linking options:
https://www.mathnet.ru/eng/tmf2020
https://doi.org/10.4213/tmf2020
https://www.mathnet.ru/eng/tmf/v147/i1/p14
This publication is cited in the following 15 articles:
Dimitry Gurevich, Pavel Saponov, “Generalized Harish-Chandra morphism on Reflection Equation algebras”, Journal of Geometry and Physics, 2025, 105435
Dimitry Gurevich, Pavel Saponov, “Quantum Schur–Weyl Duality and q-Frobenius Formula Related to Reflection Equation Algebras”, International Mathematics Research Notices, 2025:3 (2025)
Sidarth Erat, Arun S. Kannan, Shihan Kanungo, “Mixed Tensor Products, Capelli Berezinians, and Newton's Formula for gl(m|n)”, Transformation Groups, 2025
Ogievetsky O. Pyatov P., “Quantum Matrix Algebras of Bmw Type: Structure of the Characteristic Subalgebra”, J. Geom. Phys., 162 (2021), 104086
Ogievetsky O. Pyatov P., “Cayley-Hamilton Theorem For Symplectic Quantum Matrix Algebras”, J. Geom. Phys., 165 (2021), 104211
Gurevich D. Saponov P., “Braided Algebras and their Applications to Noncommutative Geometry”, Adv. Appl. Math., 51:2 (2013), 228–253
Lehrer G.I., Zhang H., Zhang R.B., “A Quantum Analogue of the First Fundamental Theorem of Classical Invariant Theory”, Comm Math Phys, 301:1 (2011), 131–174
Gurevich D., Pyatov P., Saponov P., “Braided differential operators on quantum algebras”, J Geom Phys, 61:8 (2011), 1485–1501
Gurevich D., Pyatov P., Saponov P., “Bilinear identities on Schur symmetric functions”, J. Nonlinear Math. Phys., 17, Suppl. 1 (2010), 31–48
Gurevich D., Saponov P., “Generic super-orbits in gl(m|n)∗ and their braided counterparts”, J. Geom. Phys., 60:10 (2010), 1411–1423
D. I. Gurevich, P. N. Pyatov, P. A. Saponov, “Spectral parameterization for power sums of a quantum supermatrix”, Theoret. and Math. Phys., 159:2 (2009), 587–597
Isaev, AP, “Spectral Extension of the Quantum Group Cotangent Bundle”, Communications in Mathematical Physics, 288:3 (2009), 1137
Gurevich, D, “Braided affine geometry and q-analogs of wave operators”, Journal of Physics A-Mathematical and Theoretical, 42:31 (2009), 313001
D. I. Gurevich, P. N. Pyatov, P. A. Saponov, “Representation theory of (modified) Reflection Equation Algebra of GL(m|n) type”, St. Petersburg Math. J., 20:2 (2009), 213–253
Hai PH, Kriegk B, Lorenz M, “N-homogeneous superalgebras”, Journal of Noncommutative Geometry, 2:1 (2008), 1–51
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