Action of the Quantum Matrix Spectrum in Finite-dimensional RE-algebra Representations

Student: Zaytsev Mikhail

Educational Programme: Mathematics (Bachelor)

Final Grade: 10

Year of Graduation: 2024

We consider a class of finite-dimensional decomposable Reflection Equation algebra representations. An important tool for working with them is a characteristic subalgebra. It is central and can be generated by quantum power sums or quantum elementary symmetric polynomials. We define them in two forms: as polynomials in generators of the given Reflection Equation algebra and in terms of so-called eigenvalues of the generating matrix M, defined by means of the quantum Cayley-Hamilton identity. We establish two types of quantum power sums and prove Newton identities connecting them with quantum elementary symmetric polynomials. In particular, this construction allows us to give explicit formulas for generators in terms of the quantum matrix spectrum and find eigenvalues for their images in the representations.

Full text (added May 31, 2024)

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