Cayley--Hamilton Theorem for Orthogonal Quantum Matrix Algebras
Oleg Ogievetsky, Pavel Pyatov
TL;DR
The paper develops Cayley–Hamilton type theorems for orthogonal BMW-type quantum matrix algebras, distinguishing odd and even height cases and introducing separate identities for the two even-height components O^+(2ell) and O^−(2ell). By constructing a star-product-based QM-powers algebra and resolving reciprocal relations via extended algebras, the authors obtain explicit CH identities and then achieve a spectral parameterization that factorizes these identities into eigenvalue-like factors (M − q nu I) or (M − nu I). This spectral framework is complemented by explicit Newton and Wronski parameterizations of the characteristic subalgebra elements goth p_i and goth h_i in terms of spectral variables, enabling central extensions and a clear eigenstructure for orthogonal quantum matrices. The results generalize and unify the CH–Newton framework for orthogonal and symplectic BMW-type QM-algebras and provide a concrete path to eigenvalue-based descriptions of quantum matrix invariants with potential applications in quantum groups and noncommutative geometry.
Abstract
For a family of the orthogonal $O(k)$ type Quantum Matrix algebras we establish an analogue of the Cayley--Hamilton theorem. The form of the Cayley-Hamilton identity is different in three cases. First, the cases of odd ($k=2\ell -1$) and even ($k=2\ell$) heights are different. Second, for even height orthogonal Quantum Matrix algebra we derive two versions of the Cayley--Hamilton theorem, one for its positive component $O^+(2\ell)$ and another one for the negative component $O^-(2\ell)$. In each case we introduce the spectral parameterization of the coefficients of the Cayley--Hamilton identity by the `eigenvalues' of the quantum matrices.