Cayley--Hamilton Theorem for Orthogonal Quantum Matrix Algebras

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 ($k=2\mathrm{\ell }-1$) and even ($k=2\mathrm{\ell }$) 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 $O^{+}(2\mathrm{\ell })$ and another one for the negative component $O^{-}(2\mathrm{\ell })$. In each case we introduce the spectral parameterization of the coefficients of the Cayley–Hamilton identity by the `eigenvalues’ of the quantum matrices.