
Applications: Algebraic Dependency In Quantum Algebra
This nonstandard quantum algebra arised from the theoretical physics
U_{q}(so_{3})
= < x, y, z  yx = q xy  q^{1/2} z,
zx =  (q + 1) xz  q^{1/2}(q + 1) y,
zy = q yz  q^{1/2}x >.
If we consider q as a free parameter,
U_{q}(so_{3}) has
only one central element C_{q} .
However,
if we specialize q
at the nth root of unity,
there will appear three additional central elements. If n=3, these are
C_{q} =
q^{2} x^{2}
+ y^{2}+q^{2}z^{2}
+q^{1/2}(1q^{2})xyz,

C_{1} =
1/3 (x^{3} + x),

C_{2} =
1/3 (y^{3} + y),

C_{3} =
1/3 (z^{3} + z).

Task: Compute the polynomial, desribing the algebraic dependency
of the central elements. 
Answer:
C_{q}^{3} +
81q^{1/2}(q+2)C_{1}C_{2}C_{3}  q C_{q}^{2}
 9(C_{1}^{2} +
C_{2}^{2} +
C_{3}^{2})

