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Algebraic Dependency in Quantum Algebras
The following non-standard quantum algebra arises from theoretical physics:
Uq(so3)
= < x, y, z | yx = q xy - q1/2 z,
zx = - (q+1) xz - q1/2(q+1) y,
zy = q yz - q1/2x >.
If we consider q as a free parameter,
Uq(so3) has
only one central element Cq .
However,
if we specialize q
to the n-th root of unity,
there appear three additional central elements. For
n=3, the central elements are
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Cq =
q2 x2
+ y2 +
q2z2
+
q1/2(1-q2)xyz,
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C1 =
1/3 (x3 + x),
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C2 =
1/3 (y3 + y),
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C3 =
1/3 (z3 + z).
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Task: Compute the polynomial, describing the
algebraic dependency of the central elements. |
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Answer:
Cq3 +
81q1/2(q+2)C1C2C3 - q Cq2
- 9(C12 +
C22 +
C32) .
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