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The rank-one small quantum group at a primitive fifth root has dimension 125

Example

Let $ζ$ be a primitive fifth root of unity and let $u_{ζ} ({s l}_{2})$ be the rank-one root-power quotient of De Concini–Kac, Lusztig, and small quantum-group forms at roots of unity. Its generators satisfy $E^{5} = F^{5} = 0$ and $K^{5} = 1$ .

Facts & Assumptions

Given: A primitive fifth root $ζ$ and the rank-one quotient $u_{ζ} ({s l}_{2})$ .

[F1]

For every odd $ℓ > 2$ , the quotient has basis ${{\overline{F}}^{a} {\overline{K}}^{b} {\overline{E}}^{c} : 0 \leq a, b, c < ℓ}$ and dimension $ℓ^{3}$ (The rank-one root-power quotient is a finite-dimensional Hopf algebra).

Verification

1.1givenF1algebra

Since $ζ$ has order five, $ζ^{5} = 1$ and $ζ^{2}$ also has order five. In particular $ζ - ζ^{- 1} \neq 0$ , so the rank-one root forms and the quotient in [F1] are defined at this parameter.

2.1givenF1step 1.1algebra∎

Substitute $ℓ = 5$ in the basis assertion [F1]. The exponents $a, b, c$ each range independently over ${0, 1, 2, 3, 4}$ , so this basis has $5 \cdot 5 \cdot 5 = 125$ elements. Hence $\dim_{C} u_{ζ} ({s l}_{2}) = 125$ .

Depends on

Used by

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Sources

Pavel Etingof and Mykola Semenyakin, A Brief Introduction to Quantum Groups (standard reference, not scraped)