  Let $R := R_{2}(p)=\mathbb{C}[t^{\pm 1}, u : u^2 = t(t-\alpha_1)\cdots
(t-\alpha_{2n})] $ be the coordinate ring of a nonsingular hyperelliptic curve
and let $\mathfrak{g}\otimes R$ be the corresponding current Lie algebra.
\color{black} Here $\mathfrak g$ is a finite dimensional simple Lie algebra
defined over $\mathbb C$ and \begin{equation*} p(t)= t(t-\alpha_1)\cdots
(t-\alpha_{2n})=\sum_{k=1}^{2n+1}a_kt^k. \end{equation*} In earlier work, Cox
and Im gave a generator and relations description of the universal central
extension of $\mathfrak{g}\otimes R$ in terms of certain families of
polynomials $P_{k,i}$ and $Q_{k,i}$ and they described how the center
$\Omega_R/dR$ of this universal central extension decomposes into a direct sum
of irreducible representations when the automorphism group was the cyclic group
$C_{2k}$ or the dihedral group $D_{2k}$. We give examples of $2n$-tuples
$(\alpha_1,\dots,\alpha_{2n})$, which are the automorphism groups $\mathbb
G_n=\text{Dic}_{n}$, $\mathbb U_n\cong D_n$ ($n$ odd), or $\mathbb U_n$ ($n$
even) of the hyperelliptic curves
  \begin{equation} S=\mathbb{C}[t, u: u^2 = t(t-\alpha_1)\cdots
(t-\alpha_{2n})]
  \end{equation} given in [CGLZ17]. In the work below, we describe this
decomposition when the automorphism group is $\mathbb U_n=D_n$, where $n$ is
odd.
