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Consider the automorphism of the algebra $U_q(\widehat{\mathfrak{sl}}_n)$ induced by the obvious diagram automorphism of the extended type A Dynkin diagram. More precisely, if the vertices of the Dynkin diagram are labelled $0,1,\ldots,n-1$, define $\overline{i}$ to be the number which is congruent to $-i$ modulo $n$. Then the diagram map sends the vertex labelled $i$ to the vertex labelled $\overline{i}$, and this induces an involution on the algebra which is given in terms of the Serre generators as $e_i\mapsto e_{\overline{i}}$, $f_i\mapsto f_{\overline{i}}$, $t_i^{\pm 1}\mapsto t_{\overline i}^{\pm 1}$ and $q^d\mapsto q^d$. Call this involution $\sigma$ (it is easy but tedious to check that this is indeed an involution).

Now consider the algebra $U_q(\widehat{\mathfrak{sl}}_n)^\sigma$ of points fixed by the involution $\sigma$. My question is: can this algebra be realised as $U_q(\mathfrak{g})$ for some affine Lie algebra $\mathfrak{g}$? If not, is there a known presentation of the fixed point algebra by generators and relations?

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up vote 2 down vote accepted

I'm not sure that I know a proof that there's no way to write that algebra as a quantized universal enveloping algebra, but it's definitely not the QUEA of $(\mathfrak{\widehat{sl}}_n)^\sigma$, the fixed points on the Lie algebra (since it's not generated by the elements you expect). Even worse, it doesn't even contain $U_q\big((\mathfrak{\widehat{sl}}_n)^\sigma\big)$ as a subalgebra! The degree to which this can be fixed has been studied by Berenstein and Greenstein in the finite case. I'm not sure if anyone has thought about the affine case.

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