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?



Fixed points of quantised enveloping algebra for affine $\mathfrak{sl}_n$
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](https://arxiv.org/abs/1007.4357) in the finite case. I'm not sure if anyone has thought about the affine case.


