Let \(\mathfrak{g}\) be a finite-dimensional semi-simple complex Lie algebra, and let \(\theta\) be an involutive automorphism of \(\mathfrak{g}\). G. Letzter [J. Algebra 220, No. 2, 729–767 (1999; Zbl 0956.17007)] constructs a coideal subalgebra \(B\) of the quantized universal enveloping algebra \(U_q(\mathfrak{g})\) that is a quantum analogue of the fixed-point subalgebra \(\mathfrak{k}=\mathfrak{g}^\theta\). More recently, M. Balagović and S. Kolb [J. Reine Angew. Math. 747, 299–353 (2019; Zbl 1425.81058)] have shown that \(B\) has a universal \(K\)-matrix \(\mathcal{K}\). Note that \(\theta\), \(\mathfrak{k}\), \(B\), and \(\mathcal{K}\) can be described in terms of Satake diagrams. A Satake diagram is a refinement of a Dynkin diagram obtained by coloring some of the nodes and allowing arrows between certain nodes. Satake diagrams can be used to classify the involutive automorphisms (and therefore the real forms) of \(\mathfrak{g}\). They also allow to describe the fixed-point subalgebra \(\mathfrak{k} =\mathfrak{g}^\theta\) in terms of Chevalley-type generators and Serre-type relations.
In the paper under review the constructions of \(\theta\), \(\mathfrak{k}\), \(B\), and \(\mathcal{K}\) are extended to generalized Satake diagrams first considered by A. Heck [J. Math. Soc. Japan 36, 643–658 (1984; Zbl 0533.17003)]. In type \(A\) this generalization does not yield anything new, but in all other types there are generalized Satake diagrams that are not Satake. Such a generalized Satake diagram naturally defines a semi-simple automorphism \(\theta\) of \(\mathfrak{g}\) whose restriction to the standard Cartan subalgebra \(\mathfrak{h}\) of \(\mathfrak{g}\) is an involution. Moreover, it defines a subalgebra \(\mathfrak{k}\) of \(\mathfrak{g}\) such that \(\mathfrak{k}\cap\mathfrak{h}=\mathfrak{h}^\theta\), but \(\mathfrak{k}\) is not always fixed under \(\theta\). The authors show that the subalgebra \(\mathfrak{k}\) can be quantized to a coideal subalgebra \(B\) of \(U_q(\mathfrak{g})\) which is endowed with a universal \(K\)-matrix in the sense of Balagović and Kolb, and they conjecture that conversely any such quasi-triangular coideal subalgebra arises from a generalized Satake diagram in this way.