Automorphisms of \(\check U_ q({\mathfrak b}^ +)\)
Let \(\mathfrak g\) be a simple Lie algebra over \(\mathbb C\) and \(q\) a nonzero complex number that is not a root of unity. The author computes the groups of algebra and Hopf algebra automorphisms of the augmented form \(\check U_q(\mathfrak b^+)\) of the quantized enveloping algebra of \(\mathfrak b^+\) with \(\mathfrak b^+\) a Borel subalgebra of \(\mathfrak g\). There are substantially fewer of these than there are automorphisms of \(U_q(\mathfrak g)\) or \(U(\mathfrak g)\). More precisely, it turns out the algebra automorphisms of \(\check U_q(\mathfrak b^+)\) are generated by a \(2n\)-dimensional multiplicative torus, the Dynkin diagram automorphisms, and the vector space of \(n\times n\) symmetric integer matrices, where \(n\) is the rank of \(\mathfrak g\). The Hopf algebra automorphisms are even simpler, being generated by an \(n\)-dimensional multiplicative torus and the Dynkin diagram automorphisms. By contrast, even the automorphism group of \(U(\mathfrak {sl}_2)\) is still unknown.
scientific article; zbMATH DE number 1066104