Any quantized universal enveloping algebra (in fact, any toplogically quasi-triangular Hopf algebra) has an (in its completion) an element u called the Drinfeld element which gives an isomorphism from a representation to its double dual.


However, most people prefer to use a different pivotal structure on the category of representations of the quantized universal enveloping algebra, where u is replaced by g=v-1u. Several obvious references don't seem to have a formula for this element, even though my dim recollection is that it is very simple. Is there anywhere where this is written down properly?



What is a formula for the "group-like Drinfeld element"?
Up to sign conventions it's just K\_\rho, where \rho is the Weyl vector (half the sum of the positive roots). It's easy to check that this does satisfy the basic properties of g, namely that it is grouplike and that conjugation by it acts by S^2. Thus it gives a pivotal structure. (You still need to check that it actually is the g that comes from the usual ribbon element, which is a bit more delicate.)


