On the inductive construction of quantized enveloping algebras

We consider an inductive scheme for quantized enveloping algebras, arising from certain inclusions of the associated root data. These inclusions determine an algebra-subalgebra pair with the subalgebra also a quantized enveloping algebra, and we want to understand the structure of the “difference” between the algebra and the subalgebra. Our point of view treats the background field and quantization parameter q as fixed and the root datum as being the varying parameter: we are interested in how the quantized enveloping algebras associated to different root data are related. One can think of this schematically as the addition and deletion of nodes of the associated Dynkin diagrams. By means of the Radford–Majid theorem, we show that associated to each root datum inclusion there is a graded Hopf algebra in the braided category of modules of the subalgebra. We prove that we therefore have a double-bosonisation (as introduced by Majid), this being a natural quotient of the Drinfeld double of a semi-direct product of Hopf algebras given by identifying the acting Hopf algebra and its dual. This reconstructs the full algebra from a central extension of the subalgebra, the graded Hopf algebra in the category and its dual, generalising the usual triangular decomposition. We study the structure of the graded braided Hopf algebra obtained in this way and identify a set of generators for it, establish its module structure and prove that it is an example of a Nichols algebra. Nichols algebras have recently come to prominence particularly in the study of pointed Hopf algebras and arise as quotients of braided tensor algebras. Our work adds to the point of view that certain types of Nichols algebras provide braided analogues of enveloping algebras for more general objects than just semisimple Lie algebras.