Lie groups are topological groups which are simultaneously (and compatibly) differentiable manifolds. The tangent space at the identity of a Lie group is a Lie algebra. Lie algebras may also be defined axiomatically as non-associative algebras satisfying certain identities (the identities being those satisfied when the new product [x,y]=xy-yx is introduced on an associative algebra). The finite-dimensional representations of the Lie group and its associated Lie algebra are the same: this was the initial motivation for the study of Lie algebras (where one can employ more algebraic as opposed to analytic tools).
An algebraic group is a group that is simultaneously (and compatibly) an algebraic set (i.e., a set with group operations defined by polynomial equations). To each algebraic group there is also a Lie algebra associated.
Research in this department focuses on representations of Lie algebras and the structure of their enveloping algebras, in both the classical (characteristic 0) and the modular (positive characteristic) case. The study of quantum groups is also closely linked to the study of Lie algebras and algebraic groups.
The members of the algebra group who work in this area are Ian Musson, and Yi Ming Zou, with Allen Bell having some interests along this line. All four members of the algebra group mentioned above have studied graded versions of Lie algebras known as Lie superalgebras.