Algebras with action and coaction of Hopf algebras

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My research area is Lie theory and its generalizations in the context of Hopf algebras. Lie theory describes the interplay between groups of continuous transformations, called Lie groups, and their linearizations, called Lie algebras. The theory was pioneered by the 19th century Norwegian mathematician Sophus Lie and, since then, has become ubiquitous in mathematics and physics. Examples of Lie groups include the group of all rigid motions in 3d space and the group of all Lorentz transformations in the 4d space-time of relativity theory. They also include the groups of symmetries of various objects (geometric bodies, systems of equations of motion, elasticity tensors, molecules - to name a few), that is, the collections of those transformations that preserve the given object. Hopf algebras were discovered by a topologist Heinz Hopf in the 20th century in his study of cohomology of Lie groups. They were later found useful in many branches of mathematics and physics including algebra, functional analysis, statistical mechanics, quantum field theory and string theory. Roughly speaking, while groups describe classical symmetry, Hopf algebras describe quantum symmetry and, for this reason, are sometimes called quantum groups. In algebra, they provide a unified approach to many classical problems. The focus of the proposed research program will be on actions (or coactions) of Hopf algebras on other algebras. In particular, the theory of Hopf algebras gives a unified approach to the study of graded algebras, algebras with automorphisms, and algebras with derivations by viewing them as objects in an appropriate category associated to a Hopf algebra. The proposed research deals with classical objects of mathematics such as matrix algebras and finite-dimensional simple Lie (super)algebras and is expected to contribute to the advancement of knowledge in fundamental areas of algebra. It may also find applications in other areas of mathematics and in theoretical physics.