This research explores connections among several areas of modern algebra: associative algebras, Lie algebras, their automorphism groups and gradings. The unifying theme is the theory of so-called Hopf algebras, which were discovered in algebraic topology in the 1940s and since then have been applied in many areas of mathematics and theoretical physics including knot theory, operator algebras, quantum theory and statistical mechanics.
A distinguishing feature of Hopf algebras is the presence, in addition to the multiplication operation, of the so-called comultiplication, which can be thought of as multiplication backwards. Hopf algebras have gained increasing interest since the discovery of quantum groups by V. Drinfeld and M. Jimbo in the 1980s. In spite of their name, quantum groups are in fact Hopf algebras, not groups. The name is motivated by analogy with Lie groups, which were introduced by Sophus Lie in the 19th century to study continuous symmetry in classical physics. Quantum groups, on the other hand, are believed to be more suitable to describe symmetry in quantum physics. As mentioned above, Hopf algebras (and quantum groups in particular) also find numerous applications in mathematics itself, which make the theory of Hopf algebras one of the most active research areas of modern algebra.