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Pavel Etingof studies representation theory, quantum groups and noncommutative algebra. His MIT research profile places this work at the intersection of algebra and mathematical physics.^[ref]

His Lie Groups and Lie Algebras by Pavel Etingof (2024) develops the bridge from continuous groups to their infinitesimal algebraic structures. A Lie algebra representation replaces a nonlinear group action with linear operators satisfying bracket identities; the theory then recovers structural information about Lie groups. His work on tensor categories broadens the study of representations by treating their tensor products and compatibility maps as part of the mathematical object.^[ref][ref]

A quantum group is a type of algebraic symmetry commonly encoded by a Hopf algebra deforming a classical symmetry algebra. The term covers several related constructions; it does not specify a single additional Hopf algebra axiom. A standard example over $\mathbb{C}$, for $q\ne 0,\pm 1$, is $U_q({\mathfrak{sl}}_2)$ with generators $E,F,K,K^{-1}$ and relationsIts coproduct can be taken as $\Delta K=K\otimes K$, $\Delta E=E\otimes K+1\otimes E$, and $\Delta F=F\otimes 1+K^{-1}\otimes F$.^[ref]

These formulas make the symmetry act on tensor products of representations. In particular, the coproduct is generally not unchanged by swapping its two tensor factors. Formally writing $K=q^h$ and letting $q$ approach one recovers the familiar Lie bracket relations for ${\mathfrak{sl}}_2$. Making that limit rigorous requires choosing a suitable algebra over a parameter ring.