Hopf Algebras and Their Representations

Hopf algebras blend algebraic and coalgebraic structures to generalise the symmetry frameworks of groups, Lie algebras and algebraic groups. Initially emerging in algebraic topology for the study of homology of H-spaces, they have become central to the theory of quantum groups and braided tensor categories. A Hopf algebra carries both multiplication and comultiplication maps that interact via antipode and counit axioms, endowing its module categories with monoidal and, in the quasitriangular case, braided structures. Representation theory of Hopf algebras investigates modules and comodules, elucidating how symmetry and duality are realised in algebraic, geometric and physical contexts. Key developments include the classification of finite-dimensional pointed Hopf algebras via Nichols algebras and generalised root systems, the construction of Drinfeld doubles capturing braided monoidal dualities, and applications in low-dimensional topology, statistical mechanics and quantum computing. Recent computational advances have refined character formulae and cohomological invariants, while structural results on automorphism groups, extensions and deformations continue to link abstract theory with concrete algebraic and topological models.

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Hopf Algebras and Their Representations publication trend

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Technical terms

Hopf algebra: An algebraic structure equipped with compatible operations of multiplication, unit, comultiplication, counit and antipode, generalising group and ring symmetries.
Drinfeld double: A construction producing a quasitriangular Hopf algebra from a given one by combining it with its dual, which encodes full braided monoidal category structure.
Nichols algebra: A graded Hopf algebra generated by a Yetter–Drinfeld module, whose relations are determined by a generalised root system and which underpins the classification of pointed Hopf algebras.
Ore extension: A noncommutative analogue of a polynomial extension, defined via an automorphism and derivation that deform the usual commutation relations.
Pointed Hopf algebra: A Hopf algebra whose simple subcoalgebras are one-dimensional, so that its irreducible comodules correspond to group-like elements.

References

1. The Hopf Automorphism Group of Two Classes of Drinfeld Doubles. Symmetry (2024).
2. On finite dimensional Nichols algebras of diagonal type. Bulletin of Mathematical Sciences (2017).
3. A presentation by generators and relations of Nichols algebras of diagonal type and convex orders on root systems. Journal of the European Mathematical Society (2015).
4. Ore Extensions for the Sweedler’s Hopf Algebra H4. Mathematics (2020).