An important idea in the work of G.-C. Rota is that certain combinatorial objects give rise to Hopf algebras that reflect the manner in which these objects compose and decompose. Recent work has seen the emergence of several interesting Hopf algebras of this kind, which connect diverse subjects such as combinatorics, algebra, geometry, and theoretical physics. This monograph presents a novel geometric approach using Coxeter complexes and the projection maps of Tits for constructing and studying many of these objects as well as new ones. The first three chapters introduce the necessary background ideas making this work accessible to advanced graduate students. The later chapters culminate in a unified and conceptual construction of several Hopf algebras based on combinatorial objects which emerge naturally from the geometric viewpoint. This work lays a foundation and provides new insights for further development of the subject.

To read more about Coxeter groups see The Coxeter Legacy: Reflections and Projections.

Titles in this series are co-published with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

Readership

Graduate students and research mathematicians interested in algebraic combinatorics, Coxeter groups, and Hopf algebras.

Reviews

"Despite the formidable notational complexity, the book is well-organized and quite readable. In particular, there is a useful notation index."

-- Earl J. Taft for Zentralblatt MATH

"This is a fascinating research monograph with many new and interesting ideas for the combinatorial study of algebras."

-- Mathematical Reviews