Download E-books Finite Dimensional Algebras and Quantum Groups (Mathematical Surveys and Monographs) PDF

By Bangming Deng, Jie Du, Brian Parshall, and Jianpan Wang

The interaction among finite dimensional algebras and Lie thought dates again decades. in additional contemporary instances, those interrelations became much more strikingly obvious. this article combines, for the 1st time in booklet shape, the theories of finite dimensional algebras and quantum teams. extra accurately, it investigates the Ringel-Hall algebra consciousness for the optimistic a part of a quantum enveloping algebra linked to a symmetrizable Cartan matrix and it appears heavily on the Beilinson-Lusztig-MacPherson cognizance for the whole quantum $\mathfrak {gl}_n$. The ebook starts off with the 2 realizations of generalized Cartan matrices, particularly, the graph cognizance and the foundation datum consciousness. From there, it develops the illustration thought of quivers with automorphisms and the idea of quantum enveloping algebras linked to Kac-Moody Lie algebras. those self reliant theories finally meet partially four, lower than the umbrella of Ringel-Hall algebras. Cartan matrices is additionally used to outline a huge category of groups--Coxeter groups--and their linked Hecke algebras. Hecke algebras linked to symmetric teams provide upward push to an enticing category of quasi-hereditary algebras, the quantum Schur algebras. The constitution of those finite dimensional algebras is utilized in half five to construct the full quantum $\mathfrak{gl}_n$ via a of completion means of a restrict algebra (the Beilinson-Lusztig-MacPherson algebra). The booklet is acceptable for complicated graduate scholars. each one bankruptcy concludes with a chain of workouts, starting from the regimen to sketches of proofs of modern effects from the present literature.

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