Cohomology for Quantum Groups via the Geometry of the Nullcone (eBook)

Christopher P Bendel (Autor)

eBook Download: PDF

93 Seiten
American Mathematical Society (Verlag)
978-1-4704-1531-0 (ISBN)

Infos

Let $/zeta$ be a complex $/ell$th root of unity for an odd integer $/ell>1$. For any complex simple Lie algebra $/mathfrak g$, let $u_/zeta=u_/zeta({/mathfrak g})$ be the associated "e;small"e; quantum enveloping algebra. This algebra is a finite dimensional Hopf algebra which can be realized as a subalgebra of the Lusztig (divided power) quantum enveloping algebra $U_/zeta$ and as a quotient algebra of the De Concini-Kac quantum enveloping algebra ${/mathcal U}_/zeta$. It plays an important role in the representation theories of both $U_/zeta$ and ${/mathcal U}_/zeta$ in a way analogous to that played by the restricted enveloping algebra $u$ of a reductive group $G$ in positive characteristic $p$ with respect to its distribution and enveloping algebras. In general, little is known about the representation theory of quantum groups (resp., algebraic groups) when $l$ (resp., $p$) is smaller than the Coxeter number $h$ of the underlying root system. For example, Lusztig's conjecture concerning the characters of the rational irreducible $G$-modules stipulates that $p /geq h$. The main result in this paper provides a surprisingly uniform answer for the cohomology algebra $/operatorname{H}^/bullet(u_/zeta,{/mathbb C})$ of the small quantum group.

Produkt-Details

• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• ISBN-10: 1-4704-1531-3 / 1470415313
• ISBN-13: 978-1-4704-1531-0 / 9781470415310