{"product_id":"9780821891759","title":"Cohomology for Quantum Groups via the Geometry of the Nullcone","description":"\u003cp\u003eLet $\\zeta$ be a complex $\\ell$th root of unity for an odd integer $\\ell\u0026gt;1$. For any complex simple Lie algebra $\\mathfrak g$, let $u_\\zeta=u_\\zeta({\\mathfrak g})$ be the associated ''small'' 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.\u003c\/p\u003e","brand":"AMS","offers":[{"title":"Default Title","offer_id":47031781851376,"sku":"9780821891759","price":71.0,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0737\/7593\/9824\/files\/9780821891759_p0.jpg?v=1763749980","url":"https:\/\/shop-qa.barnesandnoble.com\/products\/9780821891759","provider":"Barnes \u0026 Noble (DEV)","version":"1.0","type":"link"}