Memoirs of the American Mathematical Society 2013; 93 pp; softcover Volume: 229 ISBN10: 0821891758 ISBN13: 9780821891759 List Price: US$71 Individual Members: US$42.60 Institutional Members: US$56.80 Order Code: MEMO/229/1077
 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 "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 ConciniKac 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. Table of Contents  Preliminaries and statement of results
 Quantum groups, actions, and cohomology
 Computation of \(\Phi_{0}\) and \({\mathcal N}(\Phi_{0})\)
 Combinatorics and the Steinberg module
 The cohomology algebra \(\operatorname{H}^{\bullet}(u_{\zeta}(\mathfrak{g}),\mathbb{C})\)
 Finite generation
 Comparison with positive characteristic
 Support varieties over \(u_{\zeta}\) for the modules \(\nabla_{\zeta}(\lambda)\) and \(\Delta_{\zeta}(\lambda)\)
 Appendix A
 Bibliography
