Cohomology for quantum groups via the geometry of the nullcone. (English) Zbl 1344.20066
Mem. Am. Math. Soc. 1077, x, 93 p. (2014).
From the summary: “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 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…. The main result in this paper provides a surprisingly uniform answer for the cohomology algebra \(\mathbf H^\bullet(u_\zeta,\mathbb C)\) of the small quantum group. When \(\ell>h\), this cohomology algebra has been calculated by V. Ginzburg and S. Kumar [Duke Math. J. 69, No. 1, 179-198 (1993; Zbl 0774.17013)]. Our result requires …a detailed knowledge of the geometry of the nullcone of \(\mathfrak g\) …. Finally, we establish that if \(M\) is a finite dimensional \(u_\zeta\)-module, then \(\mathbf H^\bullet(u_\zeta,M)\) is a finitely generated \(\mathbf H^\bullet(u_\zeta,\mathbb C)\)-module, and we obtain new results on the theory of support varieties for \(u_\zeta\).”
The authors mostly work under the assumption that \(\ell\) is not divisible by a bad prime for the root system \(\Phi\). If \(\Phi\) is of type \(G_2\) they also assume that 3 does not divide \(\ell\). When \(\Phi\) is of type \(B_n\) or type \(C_n\) they often further assume that \(\ell>3\). This still leaves many cases to consider separately. They determine the nullcone in all these cases. One of the key results describes the support varieties of the standard and costandard \(U_\zeta\) modules. It turns out that such a support variety is the closure of a Richardson orbit, as long as the prime is good. To find the relevant Richardson orbit requires a case by case analysis of the interaction between \(\ell\), \(\Phi\) and a given weight \(\lambda\). Computer assistance is invoked where appropriate. With few exceptions the relevant orbit closures are normal. A new feature for \(\ell<h\) is the appearance of “Steinberg modules” in the computation.
The authors mostly work under the assumption that \(\ell\) is not divisible by a bad prime for the root system \(\Phi\). If \(\Phi\) is of type \(G_2\) they also assume that 3 does not divide \(\ell\). When \(\Phi\) is of type \(B_n\) or type \(C_n\) they often further assume that \(\ell>3\). This still leaves many cases to consider separately. They determine the nullcone in all these cases. One of the key results describes the support varieties of the standard and costandard \(U_\zeta\) modules. It turns out that such a support variety is the closure of a Richardson orbit, as long as the prime is good. To find the relevant Richardson orbit requires a case by case analysis of the interaction between \(\ell\), \(\Phi\) and a given weight \(\lambda\). Computer assistance is invoked where appropriate. With few exceptions the relevant orbit closures are normal. A new feature for \(\ell<h\) is the appearance of “Steinberg modules” in the computation.
Reviewer: Wilberd van der Kallen (Utrecht)
MSC:
| 20G10 | Cohomology theory for linear algebraic groups |
| 17B08 | Coadjoint orbits; nilpotent varieties |
| 20G42 | Quantum groups (quantized function algebras) and their representations |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
Citations:
Zbl 0774.17013Software:
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