We compute the ${ℂ}^{\ast }$-equivariant quantum cohomology ring of $Y$, the minimal resolution of the DuVal singularity ${ℂ}^2∕G$ where $G$ is a finite subgroup of $SU\left(2\right)$. The quantum product is expressed in terms of an ADE root system canonically associated to $G$. We generalize the resulting Frobenius manifold to nonsimply laced root systems to obtain an $n$ parameter family of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifold Gromov–Witten potential of $\left[{ℂ}^2∕G\right]$.

Bhargava proved a formula for counting, with certain weights, degree $n$ étale extensions of a local field, or equivalently, local Galois representations to $S_n$. This formula is motivation for his conjectures about the density of discriminants of $S_n$-number fields. We prove there are analogous “mass formulas” that count local Galois representations to any group that can be formed from symmetric groups by wreath products and cross products, corresponding to counting towers and direct sums of étale extensions. We obtain as a corollary that the above mentioned groups have rational character tables. Our result implies that $D_4$ has a mass formula for certain weights, but we show that $D_4$ does not have a mass formula when the local Galois representations to $D_4$ are weighted in the same way as representations to $S_4$ are weighted in Bhargava’s mass formula.

It is shown that for any morphism, $\varphi :\mathfrak{g}\to \mathfrak{h}$, of Lie algebras the vector space underlying the Lie algebra $\mathfrak{h}$ is canonically a $\mathfrak{g}$-homogeneous formal manifold with the action of $\mathfrak{g}$ being highly nonlinear and twisted by Bernoulli numbers. This fact is obtained from a study of the 2-coloured operad of formal homogeneous spaces whose minimal resolution gives a new conceptual explanation of both Ziv Ran’s Jacobi–Bernoulli complex and Fiorenza–Manetti’s $L_{\infty }$-algebra structure on the mapping cone of a morphism of two Lie algebras. All these constructions are iteratively extended to the case of a morphism of arbitrary $L_{\infty }$-algebras.

The algebra of a finite group over a field $k$ of characteristic zero is known to be a projective separable $k$-algebra; but these separable algebras are of a very special type, characterized by Brauer and Witt.

In contrast with that, we prove that any projective separable $k$-algebra is a quotient of the group algebra of a suitable group scheme, finite étale over $k$. In particular, any finite separable field extension $K\subset L$, even a noncyclotomic one, may be generated by a finite étale $K$-group scheme.

We develop the theory of rational ideals for arbitrary associative algebras $R$ without assuming the standard finiteness conditions, noetherianness or the Goldie property. The Amitsur–Martindale ring of quotients replaces the classical ring of quotients which underlies the previous definition of rational ideals but is not available in a general setting.

Our main result concerns rational actions of an affine algebraic group $G$ on $R$. Working over an algebraically closed base field, we prove an existence and uniqueness result for generic rational ideals in the sense of Dixmier: for every $G$-rational ideal $I$ of $R$, the closed subset of the rational spectrum $RatR$ that is defined by $I$ is the closure of a unique $G$-orbit in $RatR$. Under additional Goldie hypotheses, this was established earlier by Mœglin and Rentschler (in characteristic $0$) and by Vonessen (in arbitrary characteristic), answering a question of Dixmier.