The main result of this paper establishes a bijection between the set of equivalence classes of simple transitive 2-representations with a fixed apex ${\mathcal{J}}$ of a fiat 2-category $\mathscr{C}$ and the set of equivalence classes of faithful simple transitive 2-representations of the fiat 2-subquotient of $\mathscr{C}$ associated with a diagonal ${\mathcal{H}}$-cell in ${\mathcal{J}}$. As an application, we classify simple transitive 2-representations of various categories of Soergel bimodules, in particular, completing the classification in types $B_{3}$ and $B_{4}$.

We give an explicit construction of the cusp eigenforms on an elliptic curve defined over a finite field, using the theory of Hall algebras and the Langlands correspondence for function fields and ${\mathrm{GL} }_{n} $. As a consequence we obtain a description of the Hall algebra of an elliptic curve as an infinite tensor product of simpler algebras. We prove that all these algebras are specializations of a universal spherical Hall algebra (as defined and studied by Burban and Schiffmann  [On the Hall algebra of an elliptic curve I, Preprint (2005), arXiv:math/0505148 [math.AG]] and Schiffmann and Vasserot [The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials, Compositio Math. 147 (2011), 188–234]).

Given two Lie 2-groups, we study the problem of integrating a weak morphism between the corresponding Lie 2-algebras to a weak morphism between the Lie 2-groups. To do so, we develop a theory of butterflies for 2-term L∞-algebras. In particular, we obtain a new description of the bicategory of 2-term L∞-algebras. An interesting observation here is that the role played by 1-connected Lie groups in Lie theory is now played by 2-connected Lie 2-groups. Using butterflies, we also give a functorial construction of 2-connected covers of Lie 2 -groups. Based on our results, we expect that a similar pattern generalizes to Lie n-groups and Lie n-algebras.

We study the transfer of nondegeneracy between Lie triple systems and their standard Lie algebra envelopes as well as between Kantor pairs, their associated Lie triple systems, and their Lie algebra envelopes. We also show that simple Kantor pairs and Lie triple systems in characteristic 0 are nondegenerate.

Gaudin subalgebras are abelian Lie subalgebras of maximal dimension spanned by generators of the Kohno–Drinfeld Lie algebra . We show that Gaudin subalgebras form a variety isomorphic to the moduli space of stable curves of genus zero with n+1 marked points. In particular, this gives an embedding of in a Grassmannian of (n−1)-planes in an n(n−1)/2-dimensional space. We show that the sheaf of Gaudin subalgebras over is isomorphic to a sheaf of twisted first-order differential operators. For each representation of the Kohno–Drinfeld Lie algebra with fixed central character, we obtain a sheaf of commutative algebras whose spectrum is a coisotropic subscheme of a twisted version of the logarithmic cotangent bundle of .

For an algebraic structure A denote by d(A) the smallest size of a generating set for A, and let d(A)=(d(A),d(A2),d(A3),…), where An denotes a direct power of A. In this paper we investigate the asymptotic behaviour of the sequence d(A) when A is one of the classical structures—a group, ring, module, algebra or Lie algebra. We show that if A is finite then d(A) grows either linearly or logarithmically. In the infinite case constant growth becomes another possibility; in particular, if A is an infinite simple structure belonging to one of the above classes then d(A) is eventually constant. Where appropriate we frame our exposition within the general theory of congruence permutable varieties.