We prove the longstanding physics conjecture that there exists a unique two-parameter ${\mathcal {W}}_{\infty }$-algebra which is freely generated of type ${\mathcal {W}}(2,3,\ldots )$, and generated by the weights $2$ and $3$ fields. Subject to some mild constraints, all vertex algebras of type ${\mathcal {W}}(2,3,\ldots , N)$ for some $N$ can be obtained as quotients of this universal algebra. As an application, we show that for $n\geq 3$, the structure constants for the principal ${\mathcal {W}}$-algebras ${\mathcal {W}}^k({\mathfrak s}{\mathfrak l}_n, f_{\text {prin}})$ are rational functions of $k$ and $n$, and we classify all coincidences among the simple quotients ${\mathcal {W}}_k({\mathfrak s}{\mathfrak l}_n, f_{\text {prin}})$ for $n\geq 2$. We also obtain many new coincidences between ${\mathcal {W}}_k({\mathfrak s}{\mathfrak l}_n, f_{\text {prin}})$ and other vertex algebras of type ${\mathcal {W}}(2,3,\ldots , N)$ which arise as cosets of affine vertex algebras or nonprincipal ${\mathcal {W}}$-algebras.

Let A be a symmetrisable generalised Cartan matrix, and let $\mathfrak {g}(A)$ be the corresponding Kac–Moody algebra. In this paper, we address the following fundamental question on the structure of $\mathfrak {g}(A)$ : given two homogeneous elements $x,y\in \mathfrak {g}(A)$ , when is their bracket $[x,y]$ a nonzero element? As an application of our results, we give a description of the solvable and nilpotent graded subalgebras of $\mathfrak {g}(A)$ .

We prove duality isomorphisms of certain representations of ${\mathcal{W}}$ -algebras which play an essential role in the quantum geometric Langlands program and some related results.

Under the local Langlands correspondence, the conductor of an irreducible representation of $\text{G}{{\text{l}}_{n}}\left( F \right)$ is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection.

We prove a conjecture of Rouquier relating the decomposition numbers in category ${\mathcal{O}}$ for a cyclotomic rational Cherednik algebra to Uglov’s canonical basis of a higher level Fock space. Independent proofs of this conjecture have also recently been given by Rouquier, Shan, Varagnolo and Vasserot and by Losev, using different methods. Our approach is to develop two diagrammatic models for this category ${\mathcal{O}}$ ; while inspired by geometry, these are purely diagrammatic algebras, which we believe are of some intrinsic interest. In particular, we can quite explicitly describe the representations of the Hecke algebra that are hit by projectives under the $\mathsf{KZ}$ -functor from the Cherednik category ${\mathcal{O}}$ in this case, with an explicit basis. This algebra has a number of beautiful structures including categorifications of many aspects of Fock space. It can be understood quite explicitly using a homogeneous cellular basis which generalizes such a basis given by Hu and Mathas for cyclotomic KLR algebras. Thus, we can transfer results proven in this diagrammatic formalism to category ${\mathcal{O}}$ for a cyclotomic rational Cherednik algebra, including the connection of decomposition numbers to canonical bases mentioned above, and an action of the affine braid group by derived equivalences between different blocks.

We consider a lifting of Joseph ideals for the minimal nilpotent orbit closure to the setting of affine Kac–Moody algebras and find new examples of affine vertex algebras whose associated varieties are minimal nilpotent orbit closures. As an application we obtain a new family of lisse ( $C_{2}$ -cofinite) $W$ -algebras that are not coming from admissible representations of affine Kac–Moody algebras.

Let $k$ be a field of characteristic 0. Let $G$ be a reductive group over the ring of Laurent polynomials $R\,=\,k\left[ x_{1}^{\pm 1},\ldots ,x_{n}^{\pm 1} \right]$ . Assume that $G$ contains a maximal $R$ -torus, and that every semisimple normal subgroup of $G$ contains a two-dimensional split torus $\mathbf{G}_{m}^{2}$ . We show that the natural map of non-stable ${{K}_{1}}$ -functors, also called Whitehead groups, $K_{1}^{G}\left( R \right)\,\to \,K_{1}^{G}\left( k\left( \left( {{x}_{1}} \right) \right)\cdots \left( \left( {{x}_{n}} \right) \right) \right)$ is injective, and an isomorphism if $G$ is semisimple. As an application, we provide a way to compute the difference between the full automorphism group of a Lie torus (in the sense of Yoshii–Neher) and the subgroup generated by exponential automorphisms.

In Bennett et al. [BGG reciprocity for current algebras, Adv. Math. 231 (2012), 276–305] it was conjectured that a BGG-type reciprocity holds for the category of graded representations with finite-dimensional graded components for the current algebra associated to a simple Lie algebra. We associate a current algebra to any indecomposable affine Lie algebra and show that, in this generality, the BGG reciprocity is true for the corresponding category of representations.

Let G be an arbitrary non-zero additive subgroup of the complex number field ℂ, and let Vir[G] be the corresponding generalized Virasoro algebra over ℂ. In this paper we determine all irreducible weight modules with finite-dimensional weight spaces over Vir[G]. The classification strongly depends on the index group G. If G does not have a direct summand isomorphic to ℤ (the integers), then such irreducible modules over Vir[G] are only modules of intermediate series whose weight spaces are all one dimensional. Otherwise, there is one further class of modules that are constructed by using intermediate series modules over a generalized Virasoro subalgebra Vir[G0] of Vir[G] for a direct summand G0 of G with G = G0 ⊕ ℤb, where b ∈ G \ G0. This class of irreducible weight modules do not have corresponding weight modules for the classical Virasoro algebra.

Let $\mathfrak{n}$ be a maximal nilpotent subalgebra of a complex symmetric Kac–Moody Lie algebra. Lusztig has introduced a basis of $U(\mathfrak{n})$ called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of nilpotent modules over a preprojective algebra of the same type as $\mathfrak{n}$. We prove a formula for the product of two elements of the dual of this semicanonical basis, and more generally for the product of two evaluation forms associated to arbitrary modules over the preprojective algebra. This formula plays an important rôle in our work on the relationship between semicanonical bases, representation theory of preprojective algebras, and Fomin and Zelevinsky's theory of cluster algebras. It was inspired by recent results of Caldero and Keller.

We obtain several determinant evaluations, related to affine root systems, which provide elliptic extensions of Weyl denominator formulas. Some of these are new, also in the polynomial special case, while others yield new proofs of the Macdonald identities for the seven infinite families of irreducible reduced affine root systems.

We give a geometric construction of the Verma modules of a symmetric Kac-Moody Lie algebra g in terms of constructible functions on the varieties of nilpotent finite-dimensional modules of the corresponding preprojective algebra Λ.

We consider a class of reductive linear groups defined in terms of weighted oriented graphs of a special sort that we call signed quivers. Each of these yields a symmetric quiver, that is, a quiver endowed with an involutive anti-automorphism together with signs for the vertices and arrows fixed by the involution. The orbits of the groups can be described in terms of the indecomposable symmetric representations of symmetric quivers. We provide a general description of the indecomposable symmetric representations and prove that their dimensions in the finite and tame cases constitute root systems corresponding to certain symmetrizable generalized Cartan matrices.

Over arbitrary fields of characteristic not equal to 2, we construct three families of simple Lie algebras and six families of simple Lie superalgebras of matrices with entries chosen from different one-sided ideals of a simple associative algebra. These families correspond to the classical Lie algebras and superalgebras. Our constructions intermix the structure of the associative algebra and the structure of the matrix algebra in an essential, compatible way. Many examples of simple associative algebras without an identity element arise as a by-product. The study of conformal algebras and superalgebras often involves matrix algebras over associative algebras such as Weyl algebras, and for that reason, we illustrate our constructions by taking various one-sided ideals from a Weyl algebra or a quantum torus.

We investigate a class of Lie algebras which we call generalized reductive Lie algebras. These are generalizations of semi-simple, reductive, and affine Kac–Moody Lie algebras. A generalized reductive Lie algebra which has an irreducible root system is said to be irreducible and we note that this class of algebras has been under intensive investigation in recent years. They have also been called extended affine Lie algebras. The larger class of generalized reductive Lie algebras has not been so intensively investigated. We study them in this paper and note that one way they arise is as fixed point subalgebras of finite order automorphisms. We show that the core modulo the center of a generalized reductive Lie algebra is a direct sum of centerless Lie tori. Therefore one can use the results known about the classification of centerless Lie tori to classify the cores modulo centers of generalized reductive Lie algebras.

If G is a (split) Kac–Moody group over a field K endowed with a real valuation $\omega$, we build an action of G on a geometric object $\mathcal I$. This object is called a building, as it is an union of apartments, with the classical properties of systems of apartments. However, these apartments are more exotic: that associated to a torus T may be seen as the gluing of all Satake compactifications of affine apartments of T with respect to spherical parabolic subgroups of G containing T. Another geometric realization of these apartments makes them look more like the apartments of $\Lambda$-buildings; then the translations of the Weyl group act only on infinitely small elements of the apartment, so we call these buildings microaffine.

Twenty-five years ago, Conway and Norton published in this journal their remarkable paper ‘Monstrous Moonshine’, proposing a completely unexpected relationship between finite simple groups and modular functions. We review the progress made in broadening and understanding that relationship.

We show that the norm of a Bethe vector in the $sl_{r+1}$ Gaudin model is equal to the Hessian of the corresponding master function at the corresponding critical point. In particular the Bethe vectors corresponding to non-degenerate critical points are non-zero vectors. This result is a byproduct of functorial properties of Bethe vectors studied in this paper. As another byproduct of functoriality we show that the Bethe vectors form a basis in the tensor product of several copies of first and last fundamental $sl_{r+1}$-modules.

In this paper, we develop the crystal basis theory for quantum generalized Kac–Moody algebras. For a quantum generalized Kac–Moody algebra $U_q(\mathfrak{g})$, we first introduce the category $\mathcal{O}_{int}$ of $U_q(\mathfrak{g})$-modules and prove its semisimplicity. Next, we define the notion of crystal bases for $U_q(\mathfrak{g})$-modules in the category $\mathcal{O}_{int}$ and for the subalgebra $U_q^-(\mathfrak{g})$. We then prove the tensor product rule and the existence theorem for crystal bases. Finally, we construct the global bases for $U_q(\mathfrak{g})$-modules in the category $\mathcal{O}_{int}$ and for the subalgebra $U_q^-(\mathfrak{g})$.

A theorem of Kac on quiver representations states that the dimension vectors of indecomposable representations are precisely the positive roots of the associated symmetric Kac–Moody Lie algebra. Here this result is generalised to representations respecting an admissible quiver automorphism, and the positive roots of an associated symmetrisable Kac–Moody Lie algebra are obtained.

Also the relationship with species of valued quivers over finite fields is discussed. It is known that the number of isomorphism classes of indecomposable representations of a given dimension vector for a species is a polynomial in the size of the base field. It is shown that these polynomials are non-zero if and only if the dimension vector is a positive root of the corresponding symmetrisable Kac–Moody Lie algebra.