We introduce a generating function approach to the affine Brauer and Kauffman categories, and show how it allows one to efficiently recover important sets of relations in these categories. We use this formalism to deduce restrictions on possible categorical actions and show how this recovers admissibility results that have appeared in the literature on cyclotomic Birman–Murakami–Wenzl (BMW) algebras and their degenerate versions, also known as cyclotomic Nazarov–Wenzl algebras or VW algebras.

This article focuses on the representation theory of algebras associated with $\mathfrak {sl}_2$, including the affine Lie algebra $\widehat {\mathfrak {sl}_2}$, the affine Kac–Moody algebra $\widetilde {\mathfrak {sl}_2}$, and the affine-Virasoro algebra $\mathfrak {Vir}\ltimes \widehat {\mathfrak {sl}_2}$. First, we classify certain modules over these algebras, which are free of rank one when restricted to some specific subalgebras. We demonstrate a connection between these modules and modules over the Weyl algebras, which allows us to construct large families of modules that are free of arbitrary finite rank when restricted to the Cartan subalgebra. We then investigate the simplicity of these modules. For reducible modules, we fully characterize their composition factors. Through a comparison with existing simple modules in the literature, we have identified a novel family of simple modules over the affine Kac–Moody algebra $\widetilde {\mathfrak {sl}_2}$. Finally, we turn our attention to a class of tensor product modules over the affine-Virasoro algebra $\mathfrak {Vir}\ltimes \widehat {\mathfrak {sl}_2}$. We derive a necessary and sufficient condition for the simplicity of these modules and determine their isomorphism classes.

We construct two families of orthogonal polynomials associated with the universal central extensions of the superelliptic Lie algebras. These polynomials satisfy certain fourth-order linear differential equations, and one of the families is a particular collection of associated ultraspherical polynomials. We show that the generating functions of the polynomials satisfy fourth-order linear PDEs. Since these generating functions can be represented by superelliptic integrals, we have examples of linear PDEs of fourth order with explicit solutions without complete integrability.

Let ${\mathscr {G}} $ be a special parahoric group scheme of twisted type over the ring of formal power series over $\mathbb {C}$, excluding the absolutely special case of $A^{(2)}_{2\ell }$. Using the methods and results of Zhu, we prove a duality theorem for general ${\mathscr {G}} $: there is a duality between the level one twisted affine Demazure modules and the function rings of certain torus fixed point subschemes in affine Schubert varieties for ${\mathscr {G}} $. Along the way, we also establish the duality theorem for $E_6$. As a consequence, we determine the smooth locus of any affine Schubert variety in the affine Grassmannian of ${\mathscr {G}} $. In particular, this confirms a conjecture of Haines and Richarz.

We construct a novel family of difference-permutation operators and prove that they are diagonalized by the wreath Macdonald P-polynomials; the eigenvalues are written in terms of elementary symmetric polynomials of arbitrary degree. Our operators arise from integral formulas for the action of the horizontal Heisenberg subalgebra in the vertex representation of the corresponding quantum toroidal algebra.

Gel’fand–Dorfman algebras (GD algebras) give a natural construction of Lie conformal algebras and are in turn characterized by this construction. In this article, we define the Gel’fand–Dorfman bialgebra (GD bialgebra) and enrich the above construction to a construction of Lie conformal bialgebras by GD bialgebras. As a special case, Novikov bialgebras yield Lie conformal bialgebras. We further introduce the notion of the Gel’fand–Dorfman Yang–Baxter equation (GDYBE), whose skew-symmetric solutions produce GD bialgebras. Moreover, the notions of $\mathcal {O}$-operators on GD algebras and pre-Gel’fand–Dorfman algebras (pre-GD algebras) are introduced to provide skew-symmetric solutions of the GDYBE. The relationships between these notions for GD algebras and the corresponding ones for Lie conformal algebras are given. In particular, there is a natural construction of Lie conformal bialgebras from pre-GD algebras. Finally, GD bialgebras are characterized by certain matched pairs and Manin triples of GD algebras.

We introduce the notion of integrable modules over $\imath $quantum groups (a.k.a. quantum symmetric pair coideal subalgebras). After determining a presentation of such modules, we prove that each integrable module over a quantum group is integrable when restricted to an $\imath $quantum group. As an application, we show that the space of matrix coefficients of all simple integrable modules over an $\imath $quantum group of finite type with specific parameters coincides with Bao-Song’s coordinate ring of the $\imath $quantum group.

We give a crystal structure on the set of Gelfand–Tsetlin patterns (GTPs), which parametrize bases for finite-dimensional irreducible representations of the general linear Lie algebra. The crystal data are given in closed form and are expressed using tropical polynomial functions of the entries of the patterns. We prove that with this crystal structure, the natural bijection between GTPs and semistandard Young tableaux is a crystal isomorphism.

We study simple Lie algebras generated by extremal elements, over arbitrary fields of arbitrary characteristic. We show the following: (1) If the extremal geometry contains lines, then the Lie algebra admits a $5 \times 5$-grading that can be parametrized by a cubic norm structure; (2) If there exists a field extension of degree at most $2$ such that the extremal geometry over that field extension contains lines, and in addition, there exist symplectic pairs of extremal elements, then the Lie algebra admits a $5 \times 5$-grading that can be parametrized by a quadrangular algebra.

One of our key tools is a new definition of exponential maps that makes sense even over fields of characteristic $2$ and $3$, which ought to be interesting in its own right.

Not only was Jacques Tits a constant source of inspiration through his work, he also had a direct personal influence, notably through his threat to speak evil of our work if it did not include the characteristic 2 case.

—The Book of Involutions [KMRT98, p. xv]

In the present work, we investigate the Lie algebra of the Formanek-Procesi group $\textrm {FP}(A_{\Gamma })$ with base group $A_{\Gamma }$ a right-angled Artin group. We show that the Lie algebra $\textrm {gr}(\textrm {FP}(A_{\Gamma }))$ has a presentation that is dictated by the group presentation. Moreover, we show that if the base group $G$ is a finitely generated residually finite $p$-group, then $\textrm { FP}(G)$ is residually nilpotent. We also show that $\textrm {FP}(A_{\Gamma })$ is a residually torsion-free nilpotent group.

We describe algebraically, diagrammatically, and in terms of weight vectors, the restriction of tensor powers of the standard representation of quantum $\mathfrak {sl}_2$ to a coideal subalgebra. We realize the category as a module category over the monoidal category of type $\pm 1$ representations in terms of string diagrams and via generators and relations. The idempotents projecting onto the quantized eigenspaces are described as type $B/D$ analogues of Jones–Wenzl projectors. As an application, we introduce and give recursive formulas for analogues of $\Theta$-networks.

We prove a conjecture of Lê and Sikora by providing a comparison between various existing $SL_n$ skein theories. While doing so, we show that the full subcategory of the spider category, $\mathcal {S}p(SL_n)$, defined by Cautis–Kamnitzer–Morrison, whose objects are monoidally generated by the standard representation and its dual, is equivalent as a spherical braided category to Sikora’s quotient category. This also answers a question from Morrison’s Ph.D. thesis. Finally, we show that the skein modules associated with the CKM and Sikora’s webs are isomorphic.

Scalar relative invariants play an important role in the theory of group actions on a manifold as their zero sets are invariant hypersurfaces. Relative invariants are central in many applications, where they often are treated locally since an invariant hypersurface may not be a locus of a single function. Our aim is to establish a global theory of relative invariants.

For a Lie algebra ${\mathfrak g}$ of holomorphic vector fields on a complex manifold M, any holomorphic ${\mathfrak g}$-invariant hypersurface is given in terms of a ${\mathfrak g}$-invariant divisor. This generalizes the classical notion of scalar relative ${\mathfrak g}$-invariant. Any ${\mathfrak g}$-invariant divisor gives rise to a ${\mathfrak g}$-equivariant line bundle, and a large part of this paper is therefore devoted to the investigation of the group $\mathrm {Pic}_{\mathfrak g}(M)$ of ${\mathfrak g}$-equivariant line bundles. We give a cohomological description of $\mathrm {Pic}_{\mathfrak g}(M)$ in terms of a double complex interpolating the Chevalley-Eilenberg complex for ${\mathfrak g}$ with the Čech complex of the sheaf of holomorphic functions on M.

We also obtain results about polynomial divisors on affine bundles and jet bundles. This has applications to the theory of differential invariants. Those were actively studied in relation to invariant differential equations, but the description of multipliers (or weights) of relative differential invariants was an open problem. We derive a characterization of them with our general theory. Examples, including projective geometry of curves and second-order ODEs, not only illustrate the developed machinery but also give another approach and rigorously justify some classical computations. At the end, we briefly discuss generalizations of this theory.

We prove a theorem that computes, for any augmented operad $\mathcal{O}$, the stable homology of the Lie algebra of derivations of the free algebra $\mathcal{O}(V)$ with twisted bivariant coefficients (here stabilization occurs as $\dim(V)\to\infty$) out of the homology of the wheeled bar construction of $\mathcal{O}$; this can further be used to prove uniform mixed representation stability for the homology of the positive part of that Lie algebra with constant coefficients. This result generalizes both the Loday–Quillen–Tsygan theorem on the homology of the Lie algebra of infinite matrices and the Fuchs stability theorem for the homology of the Lie algebra of vector fields. We also prove analogous theorems for the Lie algebras of derivations with constant and zero divergence, in which case one has to consider the wheeled bar construction of the wheeled completion of $\mathcal{O}$. Similarly to how cyclic homology of an algebra A may be viewed as an additive version of the algebraic K-theory of A, our results hint at the additive K-theoretic nature of the wheeled bar construction.

This corrigendum fixes an error in the statement of Lemma 8.3 in the published version of the paper.

We prove a strong quantitative version of the Kurosh Problem, which has been conjectured by Zelmanov, up to a mild polynomial error factor, thereby extending all previously known growth rates of algebraic algebras. Consequently, we provide the first counterexamples to the Kurosh Problem over any field with known subexponential growth, and the first examples of finitely generated, infinite-dimensional, nil Lie algebras with known subexponential growth over fields of characteristic zero.

We also widen the known spectrum of the Gel’fand–Kirillov dimensions of algebraic algebras, improving the answer of Alahmadi–Alsulami–Jain–Zelmanov to a question of Bell, Smoktunowicz, Small and Young. Finally, we prove improved analogous results for graded-nil algebras.

In this paper, we show that for a large natural class of vertex operator algebras (VOAs) and their modules, the Zhu’s algebras and bimodules (and their g-twisted analogs) are Noetherian. These carry important information about the representation theory of the VOA, and its fusion rules, and the Noetherian property gives the potential for (non-commutative) algebro-geometric methods to be employed in their study.

We give a complete combinatorial classification of the parabolic Verma modules in the principal block of the parabolic category $\mathcal{O}$ associated with a minimal or a maximal parabolic subalgebra of the special linear Lie algebra for which the answer to Kostant’s problem is positive.

Let $\mathfrak{g}$ be the queer superalgebra $\operatorname {\mathfrak{q}}(n)$ over the field of complex numbers $\mathbb C$. For any associative, commutative, and finitely generated $\mathbb C$-algebra A with unity, we consider the loop Lie superalgebra $\mathfrak{g} \otimes A$. We define a class of central operators for $\mathfrak{g} \otimes A$, which generalizes the classical Gelfand invariants. We show that they generate the algebra $U(\mathfrak{g} \otimes A)^{\mathfrak{g}}$. We also show that there are no non-trivial $\mathfrak{g}$-invariants of $U(\mathfrak{g} \otimes A)$ where $\mathfrak{g}=\mathfrak{p}(n)$, the periplectic Lie superalgebra.