The quantum duality principle (QDP) by Drinfeld predicts a connection between the quantized universal enveloping algebras and the quantized coordinate algebras, where the underlying classical objects are related by the duality in Poisson geometry. The current paper gives an explicit formulization of the QDP for quantum symmetric pairs.

Let $\mathfrak {g}$ be a complex semi-simple Lie algebra, equipped with the standard Lie bialgebra structure. Let $\theta $ be a Lie algebra involution on $\mathfrak {g}$ and denote by $\mathfrak {k}=\mathfrak {g}^\theta $ the fixed point subalgebra. The quantum symmetric pair $(\mathrm {U},\mathrm {U}^\imath )$ is originally defined to be a quantization of the symmetric pair of the universal enveloping algebras $(U(\mathfrak {g}),U(\mathfrak {k}))$. In this paper, we show that an explicit specialization of $(\mathrm {U},\mathrm {U}^\imath )$ gives rise to the pair of the coordinate algebras $(\mathcal {O}(G^*),\mathcal {O}(K^\perp \backslash G^*))$, where $G^*$ is the dual Poisson-Lie group with the Lie algebra $\mathfrak {g}^*$, and $K^\perp \backslash G^*$ is a $G^*$-Poisson homogeneous space. Here $K^\perp $ is the closed subgroup of $G^*$ associated to the complementary dual of $\mathfrak {k}$. Therefore $(\mathrm {U},\mathrm {U}^\imath )$ can be viewed as a pair of quantized coordinate algebras. This generalizes the result of De Concini–Procesi [14] that the quantum group $\mathrm {U}$ provides a quantization of the coordinate algebra of $G^*$.

Joyce vertex algebras are vertex algebra structures defined on the homology of certain $\mathbb {C}$-linear moduli stacks, and are used to express wall-crossing formulae for Joyce’s homological enumerative invariants. This paper studies the generalization of this construction to settings that come from nonlinear enumerative problems. In the special case of orthosymplectic enumerative geometry, we obtain twisted modules for Joyce vertex algebras.

We expect that our construction will be useful for formulating wall-crossing formulae for enumerative invariants for nonlinear moduli stacks. We include several variants of our construction that apply to different flavours of enumerative invariants, including Joyce’s homological invariants, DT4 invariants, and a version of K-theoretic enumerative invariants.

Liouville field theory has long been a cornerstone of two-dimensional quantum field theory and quantum gravity, which has attracted much recent attention in the mathematics literature. Timelike Liouville field theory is a version of Liouville field theory where the kinetic term in the action appears with a negative sign, which makes it closer to a theory of quantum gravity than ordinary (spacelike) Liouville field theory. Making sense of this “wrong sign” requires a theory of Gaussian random variables with negative variance. Such a theory is developed in this paper, and is used to prove the timelike DOZZ formula for the $3$-point correlation function when the parameters satisfy the so-called “charge neutrality condition.” Expressions are derived also for the k-point correlation functions for all $k\ge 3$, and it is shown that these functions approach the correct semiclassical limits as the coupling constant is sent to zero.

Let $Z(\mathcal{W})$ be the center of the finite W-algebra $\mathcal{W}({\mathfrak{g}},e)$ associated with $\mathfrak{g}=\text{Lie}(G)$ and a nilpotent element $e\in\mathfrak{g}$ for a connected reductive algebraic group G over an algebraically closed field ${\unicode{x1D55C}}$ of prime characteristic p under the standard hypotheses (H1)-(H3) (see [8, section 6·3]). In this paper, we first demonstrate that our previous results in [20] on the structure and geometric properties of $Z({\mathcal{W}})$ for $p\gg0$ are still true under the present weakened restriction on p. Then we study the Zassenhaus variety $\mathscr{Z}$ of $\mathcal{W}({\mathfrak{g}},e)$, which is by definition the maximal spectrum $\text{Specm}(Z({\mathcal{W}}))$ of $Z({\mathcal{W}})$. On basis of the structure properties of $Z({\mathcal{W}})$, we describe $\mathscr{Z}$ via a good transverse slice ${\mathcal{S}}$ and show that $\mathscr{Z}$ is birationally equivalent to ${\mathcal{S}}$, thereby a rational affine scheme. In the special case when $e=0$, we reobtain one of the main results of [26] on the rationality of the Zassenhaus varieites for reductive Lie algebras in prime characteristic.

Let G be a simple algebraic group over an algebraically closed field $\Bbbk $ of positive characteristic. We consider the questions of when the tensor product of two simple G-modules is multiplicity free or completely reducible. We develop tools for answering these questions in general, and we use them to provide complete answers for the groups $G = \mathrm {SL}_3(\Bbbk )$ and $G = \mathrm {Sp}_4(\Bbbk )$.

A ${\mathcal Z}$-subalgebra $U_{\mathcal Z}^\jmath {(n)}$ (${\mathcal Z}=\mathbb Z[\upsilon ,\upsilon ^{-1}]$) for the i-quantum group ${\mathbf {U}}^{\jmath }(n)$ over the field $\mathbb Q(\upsilon )$ is constructed by two of the authors [‘A new realisation of the i-quantum group $U^\jmath {(n)}$’, J. Pure Appl. Algebra 226(1) (2022), Paper no. 106793, 27 pages, Theorem 6.5], using a Beilinson–Lusztig–MacPherson (BLM) type realisation. In this paper, we construct bases for $U_{\mathcal Z}^\jmath {(n)}$, including the monomial basis conjectured in [‘A new realisation of the i-quantum group $U^\jmath {(n)}$’, J. Pure Appl. Algebra 226(1) (2022), Paper no. 106793, 27 pages, Remark 6.6(4)]. This proves that the ${\mathcal Z}$-algebra $U_{\mathcal Z}^\jmath {(n)}$ is a free ${\mathcal Z}$-module. Hence, $U_{\mathcal Z}^\jmath {(n)}$ is in fact an integral form of Lusztig type. This construction is further extended to the i-quantum hyperalgebra over a field of any characteristic. By specialising $\upsilon $ to an l th primitive root $\varepsilon $ of $1$ with l odd, a realisation of the quotient of modulo the ideal generated by $d_i^l-1$, for all $1\leqslant i\leqslant n+1$, is also given as a by-product.

We investigated the symplectic geometry of homogeneous spaces associated with semisimple Lie groups, focusing on cotangent bundles of maximal flag manifolds. Our work provides an explicit description of the canonical symplectic structure on these spaces using connections and curvatures of principal bundles naturally associated with the underlying Lie groups. We extend classical results concerning the exactness of symplectic forms on adjoint orbits, previously known for specific Lie algebras, to arbitrary simple Lie groups. In particular, we identify conditions under which the Kostant–Kirillov–Souriau form on a regular adjoint orbit coincides with the canonical symplectic form of the cotangent bundle, yielding exact symplectic structures. The approach combines differential-geometric techniques with Lie-theoretic constructions, offering a unifying framework that connects the geometry of coadjoint orbits with symplectic structures on homogeneous spaces.

We develop a diagrammatic approach to the representation theory of the quantum symmetric pairs corresponding to orthosymplectic Lie superalgebras inside general linear Lie superalgebras. Our approach is based on the disoriented skein category, which we define as a module category over the framed HOMFLYPT skein category. The disoriented skein category admits full incarnation functors to the categories of modules over the iquantum enveloping algebras corresponding to the quantum symmetric pairs, and it can be viewed as an interpolating category for these categories of modules. We define an equivalence of module categories between the disoriented skein category and the iquantum Brauer category (also known as the q-Brauer category), after endowing the latter with the structure of a module category over the framed HOMFLYPT skein category. The disoriented skein category has some advantages over the iquantum Brauer category, possessing duality structure and allowing the incarnation functors to be strict morphisms of module categories. Finally, we construct explicit bases for the morphism spaces of the disoriented skein and iquantum Brauer categories.

Dans la lignée des résultats existants pour les groupes, on montre que le radical nilpotent existe dans les anneaux de Lie qui n’ont pas de chaîne infinie de centralisateurs. On établit également un analogue du théorème d’Engel si la caractéristique est nulle.

In this article, we consider the categorical symmetric Howe duality introduced by Khovanov, Lauda, Sussan, and Yonezawa. While originally defined from a purely diagrammatic perspective, this construction also has geometric and representation-theoretic interpretations, corresponding to certain perverse sheaves on spaces of quiver representations and the category of Gelfand–Tsetlin modules over $\mathfrak {gl}_n$. In particular, we show that the “deformed Webster algebras” discussed in [KLSY18] manifest a Koszul duality between blocks of the category of Gelfand–Tsetlin modules over $\mathfrak {gl}_n$, and the constructible sheaves on representations of a linear quiver invariant under a certain parabolic in the group that acts by changing bases. Furthermore, we show that this duality intertwines translation functors with a diagrammatic categorical action (generalizing that of [KLSY18]).

In this article, we provide a specific characterization of invariants of classical Lie superalgebras from the super-analog of the Schur–Weyl duality in a unified way. We establish $\mathfrak {g}$-invariants of the tensor algebra $T(\mathfrak {g})$, the supersymmetric algebra $S(\mathfrak {g})$, and the universal enveloping algebra $\mathrm {U}(\mathfrak {g})$ of a classical Lie superalgebra $\mathfrak {g}$ corresponding to every element in centralizer algebras and their relationship under supersymmetrization. As a byproduct, we prove that the restriction on $T(\mathfrak {g})^{\mathfrak {g}}$ of the projection from $T(\mathfrak {g})$ to $\mathrm {U}(\mathfrak {g})$ is surjective, which enables us to determine the generators of the center $\mathcal {Z}(\mathfrak {g})$ except for $\mathfrak {g}=\mathfrak {osp}_{2m|2n}$. Additionally, we present an alternative algebraic proof of the triviality of $\mathcal {Z}(\mathfrak {p}_n)$. The key ingredient involves a technique lemma related to the symmetric group and Brauer diagrams.

Over an algebraically closed field $\mathbb F$ of characteristic $p \gt 0$, the restricted twisted Heisenberg Lie algebras are studied. We use the Hochschild–Serre spectral sequence relative to its Heisenberg ideal to compute the trivial cohomology. The ordinary 1- and 2-cohomology spaces are used to compute the restricted 1- and 2-cohomology spaces and describe the restricted one-dimensional central extensions, including explicit formulas for the Lie brackets and $-^{[p]}$-operators.

In the article “Irreducible modules of modular Lie superalgebras and super version of the first Kac-Weisfeiler conjecture, Canad. Math. Bull. 67 (2024), no. 3, 554–573.” The statement in Theorem 4.7 is improper, which is fixed here. Theorem 4.7 is an isolated result in the article. This correction does not influence any arguments and any main results after that in the original article.

In this paper, we first describe the cohomology theory of Lie supertriple systems by using the cohomology theory of the associated Leibniz superalgebras. Then we focus on Lie supertriple systems with superderivations, called LSTSDer pairs. We introduce the notion of representations of LSTSDer pairs and investigate their corresponding cohomology theory. We also construct a differential graded Lie algebra whose Maurer–Cartan elements are LSTSDer pairs. Moreover, we consider the relationship between a LSTSDer pair and the associated LeibSDer pair. Furthermore, we develop the 1-parameter formal deformation theory of LSTSDer pairs and prove that it is governed by the cohomology groups. At last, we study abelian extensions of LSTSDer pairs and show that equivalent abelian extensions of LSTSDer pairs are classified by the third cohomology groups.

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.