Cuspidal systems for affine Khovanov–Lauda–Rouquier algebras

A cuspidal system for an affine Khovanov–Lauda–Rouquier algebra $R_\alpha $ yields a theory of standard modules. This allows us to classify the irreducible modules over $R_\alpha $ up to the so-called imaginary modules. We describe minuscule imaginary modules, laying the groundwork for future study of imaginary Schur–Weyl duality. We introduce colored imaginary tensor spaces and reduce a classification of imaginary modules to one color. We study the characters of cuspidal modules. We show that under the Khovanov–Lauda–Rouquier categorification, cuspidal modules correspond to dual root vectors.     

Representations of Gan–Ginzburg algebras

Given a quiver, a fixed dimension vector, and a positive integer n, we construct a functor from the category of D-modules on the space of representations of the quiver to the category of modules over a corresponding Gan–Ginzburg algebra of rank n. When the quiver is affine Dynkin, we obtain an explicit construction of representations of the corresponding wreath product symplectic reflection algebra of rank n. When the quiver is star-shaped, but not finite Dynkin, we use this functor to obtain a Lie-theoretic construction of representations of a “spherical” subalgebra of the Gan–Ginzburg algebra isomorphic to a rational generalized double affine Hecke algebra of rank n. Our functors are a generalization of the type A and type BC functors from [1] and [4], respectively.     

Khovanov–Seidel quiver algebras and bordered Floer homology

We discuss a relationship between Khovanov- and Heegaard Floer-type homology theories for braids. Explicitly, we define a filtration on the bordered Heegaard–Floer homology bimodule associated to the double-branched cover of a braid and show that its associated graded bimodule is equivalent to a similar bimodule defined by Khovanov and Seidel.     

Representations of polynomial Rota–Baxter algebras

A Rota–Baxter operator is an algebraic abstraction of integration, which is the typical example of a weight zero Rota–Baxter operator. We show that studying the modules over the polynomial Rota–Baxter algebra $(\mathbf{k}[x],P)$ is equivalent to studying the modules over the Jordan plane, and we generalize the direct decomposability results for the $(\mathbf{k}[x],P)$-modules in [13] from algebraically closed fields of characteristic zero to fields of characteristic zero. Furthermore, we provide a classification of Rota–Baxter modules up to isomorphism based on indecomposable $\mathbf{k}[x]$-modules.     

Endomorphism algebras and Igusa–Todorov algebras

Let A be an Artin algebra. If $V\in \operatorname{mod} A$ such that the global dimension of  $\operatorname{End}_{A}V$ is at most 3, then for any ${M\in \operatorname{add}_{A}V}$ , both B and B ^op are 2-Igusa–Todorov algebras, where ${B=\operatorname{End}_{A}M}$ . Let ${P\in \operatorname{mod} A}$ be projective and ${B=\operatorname{End}_{A}P}$ such that the projective dimension of P as a right B-module is at most n(<∞). If A is an m-syzygy-finite algebra (resp. an m-Igusa–Todorov algebra), then B is an (m+n)-syzygy-finite algebra (resp. an (m+n)-Igusa–Todorov algebra); in particular, the finitistic dimension of B is finite in both cases. Some applications of these results are given.     

Khovanov–Rozansky homology and directed cycles

We determine the cycle packing number of a directed graph using elementary projective algebraic geometry. Our idea is rooted in the Khovanov–Rozansky theory. In fact, using the Khovanov–Rozansky homology of a graph, we also obtain algebraic methods of detecting directed and undirected cycles containing a particular vertex or edge.     

Cohen–Macaulay Auslander algebras of skewed-gentle algebras

For any skewed-gentle algebra, we characterize its indecomposable Gorenstein projective modules explicitly and describe its Cohen–Macaulay Auslander algebra. We prove that skewed-gentle algebras are always Gorenstein, which is independent of the characteristic of the ground field, and the Cohen–Macaulay Auslander algebras of skewed-gentle algebras are also skewed-gentle algebras.     

Almost Koszul algebras and stably Calabi–Yau algebras

We discuss different properties of Frobenius almost Koszul algebras of periodic type. We describe their bimodule projective resolutions and their relations with twisted superpotentials. We give a sufficient condition for a Frobenius almost Koszul algebra of periodic type to be stably Calabi–Yau. We also discuss the stably Calabi–Yau property of skew group algebras.     

Involutions of Iwahori–Hecke Algebras and Representations of Fixed Subalgebras

We establish branching rules between some Iwahori–Hecke algebra of type B and their subalgebras which are defined as fixed subalgebras by involutions including Goldman involution. The Iwahori–Hecke algebra of type D is one of such fixed subalgebras. We also obtain branching rules between those fixed subalgebras and their intersection subalgebra. We determine basic sets of irreducible representations of those fixed subalgebras and their intersection by making use of generalized Clifford theory.     

Representations and T*-Extensions of Hom–Jordan–Lie Algebras

The purpose of this article is to study representations and T*-extensions of δ-hom–Jordan–Lie algebras. In particular, adjoint representations, trivial representations, deformations, and many properties of T*-extensions of δ-hom–Jordan–Lie algebras are studied in detail. Derivations and central extensions of δ-hom–Jordan–Lie algebras are also discussed as an application.     

Decomposition of effect algebras and the Hammer–Sobczyk theorem

We prove an algebraic and a topological decomposition theorem for complete D-lattices (i.e., lattice-ordered effect algebras). As a consequence, we obtain a Hammer–Sobczyk type decomposition theorem for modular measures on D-lattices. Dedicated to Prof. Paolo De Lucia on the occasion of his birthday.     

Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras

We construct an explicit isomorphism between blocks of cyclotomic Hecke algebras and (sign-modified) cyclotomic Khovanov-Lauda algebras in type A. These isomorphisms connect the categorification conjecture of Khovanov and Lauda to Ariki’s categorification theorem. The Khovanov-Lauda algebras are naturally graded, which allows us to exhibit a non-trivial ℤ-grading on blocks of cyclotomic Hecke algebras, including symmetric groups in positive characteristic.     

Quantizations of Kac–Moody algebras

We analyze the extent to which a quantum universal enveloping algebra of a Kac–Moody algebra g$\mathfrak{g}$ is defined by multidegrees of its defining relations. To this end, we consider a class of character Hopf algebras defined by the same number of defining relations of the same degrees as the Kac–Moody algebra g$\mathfrak{g}$. We demonstrate that if the generalized Cartan matrix A$A$ of g$\mathfrak{g}$ is connected then the algebraic structure, up to a finite number of exceptional cases, is defined by just one “continuous” parameter q$q$ related to a symmetrization of A$A$, and one “discrete” parameter m$\mathfrak{m}$ related to the modular symmetrizations of A$A$. The Hopf algebra structure is defined by n(n−1)/2$n(n-1)/2$ additional “continuous” parameters. We also consider the exceptional cases for Cartan matrices of finite or affine types in more detail, establishing the number of exceptional parameter values in terms of the Fibonacci sequence.     

Localization of Rota–Baxter algebras

A commutative Rota–Baxter algebra can be regarded as a commutative algebra that carries an abstraction of the integral operator. With the motivation of generalizing the study of algebraic geometry to Rota–Baxter algebras, we extend the central concept of localization for commutative algebras to commutative Rota–Baxter algebras. The existence of such a localization is proved and, under mild conditions, its explicit construction is obtained. The existence of tensor products of commutative Rota–Baxter algebras is also proved and the compatibility of localization and the tensor product of Rota–Baxter algebras is established. We further study Rota–Baxter coverings and show that they form a Grothendieck topology.