Related papers: Highest weight theory for finite-dimensional grade…

Let $\mathfrak g$ be a classical Lie superalgebra of type I or a Cartan-type Lie superalgebra {\bf W}$(n)$. We study weight $\mathfrak g$-modules using a method inspired by Mathieu's classification of the simple weight modules with finite…

Relaxed highest-weight modules play a central role in the study of many important vertex operator (super)algebras and their associated (logarithmic) conformal field theories, including the admissible-level affine models. Indeed, their…

Let Q be a Dynkin quiver of type A. The bounded derived category of the path algebra of Q has an autoequivalence given by the composition of the Auslander-Reiten translate and the square of the shift functor. We classify the maximal rigid…

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$. A $\operatorname{Y}(\mathfrak{g})$-module is said to be weight if it is a weight $\mathfrak{g}$-module. We give a complete classification of simple weight…

We classify integrable irreducible highest weight representations of non-twisted affine Lie superalgebras. We give a free field construction in the level~1 case. The analysis of this construction shows, in particular, that in the simplest…

For any finite-dimensional algebra $A$ over a field $k$ with finite global dimension, we investigate the root category $\cR_A$ as the triangulated hull of the 2-periodic orbit category of $A$ via the construction of B. Keller in "On…

We define a notion of pseudo-unitarizability for weight modules over a generalized Weyl algebra (of rank one, with commutative coeffiecient ring $R$), which is assumed to carry an involution of the form $X^*=Y$, $R^*\subseteq R$. We prove…

Let $\Lambda$ be a finite-dimensional algebra with finite global dimension, $R_k=K[X]/(X^k)$ be the $\mathcal{Z}$-graded local ring with $k\geq1$, and $\Lambda_k=\Lambda\otimes_K R_k$. We consider the singularity category…

Let $\Lambda$ be a basic finite dimensional algebra over an algebraically closed field, presented as a path algebra modulo relations; further, assume that $\Lambda$ is graded by lengths of paths. The paper addresses the classifiability, via…

In this paper we study categories of tilting modules. Our starting point is the tilting modules for a reductive algebraic group G in positive characteristic. Here we extend the main result in [8] by proving that these tilting modules form a…

In this note, we survey two instances in the representation theory of finite-dimensional algebras where the quantity of a type of structures is intimately related to the size of those same structures. More explicitly, we review the fact…

It is known that the category of Lie algebras over a ring admits algebraic exponents. The aim of this paper is to show that the same is true for the category of internal Lie algebras in an additive, cocomplete, symmetric, closed, monoidal…

We present a necessary and sufficient condition for a finite-dimensional highest weight representation of the $sl_2$ loop algebra to be irreducible. In particular, for a highest weight representation with degenerate parameters of the…

In this paper, we present a uniform formula of Lusztig's $ \mathbf{a}$-functions on classical Weyl groups. Then we obtain an efficient algorithm for the Gelfand-Kirillov dimensions of simple highest weight modules of classical Lie algebras,…

We describe the isomorphism classes of infinite-dimensional graded Lie algebras of maximal class, generated by elements of weight one, over fields of odd characteristic.

We investigate the representations of a class of conformal Galilei algebras in one spatial dimension with central extension. This is done by explicitly constructing all singular vectors within the Verma modules, proving their completeness…

In this paper, we propose a conjectural formula for the highest $\ell$-weight monomial of an arbitrary real module over a simply-laced quantum affine algebra. We verify the conjecture under a multiplicative reachability condition, answering…

Let $R$ be an algebra over a ring $\Bbbk$, $T$ an $R$-algebra, $M$ a finitely generated projective $R$-module, and $N$ a $T$-module. Let $G$ be a linearly reductive group scheme over $\Bbbk$ equipped with a representation…

We examine the structure of the insertion-elimination Lie algebra on rooted trees introduced in \cite{CK}. It possesses a triangular structure $\g = \n_+ \oplus \mathbb{C}.d \oplus \n_-$, like the Heisenberg, Virasoro, and affine algebras.…

We study infinite dimensional tilting modules over a concealed canonical algebra of domestic or tubular type. In the domestic case, such tilting modules are constructed by using the technique of universal localization, and they can be…