Let $G = \mathbf{GL}_d(\mathbb{F})$ be the general linear group over a field of cardinal $q$, and let $\mathbb{k}$ be a field of positive characteristic which does not divide $q(q-1)$. Building on the works of Cline, Parshall, and Scott, we show how to compute Ext-groups between $\mathbb{k}G$-modules using the quantum Schur algebra. The main novelty is our ability to compute these Ext-groups in higher degree than what was done before. More precisely, let $\ell$ be the order of $q$ in $\mathbb{k}$. In previous work, this method enabled the computation of the cohomology groups $H^*(\mathbf{GL}_d,M)$ in degree $*\leq \ell-1$. We show that for a lot of modules $M$, we can compute these cohomology groups in higher degree, with an example where we can compute until degree $3(\ell-1)$. We also show some new result on Ext-groups between modules over the quantum Schur algebra along the way.

The construction of relations among relations is one ingredient in the Groebner-like basis construction of the maximal ideal of the universal vertex operator algebra $V^k_{\mathfrak g}$ for affine Lie algebras. For affine Lie algebras of type $C_n^{(1)}$, such combinatorially parametrized relations among relations were constructed in earlier work for level $2$ standard modules \cite{PS3}, and for $C_2^{(1)}$-standard modules at higher levels \cite{S}. This article presents two further examples in which the same counting method can be carried out. The first treats $C_n^{(1)}$-standard modules at the fixed level $k=5$, with $n$ arbitrary. The second treats $C_3^{(1)}$-standard modules for arbitrary level $k$. In both cases the calculation compares the number of required relations among relations in a trapezoid of the array of negative root vectors with the corresponding representation-theoretic dimension.

In this paper, we study the transition matrix between the PBW basis and the canonical basis for the negative part of the quantized enveloping algebra of the Kronecker quiver from a geometric viewpoint. Building on Lusztig's geometric construction of the canonical basis, we construct sheaf-complex realizations of PBW basis elements by means of flag sheaf complexes over the strata $X(α,m)$ of representation varieties. Our first goal is to give a geometric description of the simple constituents appearing in the restrictions of these flag sheaf complexes to the strata $X(α,m)$. This allows us to compare the PBW-type sheaf complexes with the simple perverse sheaves $IC(X(α),L_χ)$ arising in Lusztig's construction. Using this description together with a purity result for the relevant $\mathbb{F}_q$-structures, we obtain another proof that the elements defined by Lusztig's perverse sheaves indeed form a basis of the composition algebra.Our second goal is to make the transition coefficients between the PBW basis and the canonical basis geometrically explicit. More precisely, we show that these coefficients are governed by the multiplicities of local systems in the restrictions of intersection cohomology complexes to smaller strata. As a consequence, the transition matrix from the canonical basis to the PBW basis is upper triangular with diagonal entries equal to $1$, and its coefficients admit a direct geometric interpretation. In particular, in the Kronecker quiver case we recover the triangularity of the transition matrix and obtain positivity properties of the corresponding coefficient polynomials.

Twisted quantum loop algebras are a generalization of twisted quantum affine algebras in Drinfeld new presentation. The Hall algebras of Geigle--Lenzing's weighted projective lines are used to realize (untwisted) quantum loop algebras of simply-laced type associated to star-shaped graphs by Schiffmann and Dou--Jiang--Xiao. In this paper, we use the semi-derived Ringel-Hall algebras of more general weighted projective lines to realize the twisted quantum loop algebras associated to the valued star-shaped graphs, including the twisted quantum affine algebras in Drinfeld new presentation.

We study the space of one-point functions on the torus for a possibly nonrational $C_2$-cofinite vertex operator algebra $V$ by relating it to a trace object of the subcategory of projective objects in the representation category of $V$. We identify the dual of the trace space with symmetric functions on the endomorphism algebra $E$ of a projective generator. Motivated by the Gainutdinov-Runkel conjecture, recently established using different methods by Gui and Zhang, we present a complementary representation-theoretic approach based on Arike-Nagatomo pseudo-traces. In this framework, we prove surjectivity of the Gainutdinov-Runkel map from symmetric functions on $E$ to one-point functions. Under the additional assumption of separated conformal weights modulo $\mathbb{Z}$, we also prove injectivity, using projective-cover techniques inspired by Huang.

We study Rota--Baxter operators on vertex algebras using the integrated $λ$-bracket formalism. A Rota--Baxter operator produces a deformed vertex algebra structure, and the difference between the deformed and original brackets yields a two-cocycle in vertex algebra cohomology. This generalizes the classical relation between Rota--Baxter operators and Hochschild two-cocycles. We also characterize when this two-cocycle is trivial, showing that non-scalar operators give rise to non-trivial cohomology classes.

We give an explicit description of the BPS Lie algebra of any quiver with zero potential, by relating the perverse filtration on the cohomological Hall algebra with certain limit conditions on polynomials. Our results also give a partial description of the perverse filtration for arbitrary potential, which we conjecture is complete in the case of tripled quivers with canonical cubic potential.

Let $\g$ be a simple complex Lie algebra of type $G_2$, $F_4$, or $E_8$, and let $G$ be the unique connected simply connected complex Lie group with $\mathrm{Lie}(G)=\g$ and compact real form $K$. We prove a triangular decomposition theorem for the lower crystal lattice $\OAztG$ of the quantized function algebra $\OtG$, establishing that $\OAztG=A_0\text{-alg}<\RAzp \cup \RAzm>.$ This extends the triangular decomposition recently obtained for types $A_n, B_n, C_n, D_n, E_6$, and $E_7$ in~\cite{DDPa} to all simple complex Lie algebras. As a consequence, we obtain: (i) the inclusion $\OAztG\subseteq\OAztK$ conjectured by Matassa-Yuncken and (ii) the crystal limit $\CpKo$ is a compact quantum semigroup with a unique bi-invariant (Haar) state.