Let $k$ be an algebraically closed field and let $\Lambda$ be a finite-dimensional associative $k$-algebra. We apply Joyce and Brideland's notion of Ringel--Hall algebra to prove results about picture spaces and picture groups. For instance, we construct a faithful group functor for $\Lambda$, i.e. a faithful functor from the $\tau$-cluster morphism category of $\Lambda$ into a groupoid. Consequently, by results of Hanson--Igusa, the picture space of $\Lambda$ is locally CAT(0) provided that the $\tau$-cluster morphism category of $\Lambda$ admits compatibility of last factors. We also show that green sequences are in bijection with certain positive expressions in the picture group, which generalizes a result of Igusa--Todorov beyond hereditary $k$-algebras.
We develop a spectrum-level graded $K$-theory for real super Banach algebras. Our construction is categorical and homotopy theoretic, in the style of algebraic $K$-theory: the graded $K$-theory spectrum is obtained by a (co)fiber sequence from the $(\infty,1)$-categorical group completion of topological groupoids of fi...
We take a novel lattice-theoretic approach to the $τ$-cluster morphism category $\mathfrak{T}(A)$ of a finite-dimensional algebra $A$ and define the category via the lattice of torsion classes $\mathrm{tors } A$. Using the lattice congruence induced by an ideal $I$ of $A$ we establish a functor $F_I: \mathfrak{T}(A) \t...
Let $R$ be a commutative ring and $\Phi$ an inverse family of ideals with greatest proper member $\mathcal{L}$. We introduce and study $\Phi$-reduced and $\Phi$-coreduced modules, extending the corresponding notions for powers of a single ideal. We show that, on these classes of modules, the generalized torsion and com...
For $n\ge1$, we study representations of a simplicial group $G$ in the full $\infty$-subcategory of simplicial vector spaces whose normalized homology is concentrated in degrees $0,\ldots,n-1$, retaining its full mapping spaces. The resulting functor category depends only on the $n$-truncated homotopy type of $\mathbf{...
Let $K/\Qp$ be a finite extension, $\bG$ a connected reductive group over $K$, and set $G = \bG(K)$, considered as a $p$-adic Lie group. We show first that the Bernstein center $\cC_G$ of the category of solid locally $\Qp$-analytic representations of $G$ is isomorphic to the center of the locally analytic distribution...
Daniel Le, Matthias Strauch, Zi-Chuan Wang· 0 citations
We study when a finite-dimensional pre-Nichols algebra $R$ of $V$ over an algebraically closed field of characteristic $p>0$, with $p\nmid\dim R$, is necessarily isomorphic to the Nichols algebra $\B(V)$. A trace identity gives a rigidity criterion involving the twist on minimal defining relations. Using integral forms...
Rong-Chuan Xiong· 0 citations
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