arXiv subjects

Lleonard Rubio y Degrassi

Publications and source records attributed to Lleonard Rubio y Degrassi.

13 recordsLinked to original sources

arXiv2026

Approximation of graded bialgebras

Motivated by an equivalence of categories established by Kapranov and Schechtman, we introduce, for each non-negative integer d, the category of connected bialgebras modulo d+1. We show that these categories fit into an inverse system of categories whose inverse limit category is equivalent to the category of connected bialgebras. In addition, we extend the notion of approximation of connected bialgebras to those that are not necessarily generated in degree 1 and show that, for connected bialgebras in the category of Yetter-Drinfeld modules over a Hopf algebra, approximation is compatible with cocycle twisting.

math.QA↗

arXiv2026

On the first relative Hochschild cohomology

In this paper we investigate the Lie algebra structure of the first relative Hochschild cohomology. Let $A,B$ be finite-dimensional basic $k$-algebras over an algebraically closed field of characteristic zero, such that $Q_B$ is a subquiver of $Q_A$. We show that if the complement of $Q_A$ by the arrows of $Q_B$ is a simple directed graph, then the first relative Hochschild cohomology $\mathrm{HH}^1(A|B)$ is a solvable Lie algebra. We also compute the Lie algebra structure of the first relative Hochschild cohomology for radical square zero algebras and for dual extension algebras of directed monomial algebras.

math.RT↗

arXiv2026

On the first Hochschild cohomology of finite dimensional quiver algebras under gluing idempotents

We compare the Lie algebra structures of the first Hochschild cohomology groups of a quiver algebra $A$ and a radical embedding $B$ obtained by gluing two idempotents of $A$. Under a mild assumption, we show that the first Hochschild cohomology groups of $A$ and $B$ are either isomorphic as Lie algebras or they differ by a one-dimensional Lie ideal. In particular, in the case of stable equivalences obtained by gluing a source and a sink vertex, we prove that either the first Hochschild cohomology groups of $A$ and $B$ are isomorphic or $HH^1(B)$ is a central extension of $HH^1(A)$ by a one-dimensional ideal. As a consequence, we obtain a new invariant under stable equivalences induced by gluing a source and a sink. We also compare the dimensions of $HH^1(A)$ and $HH^1(B)$, as well as the centers of $A$ and $B$, when gluing two arbitrary idempotents.

math.RT↗

arXiv2026

Outer derivations on blocks of group algebras

Let $B$ be a block of a finite group algebra $kG$. Linckelmann has conjectured that if the defect group $P$ of $B$ is nontrivial then $B$ admits a $k$-linear outer derivation, or in other words $\operatorname{HH}^1(B)\neq 0$. We provide various criteria for the non-vanishing of $\operatorname{HH}^1(B)$ in terms of the subgroup $P\subseteq G$. In particular, we show that Linckelmann's conjecture holds for principal blocks having abelian defect group, for all blocks of the symmetric and alternating groups, for blocks of finite groups of Lie type in defining characteristic, and for blocks of general linear groups in any characteristic. Our main tool relates the non-vanishing of the Batalin-Vilkovisky operator $Δ\colon \operatorname{HH}^1(B)\to \operatorname{HH}^0(B)$ to the existence of a certain elements of $P$, called extra-strong non-Schur elements. We show that extra-strong non-Schur $p$-elements exist for all groups having order divisible by a prime $p>5$, extending work of Fleischmann, Janiszczak, and Lempken. Using this, we prove that if $k$ has characteristic $p>5$, then Linckelmann's conjecture holds for all blocks of $kG$ with Sylow defect group. Likewise, when $k$ has characteristic $p>5$ we deduce that the first Hochschild cohomology is non-zero for any nontrivial twisted group algebra over $k$.

math.RT↗

arXiv2026

Truncated factorized perverse sheaves on Sym(C)

Kapranov and Schechtman defined the category FP of factorized perverse sheaves on Sym(C) smooth along the stratification given by multiplicities and with values in a braided monoidal category V. We define for each d in N the category FP^{\leq d} of factorized perverse sheaves on U_{n\leq d}Sym^{n}(C) and the category FP_{\leq d} of factorized perverse sheaves on the open subset of Sym(C) consisting of multi-sets with multiplicities bounded by d. We prove that the natural restriction functor from FP_{\leq d} to \FP^{\leq d} is an equivalence for any d in N, and that FP^{\leq 1} and \FP_{\leq 1} are equivalent to V. We show that the full direct image *, the extension by zero ! and the intermediate extension !* induce functors from FP_{\leq d} to \FP. In addition, we show that the families (FP^{\leq d})_{d in N} and (\FP_{\leq d})_{d in N} fit into systems of categories, compatible with restrictions and extensions, whose inverse limit is FP.

math.AG↗

arXiv2024

The Hochschild cohomology groups under gluing arrows

In a previous paper arXiv:2211.05435 we have compared the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two idempotents. In the present paper, we compare the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two arrows.

math.KT↗

arXiv2022

On the Lie algebra structure of integrable derivations

Building on work of Gerstenhaber, we show that the space of integrable derivations on an Artin algebra $A$ forms a Lie algebra, and a restricted Lie algebra if $A$ contains a field of characteristic $p$. We deduce that the space of integrable classes in $\HH^1(A)$ forms a (restricted) Lie algebra that is invariant under derived equivalences, and under stable equivalences of Morita type between self-injective algebras. We also provide negative answers to questions about integrable derivations posed by Linckelmann and by Farkas, Geiss and Marcos.

math.RA↗

arXiv2021

Maximal tori in $HH^1$ and the fundamental group

We investigate maximal tori in the Hochschild cohomology Lie algebra $HH^1(A)$ of a finite dimensional algebra $A$, and their connection with the fundamental groups associated to presentations of $A$. We prove that every maximal torus in $HH^1(A)$ arises as the dual of some fundamental group of $A$, extending work of Farkas, Green and Marcos; de la Peña and Saorín; and Le Meur. Combining this with known invariance results for Hochschild cohomology, we deduce that (in rough terms) the largest rank of a fundamental group of $A$ is a derived invariant quantity, and among self-injective algebras, an invariant under stable equivalences of Morita type. Using this we prove that there are only finitely many monomial algebras in any derived equivalence class of finite dimensional algebras; hitherto this was known only for very restricted classes of monomial algebras.

math.RT↗

arXiv2021

Stable invariance of the restricted Lie algebra structure of Hochschild cohomology

We show that the restricted Lie algebra structure on Hochschild cohomology is invariant under stable equivalences of Morita type between self-injective algebras. Thereby we obtain a number of positive characteristic stable invariants, such as the $p$-toral rank of $\mathrm{HH}^1(A,A)$. We also prove a more general result concerning Iwanaga-Gorenstein algebras, using a more general notion of stable equivalences of Morita type. Several applications are given to commutative algebra and modular representation theory. These results are proven by first establishing the stable invariance of the $B_\infty$-structure of the Hochschild cochain complex. In the appendix we explain how the $p$-power operation on Hochschild cohomology can be seen as an artifact of this $B_\infty$-structure. In particular, we establish well-definedness of the $p$-power operation, following some -- originally topological -- methods due to May, Cohen and Turchin, using the language of operads.

math.RT↗

arXiv2020

The first Hochschild cohomology as a Lie algebra

In this paper we study sufficient conditions for the solvability of the first Hochschild cohomology of a finite dimensional algebra as a Lie algebra in terms of its Ext-quiver in arbitrary characteristic. In particular, we show that if the quiver has no parallel arrows and no loops then the first Hochschild cohomology is solvable. For quivers containing loops, we determine easily verifiable sufficient conditions for the solvability of the first Hochschild cohomology. We apply these criteria to show the solvability of the first Hochschild cohomology space for large families of algebras, namely, several families of self-injective tame algebras including all tame blocks of finite groups and some wild algebras including most quantum complete intersections.

math.RT↗

arXiv2019

On the Lie algebra structure of $HH^1(A)$ of a finite-dimensional algebra $A$

Let $A$ be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the Ext-quiver of $A$ is a simple directed graph, then $HH^1(A)$ is a solvable Lie algebra. The second main result shows that if the Ext-quiver of $A$ has no loops and at most two parallel arrows in any direction, and if $HH^1(A)$ is a simple Lie algebra, then char(k) is not equal to $2$ and $HH^1(A)\cong$ $sl_2(k)$. The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.

math.RT↗

arXiv2016

Block algebras with HH1 a simple Lie algebra

We show that if $B$ is a block of a finite group algebra $kG$ over an algebraically closed field $k$ of prime characteristic $p$ such that $\HH^1(B)$ is a simple Lie algebra and such that $B$ has a unique isomorphism class of simple modules, then $B$ is nilpotent with an elementary abelian defect group $P$ of order at least $3$, and $\HH^1(B)$ is in that case isomorphic to the Jacobson-Witt algebra $\HH^1(kP)$. In particular, no other simple modular Lie algebras arise as $\HH^1(B)$ of a block $B$ with a single isomorphism class of simple modules.

math.RT↗

arXiv2015

Invariance of the restricted $p$-power map on integrable derivations under stable equivalences

We show that the $p$-power maps in the first Hochschild cohomology space of finite-dimensional selfinjective algebras over a field of prime characteristic $p$ commute with stable equivalences of Morita type on the subgroup of classes represented by integrable derivations. We show, by giving an example, that the $p$-power maps do not necessarily commute with arbitrary transfer maps in the Hochschild cohomology of symmetric algebras.

math.RT↗