Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0701283

On maximal diagonalizable Lie subalgebras of the first Hochschild cohomology

Patrick Le Meur
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let A be a basic connected finite dimensional algebra over an algebraically closed field k and with ordinary quiver Q without oriented cycle. To any presentation of A by quiver and admissible relations, Martinez-Villa and de La Pena have associated the fundamental group of the presentation. Assem and de La Pena have constructed an injective mapping from the additive characters of this fundamental group (with values in the ground field) to the first Hochschild cohomology group HH^1(A). We study the image of these mappings associated to the different presentations of A in terms of diagonalizable Lie subalgebras of HH^1(A). Then we characterise the maximal diagonalisable subalgebras of HH^1(A) when A is monomial and Q has no multiple arrows and also when car(k)=0 and Q has no double bypass.

Explore related subjects

math.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Patrick Le Meur. 2009-03-27. On maximal diagonalizable Lie subalgebras of the first Hochschild cohomology. https://doi.org/10.1080/00927870902915798

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the derived Hall algebra of a graded gentle one-cycle algebra I: the triangle structure

Under a mild condition, the perfect derived category and the finite-dimensional derived category of a graded gentle one-cycle algebra are described as twisted root categories of certain infinite quivers of type $\mathbb{A}_\infty^\infty$. As a consequence, it is shown that if $\ct$ is the perfect (respectively, finite-dimensional) derived category of such a graded gentle one-cycle algebra, then its triangle structure is up to triangle equivalence determined by the underlying additive category.

math.RT↗
arXiv2026

Towards Monoidal Categorifications of Twisted Products of Flag Varieties

Let $G$ be a simple, simply connected algebraic group of simply-laced type. For a positive braid word $β$ and $v\leδ(β)$, we study the cluster algebra associated with the twisted product of flag varieties $\mathring{\mathcal Z}_{v,β}$. We compare its Bao--Ye seed with a right-inductive weave seed and obtain local acyclicity and equality of the cluster and upper cluster algebras. Using Lusztig parameters in a bosonic extension algebra, we construct a monoidal subcategory $\mathscr C_{v,β}$ of a Hernandez--Leclerc category and prove that its Grothendieck ring contains the integral cluster algebra with noninvertible frozen variables. Every cluster monomial is the class of a real simple object of $\mathscr C_{v,β}$. The reverse inclusion, which would give a full monoidal categorification, is left as a conjecture.

math.RT↗
arXiv2026

Linear independence of global monomials on positive spaces

In this paper, we prove that global monomials on positive spaces are linearly independent, extending the basic fact that Laurent monomials in a Laurent polynomial algebra are linearly independent to a much more general setting. We also establish a global monomial avoidance phenomenon for positive spaces. Our approach is based on the study of Newton polytopes of Laurent expansions. These general results apply to positive spaces arising from cluster algebras (including the totally sign-skew-symmetric case), $Y$-patterns, and Laurent phenomenon algebras whose clusters are related by subtraction-free birational transformations. In particular, we obtain the proper Laurent monomial property and the linear independence of cluster monomials for all cluster algebras and Laurent phenomenon algebras under consideration. Notably, the proper Laurent monomial property follows from the global monomial avoidance phenomenon for positive spaces.

math.RT↗