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Jan Schröer

Publications and source records attributed to Jan Schröer.

36 records · Page 2Linked to original sources
arXiv2013

Caldero-Chapoton algebras

Motivated by the representation theory of quivers with potentials introduced by Derksen, Weyman and Zelevinsky and by work of Caldero and Chapoton, who gave explicit formulae for the cluster variables of Dynkin quivers, we associate a Caldero-Chapoton algebra to any (possibly infinite dimensional) basic algebra. By definition, the Caldero-Chapoton algebra is (as a vector space) generated by the Caldero-Chapoton functions of the decorated representations of the basic algebra. The Caldero-Chapoton algebra associated to the Jacobian algebra of a quiver with potential is closely related to the cluster algebra and the upper cluster algebra of the quiver. The set of generic Caldero-Chapoton functions, which conjecturally forms a basis of the Caldero-Chapoton algebra) is parametrized by the strongly reduced components of the varieties of representations of the Jacobian algebra and was introduced by Geiss, Leclerc and Schröer. Plamondon parametrized the strongly reduced components for finite-dimensional basic algebras. We generalize this to arbitrary basic algebras. Furthermore, we prove a decomposition theorem for strongly reduced components. Thanks to the decomposition theorem, all generic Caldero-Chapoton functions can be seen as generalized cluster monomials. As another application, we obtain a new proof for the sign-coherence of g-vectors. Caldero-Chapoton algebras lead to several general conjectures on cluster algebras.

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arXiv2012

Factorial cluster algebras

We show that cluster algebras do not contain non-trivial units and that all cluster variables are irreducible elements. Both statements follow from Fomin and Zelevinsky's Laurent phenomenon. As an application we give a criterion for a cluster algebra to be a factorial algebra. This can be used to construct cluster algebras, which are isomorphic to polynomial rings. We also study various kinds of upper bounds for cluster algebras, and we prove that factorial cluster algebras coincide with their upper bounds.

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arXiv2012

Cluster algebras in algebraic Lie theory

We survey some recent constructions of cluster algebra structures on coordinate rings of unipotent subgroups and unipotent cells of Kac-Moody groups. We also review a quantized version of these results.

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arXiv2011

Generic bases for cluster algebras and the Chamber Ansatz

Let Q be a finite quiver without oriented cycles, and let $Λ$ be the corresponding preprojective algebra. Let g be the Kac-Moody Lie algebra with Cartan datum given by Q, and let W be its Weyl group. With w in W is associated a unipotent cell N^w of the Kac-Moody group with Lie algebra g. In previous work we proved that the coordinate ring \C[N^w] of N^w is a cluster algebra in a natural way. A central role is played by generating functions \vphi_X of Euler characteristics of certain varieties of partial composition series of X, where X runs through all modules in a Frobenius subcategory C_w of the category of nilpotent $Λ$-modules. We show that for every X in C_w, \vphi_X coincides after appropriate changes of variables with the cluster characters of Fu and Keller associated with any cluster-tilting module T of C_w. As an application, we get a new description of a generic basis of the cluster algebra obtained from \C[N^w] via specialization of coefficients to 1. For the special case of coefficient-free acyclic cluster algebras this proves a conjecture by Dupont.

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arXiv2011

Kac-Moody groups and cluster algebras

Let Q be a finite quiver without oriented cycles, let Λbe the associated preprojective algebra, let g be the associated Kac-Moody Lie algebra with Weyl group W, and let n be the positive part of g. For each Weyl group element w, a subcategory C_w of mod(Λ) was introduced by Buan, Iyama, Reiten and Scott. It is known that C_w is a Frobenius category and that its stable category is a Calabi-Yau category of dimension two. We show that C_w yields a cluster algebra structure on the coordinate ring \CC[N(w)] of the unipotent group N(w) := N \cap (w^{-1}N_-w). Here N is the pro-unipotent pro-group with Lie algebra the completion of n. One can identify \CC[N(w)] with a subalgebra of the graded dual of the universal enveloping algebra U(n) of n. Let S^* be the dual of Lusztig's semicanonical basis S of U(n). We show that all cluster monomials of \CC[N(w)] belong to S^*, and that S^* \cap \CC[N(w)] is a basis of \CC[N(w)]. Moreover, we show that the cluster algebra obtained from \CC[N(w)] by formally inverting the generators of the coefficient ring is isomorphic to the algebra \CC[N^w] of regular functions on the unipotent cell N^w := N \cap (B_-wB_-) of the Kac-Moody group G with Lie algebra g. We obtain a corresponding dual semicanonical basis of \CC[N^w]. As one application we obtain a basis for each acyclic cluster algebra, which contains all cluster monomials in a natural way.

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arXiv2010

Cluster algebra structures and semicanonical bases for unipotent groups

Let Q be a finite quiver without oriented cycles, and let $Λ$ be the associated preprojective algebra. To each terminal representation M of Q (these are certain preinjective representations), we attach a natural subcategory $C_M$ of $mod(Λ)$. We show that $C_M$ is a Frobenius category,and that its stable category is a Calabi-Yau category of dimension 2. Then we develop a theory of mutations of maximal rigid objects of $C_M$, analogous to the mutations of clusters in Fomin and Zelevinsky's theory of cluster algebras. We show that $C_M$ yields a categorification of a cluster algebra $A(C_M)$, which is not acyclic in general. We give a realization of $A(C_M)$ as a subalgebra of the graded dual of the enveloping algebra $U(\n)$, where $\n$ is a maximal nilpotent subalgebra of the symmetric Kac-Moody Lie algebra $\g$ associated to the quiver Q. Let $S^*$ be the dual of Lusztig's semicanonical basis $S$ of $U(\n)$. We show that all cluster monomials of $A(C_M)$ belong to $S^*$, and that $S^* \cap A(C_M)$ is a basis of $A(C_M)$. Next, we prove that $A(C_M)$ is naturally isomorphic to the coordinate ring of the finite-dimensional unipotent subgroup $N(w)$ of the Kac-Moody group $G$ attached to $\g$. Here w = w(M) is the adaptable element of the Weyl group of $\g$ which we associate to each terminal representation M of Q. Moreover, we show that the cluster algebra obtained from $A(C_M)$ by formally inverting the generators of the coefficient ring is isomorphic to the coordinate ring of the unipotent cell $N^w := N \cap (B_-wB_-)$ of G. We obtain a corresponding dual semicanonical basis of this coorindate ring.

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arXiv2007

Partial flag varieties and preprojective algebras

Let L be a preprojective algebra of Dynkin type, and let G be the corresponding complex semisimple simply connected algebraic group. We study rigid modules in subcategories sub(Q) for Q an injective L-module, and we introduce a mutation operation between complete rigid modules in sub(Q). This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to G.

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arXiv2007

Semicanonical bases and preprojective algebras II: A multiplication formula

Let $n$ be a maximal nilpotent subalgebra of a complex symmetric Kac-Moody Lie algebra. Lusztig has introduced a basis of U(n) called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of nilpotent modules over a preprojective algebra of the same type as $n$. We prove a formula for the product of two elements of the dual of this semicanonical basis, and more generally for the product of two evaluation forms associated to arbitrary modules over the preprojective algebra. This formula plays an important role in our work on the relationship between semicanonical bases, representation theory of preprojective algebras, and Fomin and Zelevinsky's theory of cluster algebras. It was inspired by recent results of Caldero and Keller.

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arXiv2006

Auslander algebras and initial seeds for cluster algebras

Let $Q$ be a Dynkin quiver and $Π$ the corresponding set of positive roots. For the preprojective algebra $Λ$ associated to $Q$ we produce a rigid $Λ$-module $I_Q$ with $r=|Π|$ pairwise non-isomorphic indecomposable direct summands by pushing the injective modules of the Auslander algebra of $kQ$ to $Λ$. If $N$ is a maximal unipotent subgroup of a complex simply connected simple Lie group of type $|Q|$, then the coordinate ring $C[N]$ is an upper cluster algebra. We show that the elements of the dual semicanonical basis which correspond to the indecomposable direct summands of $I_Q$ coincide with certain generalized minors which form an initial cluster for $C[N]$, and that the corresponding exchange matrix of this cluster can be read from the Gabriel quiver of $End_Λ(I_Q)$. Finally, we exploit the fact that the categories of injective modules over $Λ$ and over its covering $\tildeΛ$ are triangulated in order to show several interesting identities in the respective stable module categories.

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arXiv2006

Rigid modules over preprojective algebras

Let $\la$ be a preprojective algebra of simply laced Dynkin type $Δ$. We study maximal rigid $\la$-modules, their endomorphism algebras and a mutation operation on these modules. This leads to a representation-theoretic construction of the cluster algebra structure on the ring $\C[N]$ of polynomial functions on a maximal unipotent subgroup $N$ of a complex Lie group of type $Δ$. As an application we obtain that all cluster monomials of $\C[N]$ belong to the dual semicanonical basis.

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arXiv2005

Verma modules and preprojective algebras

We give a geometric construction of the Verma modules of a symmetric Kac-Moody Lie algebra in terms of constructible functions on the varieties of nilpotent finite-dimensional modules of the corresponding preprojective algebra.

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arXiv2005

Semicanonical bases and preprojective algebras

We study the multiplicative properties of the dual of Lusztig's semicanonical basis.The elements of this basis are naturally indexed by theirreducible components of Lusztig's nilpotent varieties, whichcan be interpreted as varieties of modules over preprojective algebras.We prove that the product of two dual semicanonical basis vectorsis again a dual semicanonical basis vector provided the closure ofthe direct sum of thecorresponding two irreducible components is again an irreducible component.It follows that the semicanonical basis and the canonical basiscoincide if and only if we are in Dynkin type $A_n$ with $n \leq 4$.Finally, we provide a detailed study of the varieties of modules over the preprojectivealgebra of type $A_5$.We show that in this case the multiplicative properties ofthe dual semicanonical basis are controlled by the Ringel form of a certain tubular algebra of type (6,3,2) and by thecorresponding elliptic root system of type $E_8^{(1,1)}$.

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arXiv2002

Radical embeddings and representation dimension

Given a representation-finite algebra B and a subalgebra A of B such that the Jacobson radicals of A and B coincide, we prove that the representation dimension of A is at most three. By a result of Igusa and Todorov, this implies that the finitistic dimension of A is finite.

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arXiv2002

Extension-orthogonal components of nilpotent varieties

Let Q be a Dynkin quiver, and let P(Q) be the corresponding preprojective algebra. Let I be a set of pairwise different indecomposable irreducible components of varieties of P(Q)-modules such that generically there are no extensions between these components. We show that the number of elements in I is at most the number of positive roots of Q. Furthermore, we give a module theoretic interpretation of Leclerc's counterexample to a conjecture of Berenstein and Zelevinsky.

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arXiv2002

Irreducible components of varieties of modules

We prove some basic results about irreducible components of varieties of modules for an arbitrary finitely generated associative algebra. Our work generalizes results of Kac and Schofield on representations of quivers, but our methods are quite different, being based on deformation theory.

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