We study the connection between *-representations of algebras associated with graphs, locally-scalar graph representations and the problem about the spectrum of a sum of two Hermitian operators. For algebras associated with Dynkin graphs we give an explicit description of the parameters for which there are irreducible representations and an algorithm for contructing these representations.
We prove that every deformed preprojective algebra of Dynkin type $\mathbb{E}_6$ is isomorphic to the preprojective algebra of Dynkin type $\mathbb{E}_6$.
We provide a complete classification of finite-dimensional self-injective algebras which are socle equivalent to preprojective algebras of generalized Dynkin type. In particular, we conclude that these algebras are deformed preprojective algebras of generalized Dynkin type (in the sense of [5, 12]), and hence are periodic algebras.
This paper studies silted algebras, namely, endomorphism algebras of 2-term silting complexes, over path algebras of Dynkin quivers. We will describe an algorithm to produce all basic 2-term silting complexes over the path algebra of a Dynkin quiver, and use this algorithm to compute some examples.
We present a study on the description of incidence algebras that are piecewise hereditary, which we denominate Phia algebras. We describe the quiver with relations of the Phia algebras of Dynkin type and introduce a new family of Phia algebras of extended Dynkin type, which we call ANS family, in reference to Assem, Nehring, and Skowroński. In this description, the important method was the one of cutting sets on trivial extensions, inspired by this we made of a computer program which shows exactly the cutting sets on the given trivial extension that result on incidence algebras.
There is a connection between *-representations of algebras associated with graphs and the problem about the spectrum of a sum of Hermitian operators (spectral problem). For algebras associated with extended Dynkin graphs we give an explicit description of the parameters for which there are $*$-representations and an algorithm for constructing these representations.
In this paper, we use the stable categories of some selfinjective algebras to describe the singularity categories of the cluster-tilted algebras of Dynkin type. Furthermore, in this way, we settle the problem of singularity equivalence classification of the cluster-tilted algebra of type $A$, $D$ and $E$ respectively.
We consider degenerations of all simple Lie algebras of exceptional type obtained by embedding into affine Lie algebras. We give a filtration to consider this as an abelianisation of the original Lie algebra. We then show that the associated graded of simple, finite-dimensional modules are isomorphic to Demazure modules.
We construct a numeric algorithm for completing the proof of a conjecture asserting that all deformed preprojective algebras of generalized Dynkin type are periodic. In particular, we obtain an algorithmic procedure showing that non-trivial deformed preprojective algebras of Dynkin types $\mathbb{E}_7$ and $\mathbb{E}_8$ exist only in characteristic 2. As a consequence, we show that deformed preprojective algebras of Dynkin types $\mathbb{E}_6$, $\mathbb{E}_7$ and $\mathbb{E}_8$ are periodic and we obtain an algorithm for a classification of such algebras, up to algebra isomorphism. We do it b…
In this paper we study gradings on simple Lie algebras arising from nilpotent elements. Specifically, we investigate abelian subalgebras which are degree 1 homogeneous with respect to these gradings. We show that for each odd nilpotent element there always exists canonically associated "strictly" odd nilpotent element, which allows us to reduce our investigations to the latter.
In this paper, we study two-term tilting complexes for preprojective algebras of non-Dynkin type. We show that there exist two families of two-term tilting complexes, which are respectively parameterized by the elements of the corresponding Coxeter group. Moreover, we provide the complete classification in the case of affine type by showing that any two-term silting complex belongs one of them. For this purpose, we also discuss the Krull-Schmidt property for the homotopy category of finitely generated projective modules over a complete ring.
We provide a far reaching derived equivalence classification of the cluster-tilted algebras of Dynkin type D and suggest standard forms for the derived equivalence classes. We believe that the classification is complete, but some subtle questions remain open. We introduce another notion of equivalence called good mutation equivalence which is slightly stronger than derived equivalence but is algorithmically more tractable, and give a complete classification together with standard forms.
In this paper, we study two-sided tilting complexes of preprojective algebras of Dynkin type. We construct the most fundamental class of two-sided tilting complexes, which has a group structure by derived tensor products and induces a group of auto-equivalences of the derived category. We show that the group structure of the two-sided tilting complexes is isomorphic to the braid group of the corresponding folded graph. Moreover we show that these two-sided tilting complexes induce tilting mutation and any tilting complex is given as the derived tensor products of them. Using these results, we …
This is a continuation of arXiv:0903.0398 [math.RT]. Let g be a simple Lie algebra. In this note, we provide simple formulae for the index of sl(2)-subalgebras in the classical Lie algebras and a new formula for the index of the principal sl(2). We also compute the difference, D, of the indices of principal and subregular sl(2)-subalgebras. Our formula for D involves some data related to the McKay correspondence for g. Using the index of sl(2)-subalgebras of classical Lie algebras, we also obtain three series of interesting combinatorial identities parameterised by partitions.
We show that the mutation class of a finite quiver without oriented cycles is finite if and only is the quiver is either Dynkin, extended Dynkin or has at most two vertices.
For a symmetrizable GCM $C$ and its symmetrizer $D$, Geiss-Leclerc-Schröer [Invent. Math. 209 (2017)] has introduced a generalized preprojective algebra $Π$ associated to $C$ and $D$, that contains a class of modules, called locally free modules. We show that any basic support $τ$-tilting $Π$-module is locally free and gives a classification theorem of torsion-free classes in $\operatorname{\mathbf{rep}}Π$ as the generalization of the work of Mizuno [Math. Z. 277 (2014)].
The irreducible components of the variety of all modules over the preprojective algebra and MV cycles both index bases of the universal enveloping algebra of the positive part of a semisimple Lie algebra canonically. To relate these two objects Baumann and Kamnitzer associate a cycle in the affine Grassmannian for a given module. It is conjectured that the ring of functions of the T-fixed point subscheme of the associated cycle is isomorphic to the cohomology ring of the quiver Grassmannian of the module. I give a proof of part of this conjecture. Given this conjecture, I give a proof of the r…
We show how a cluster-tilted algebra of finite representation type is related to the corresponding tilted algebra, in the case of algebras defined over an algebraically closed field.
We obtain a complete derived equivalence classification of the cluster-tilted algebras of Dynkin type E. There are 67, 416, 1574 algebras in types E6, E7 and E8 which turn out to fall into 6, 14, 15 derived equivalence classes, respectively. This classification can be achieved computationally and we outline an algorithm which has been implemented to carry out this task. We also make the classification explicit by giving standard forms for each derived equivalence class as well as complete lists of the algebras contained in each class; as these lists are quite long they are provided as suppleme…
Consider a simple Lie algebra $\mathfrak{g}$ and $\overline{\mathfrak{g}}% \subset \mathfrak{g}$ a Levi subalgebra. Two irreducible $\overline{% \mathfrak{g}}$-modules yield isomorphic inductions to $\mathfrak{g}$ when their highest weights coincide up to conjugation by an element of the Weyl group $W$ of $\mathfrak{g}$ which is also a Dynkin diagram automorphism of $% \overline{\mathfrak{g}}$. In this paper we study the converse problem: given two irreducible $\overline{\mathfrak{g}}$-modules of highest weight $μ$ and $ν$ whose inductions to $\mathfrak{g}$ are isomorphic, can we conclude that…
We consider Dynkin algebras, these are the hereditary artin algebras of finite representation type. The indecomposable modules for a Dynkin algebra correspond bijectively to the positive roots of a Dynkin diagram. Given a Dynkin algebra with n simple modules, a complete exceptional sequence is a sequence M_1,..., M_n of indecomposable modules such that Hom(M_i,M_j) = 0 = Ext(M_i,M_j) for i > j. The aim of this paper is to determine the number of complete exceptional sequences for any Dynkin algebra. There are direct connections between the representation theory of a Dynkin algebra A and the la…
We describe a method for an explicit determination of indecomposable preprojective and preinjective representations for extended Dynkin quivers by vector spaces and matrices. This method uses tilting theory and the explicit knowledge of indecomposable modules over the corresponding canonical algebra of domestic type.
We show that for a Dynkin quiver $Q$ of type $E_7$ with a specific orientation, the path algebra $KQ$ has no slope function of the form $μ=\fracθ{\dim}$ that defines a total stability condition. This gives a counterexample to a conjecture of Reineke.
In this paper, we compute the product structure of the Hochschild cohomology of preprojective algebras of quivers of type D and E over a field of characteristic zero.
The Hochschild homology/cohomology an associative algebra, together with the Connes differential, the contraction map and the Lie derivative, forms the structure of calculus. In this paper we compute explicitely the calculus structure of preprojective algebras of Dynkin quivers over a field of characteristic zero. This also completes the work in math.AG/0502301, where the Batalin-Vilkovisky structure of the Hochschild cohomology of preprojective algebras of non-Dynkin quivers are computed and the calculus can be easily computed from that.
The Dynkin algebras are the hereditary artin algebras of finite representation type. The paper exhibits the number of support-tilting modules for any Dynkin algebra. Since the support-tilting modules for a Dynkin algebra correspond bijectively to the non-crossing partitions of the same type, the calculations presented here may also be considered as a categorification of results concerning non-crossing partitions.
We study tilting complexes over preprojective algebras of Dynkin type. We classify all tilting complexes by giving a bijection between tilting complexes and the braid group of the corresponding folded graph. In particular, we determine the derived equivalence class of the algebra. For the results, we develop the theory of silting-discrete triangulated categories and give a criterion of silting-discreteness.
For Dynkin quivers, we find the Laurent polynomials $\widetilde{X}_{a, c}^{b}(v)$ and use $\widetilde{X}_{a, c}^{b}(v)$ to construct the Hall algebra $\hc_v(\cc(\cp))$ over $\mz[v, v^{-1}]$, where $\widetilde{X}_{a, c}^{b}(|\mf_q|)$'s are structure constants used by Bridgeland. The Laurent polynomials $\widetilde{X}_{a, c}^{b}(v)$ are explicitly given in $A_1$ case. As an application, we obtain the full quantum groups $U_t(\sg)$ associated to the Dynkin quivers for arbitrary $t\not=0,\pm1$.
The purpose of my Ph.D. research is to define and study an analogue of the classical Bernstein-Gelfand-Gelfand (BGG) category $\mathcal{O}$ for the Lie algebra $\mathfrak{g}$, where $\mathfrak{g}$ is one of the finitary, infinite-dimensional Lie algebras $\mathfrak{gl}_\infty(\mathbb{K})$, $\mathfrak{sl}_\infty(\mathbb{K})$, $\mathfrak{so}_\infty(\mathbb{K})$, and $\mathfrak{sp}_\infty(\mathbb{K})$. Here, $\mathbb{K}$ is an algebraically closed field of characteristic $0$. We call these categories "extended categories $\mathcal{O}$" and use the notation $\bar{\mathcal{O}}$. While the categorie…
The Spectral Problem is to describe possible spectra $σ(A_j)$ for an irreducible $n$-tuple of Hermitian operators s.t. $A_1+...+A_n$ is a scalar operator. In case when $m_j= | σ(A_j)|$ are finite and a rooted tree ${\rm T}_{m_1,..., m_n}$ with $n$ branches of lengths $m_1, ..., m_n$ is a Dynkin graph the explicit answer to the Spectral Problem was given recently by S. A. Kruglyak, S. V. Popovych, and Yu. S. Samo\vılenko. In present work the solution of the Spectral Problem for all star-shaped simply laced extended Dynkin graphs, i.e. when $(m_1, ..., m_n) \in \{(2,2,2,2), (3,3,3), (4,4,2)$, $(…
We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi-Yau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2-Calabi-Yau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of …
A desingularization of arbitrary quiver Grassmannians for representations of Dynkin quivers is constructed in terms of quiver Grassmannians for an algebra derived equivalent to the Auslander algebra of the quiver.
In this paper, we describe a general setting for dimer models on cylinders over Dynkin diagrams which in type A reduces to the well studied case of dimer models on a disc. We prove that all Berenstein--Fomin--Zelevinsky quivers for Schubert cells in a symmetric Kac--Moody algebra give rise to dimer models on the cylinder over the corresponding Dynkin diagram. We also give an independent proof of a result of Buan, Iyama, Reiten and Smith that the corresponding superpotentials are rigid using the dimer model structure of the quivers.
For every cluster-tilted algebra of simply-laced Dynkin type we provide a companion basis which is strong, i.e. gives the set of dimension vectors of the finitely generated indecomposable modules for the cluster-tilted algebra. This shows in particular that every companion basis of a cluster-tilted algebra of simply-laced Dynkin type is strong. Thus we give a proof of Parsons's conjecture.
We consider the dual space of linear groups over Dynkinian and Euclidean algebras, i.e. finite dimensional algebras derived equivalent to the path algebra of Dynkin or Euclidean quiver. We prove that this space contains an open dense subset isomorphic to the product of dual spaces of full linear groups and, perhaps, one more (explicitly described) space. The proof uses the technique of bimodule categories, deformations and representations of quivers.
We compute the minimal model for Ginzburg algebras associated to acyclic quivers $Q$. In particular, we prove that there is a natural grading on the Ginzburg algebra making it formal and quasi-isomorphic to the preprojective algebra in non-Dynkin type, and in Dynkin type is quasi-isomorphic to a twisted polynomial algebra over the preprojective with a unique higher $A_\infty$-composition. To prove these results, we construct and study the minimal model of an $A_\infty$-envelope of the derived category $\mathcal{D}^b(Q)$ whose higher compositions encode the triangulated structure of $\mathcal{D…
$\frak g$-endomorphism algebras form an interesting class of associative algebras related to the adjoint representation of a semisimple Lie algebra $\frak g$. These algebras were recently introduced by A.Kirillov, who used the term `family algebras'. Let $C_λ$ denote the $\frak g$-endomorphism algebra associated with a simple $\frak g$-module $V_λ$. Most of our results concern the case in which $C_λ$ is commutative, i.e., $V_λ$ is a weight multiplicity free $\frak g$-module. It is proved that $C_λ$ is a polynomial algebra if and only if $λ$ is minuscule. We also characterise in general the num…
We study support $τ$-tilting modules over preprojective algebras of Dynkin type. We classify basic support $τ$-tilting modules by giving a bijection with elements in the corresponding Weyl groups. Moreover we show that they are in bijection with the set of torsion classes, the set of torsion-free classes and other important objects in representation theory. We also study $g$-matrices of support $τ$-tilting modules, which show terms of minimal projective presentations of indecomposable direct summands. We give an explicit description of $g$-matrices and prove that cones given by $g$-matrices co…
We show that the stable module categories of certain selfinjective algebras of finite representation type having tree class A_n, D_n, E_6, E_7 or E_8 are triangulated equivalent to u-cluster categories of the corresponding Dynkin type. The proof relies on the 'Morita' theorem for u-cluster categories by Keller and Reiten, along with the recent computation of Calabi-Yau dimensions of stable module categories by Dugas.
Let $\mathfrak{g}$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$. It is conjectured that for each Dynkin node $k \in I \setminus \{0\}$ the affine Lie algebra $\mathfrak{g}$ has a positive geometric crystal. In this paper we construct a positive geometric crystal for the affine Lie algebra $D_6^{(1)}$ corresponding to the Dynkin spin node $k= 6$.
We give a characterization of the Dynkin elements of a simple Lie algebra. Namely, we prove that one-half of a Dynkin element is the unique point of minimal length in its N-region. In type A_n this translates into a statement about the regions determined by the canonical left Kazhdan-Lusztig cells. Some possible generalizations are explored in the last section.
Let $A$ be a finite dimensional hereditary algebra over an algebraically closed field $k$, $T_2(A)=(\begin{array}{cc}A&0 A&A\end{array})$ be the triangular matrix algebra and $A^{(1)}=(\begin{array}{cc}A&0 DA&A\end{array})$ be the duplicated algebra of $A$ respectively. We prove that ${\rm rep.dim}\ T_2(A)$ is at most three if $A$ is Dynkin type and ${\rm rep.dim}\ T_2(A)$ is at most four if $A$ is not Dynkin type. Let $T$ be a tilting A-$\module$ and $\ol{T}=T\oplus\ol{P}$ be a tilting $A^{(1)}$-$\module$. We show that $\End_{A^{(1)}} \ol{T}$ is representation finite if and only if the full s…
We investigate the structure of electrical Lie algebras of finite Dynkin type. These Lie algebras were introduced by Lam-Pylyavskyy in the study of \textit{circular planar electrical networks}. The corresponding Lie group acts on such networks via some combinatorial operations studied by Curtis-Ingerman-Morrow and Colin de Verdière-Gitler-Vertigan. Lam-Pylyavskyy studied the electrical Lie algebra of type $A$ of even rank in detail, and gave a conjecture for the dimension of electrical Lie algebras of finite Dynkin types. We prove this conjecture for all classical Dynkin types, that is, $A$, $…
For a finite subgroup $G$ of the special unitary group $SU_2$, we study the centralizer algebra $Z_k(G) = End_G(V^{\otimes k})$ of $G$ acting on the $k$-fold tensor product of its defining representation $V= \mathbb{C}^2$. These subgroups are in bijection with the simply-laced affine Dynkin diagrams. The McKay correspondence relates the representation theory of these groups to the associated Dynkin diagram, and we use this connection to show that the structure and representation theory of $Z_k(G)$ as a semisimple algebra is controlled by the combinatorics of the corresponding Dynkin diagram.
Let G be a connected almost simple algebraic group with a Dynkin automorphism σ. Let G_σ be the connected almost simple algebraic group associated to G and σ. We prove that the dimension of the tensor invariant space of G_σ is equal to the trace of σ on the corresponding tensor invariant space of G. We prove that if G has the saturation property then so does Gσ. As a consequence, we show that the spin group Spin(2n + 1) is of saturation property with factor 2, which strengthens the results of Belkale-Kumar and Sam in the case of type B_n.
For a commutative ring $R$ and an ADE Dynkin quiver $Q$, we prove that the multiplicative preprojective algebra of Crawley-Boevey and Shaw, with parameter $q=1$, is isomorphic to the (additive) preprojective algebra as $R$-algebras if and only if the bad primes for $Q$ (2 in type D, 2 and 3 for $Q = E_{6}$, $E_{7}$ and 2, 3 and 5 for $Q= E_{8}$) are invertible in $R$. We construct an explicit isomorphism over $\mathbb{Z}[1/2]$ in type D, over $\mathbb{Z}[1/2, 1/3]$ for $Q = E_{6}$, $E_{7}$ and over $\mathbb{Z}[1/2, 1/3, 1/5]$ for $Q=E_{8}$. Conversely, if some bad prime is not invertible in $R…
Given a quiver, a fixed dimension vector, and a positive integer n, we construct a functor from the category of D-modules on the space of representations of the quiver to the category of modules over a corresponding Gan-Ginzburg algebra of rank n. When the quiver is affine Dynkin we obtain an explicit construction of representations of the corresponding wreath-product symplectic reflection algebra of rank n. When the quiver is star-shaped, but not finite Dynkin, we use this functor to obtain a Lie theoretic construction of representations of a "spherical" subalgebra of the Gan-Ginzburg algebra…
In this paper the relationship between iterated tilted algebras and cluster-tilted algebras and relation-extensions is studied. In the Dynkin case, it is shown that the relationship is very strong and combinatorial.
