Algebras and Representation Theory最新文献

We show that a semi-stable moduli space of representations of an acyclic quiver can be identified with two GIT quotients by reductive groups. One of a quiver Grassmannian of a projective representation, the other of a quiver Grassmannian of an injective representation. This recovers as special cases the classical Gelfand–MacPherson correspondence and its generalization by Hu and Kim to bipartite quivers, as well as the Zelevinsky map for a quiver of Dynkin type A with the linear orientation.

The aim of this paper is to study the natural action of the real symplectic group, ({text {Sp}}(4, mathbb {R})), on the algebraic set of 4-dimensional Lie algebras admitting symplectic structures and to give a complete classification of orbit closures. We present some applications of such classification to the study of the Ricci curvature of left-invariant almost Kähler structures on four dimensional Lie groups.

A Nakayama algebra with almost separate relations is one where the overlap between any pair of relations is at most one arrow. In this paper we give a derived equivalence between such Nakayama algebras and path algebras of quivers of a special form known as quipu quivers. Furthermore, we show how this derived equivalence can be used to produce a complete classification of linear Nakayama algebras with almost separate relations. As an application, we include a list of the derived equivalence classes of all Nakayama algebras of length (le 8) with almost separate relations.

We study and classify a class of representations (called generalized geometric representations) of a Coxeter group of finite rank. These representations can be viewed as a natural generalization of the geometric representation. The classification is achieved by using characters of the integral homology group of certain graphs closely related to the Coxeter graph. On this basis, we also provide an explicit description of those representations on which the defining generators of the Coxeter group act by reflections.

For a split quasireductive supergroup (mathbbm {G}) defined over a field, we study structure and representation of Frobenius kernels (mathbbm {G}_r) of (mathbbm {G}) and we give a necessary and sufficient condition for (mathbbm {G}_r) to be unimodular in terms of the root system of (mathbbm {G}). We also establish Steinberg’s tensor product theorem for (mathbbm {G}) under some natural assumptions.

Let ((T(n),Omega )) be the covering of the generalized Kronecker quiver K(n), where (Omega ) is a bipartite orientation. Then there exists a reflection functor (sigma ) on the category ({{,textrm{mod},}}(T(n),Omega )). Suppose that (0rightarrow Xrightarrow Yrightarrow Zrightarrow 0) is an AR-sequence in the regular component (mathcal {D}) of ({{,textrm{mod},}}(T(n),Omega )), and b(Z) is the number of flow modules in the (sigma )-orbit of Z. Then the middle term Y is a sink (source or flow) module if and only if (sigma Z) is a sink (source or flow) module. Moreover, their radii and centers satisfy (r(Y)=r(sigma Z)+1) and (C(Y)=C(sigma Z)).

We give a new and intrinsic proof of the transitivity of the braid group action on the set of full exceptional sequences of coherent sheaves on a weighted projective line. We do not use the corresponding result of Crawley-Boevey for modules over hereditary algebras. As an application we prove that the strongest global dimension of the category of coherent sheaves on a weighted projective line (mathbb {X}) does not depend on the parameters of (mathbb {X}). Finally we prove that the determinant of the matrix obtained by taking the values of n (mathbb {Z})-linear functions defined on the Grothendieck group (textrm{K}_0(mathbb {X}) simeq mathbb {Z}^n ) of the elements of a full exceptional sequence is an invariant, up to sign.

Holm introduced m-free (ell )-arrangements which is a generalization of free arrangements, while he asked whether all (ell )-arrangements are m-free for m large enough. Recently Abe and the author gave a negative answer to this question when (ell ge 4). In this paper we verify that 3-arrangements (mathscr {A}) are m-free and compute the m-exponents for all (mge |mathscr {A}|+2), where (|mathscr {A}|) is the cardinality of (mathscr {A}). Hence Holm’s question has a positive answer when (ell =3). Finally we prove that 3-dimensional Weyl arrangements of types A and B are m-free for all (mge 0).

Presilting and silting subcategories in extriangulated categories were introduced by Adachi and Tsukamoto recently, which are generalizations of those concepts in triangulated categories. Exact categories and triangulated categories are extriangulated categories. In this paper, we prove that the Gabriel-Zisman localization (mathcal {B}/(textsf{thick}hspace{.01in}mathcal W)) of an exact category (mathcal {B}) with respect to a presilting subcategory (mathcal W) satisfying certain condition can be realized as a subfactor category of (mathcal {B}). Afterwards, we discuss the relation between silting subcategories and tilting subcategories in exact categories, which gives us a kind of important examples of our results. In particular, for a finite dimensional Gorenstein algebra, we get the relative version of the description of the singularity category due to Happel and Chen-Zhang.

This present paper is devoted to the study of a class of Nakayama algebras (N_n(r)) given by the path algebra of the equioriented quiver (mathbb {A}_n) subject to the nilpotency degree r for each sequence of r consecutive arrows. We show that the Nakayama algebras (N_n(r)) for certain pairs (n, r) can be realized as endomorphism algebras of tilting objects in the bounded derived category of coherent sheaves over a weighted projective line, or in its stable category of vector bundles. Moreover, we classify all the Nakayama algebras (N_n(r)) of Fuchsian type, that is, derived equivalent to the bounded derived categories of extended canonical algebras. We also provide a new way to prove the classification result on Nakayama algebras of piecewise hereditary type, which have been done by Happel–Seidel before.