Algebras and Representation Theory最新文献

The aim of this paper is to develop a framework for localization theory of triangulated categories (mathcal {C}), that is, from a given extension-closed subcategory (mathcal {N}) of (mathcal {C}), we construct a natural extriangulated structure on (mathcal {C}) together with an exact functor (Q:mathcal {C}rightarrow widetilde{mathcal {C}}_mathcal {N}) satisfying a suitable universality, which unifies several phenomena. Precisely, a given subcategory (mathcal {N}) is thick if and only if the localization (widetilde{mathcal {C}}_mathcal {N}) corresponds to a triangulated category. In this case, Q is nothing other than the usual Verdier quotient. Furthermore, it is revealed that (widetilde{mathcal {C}}_mathcal {N}) is an exact category if and only if (mathcal {N}) satisfies a generating condition (textsf{Cone}(mathcal {N},mathcal {N})=mathcal {C}). Such an (abelian) exact localization (widetilde{mathcal {C}}_mathcal {N}) provides a good understanding of some cohomological functors (mathcal {C}rightarrow textsf{Ab}), e.g., the heart of t-structures on (mathcal {C}) and the abelian quotient of (mathcal {C}) by a cluster-tilting subcategory (mathcal {N}).

We study the closure of the graph of the birational map from a projective space to a Grassmannian. We provide explicit description of the graph closure and compute the fibers of the natural projection to the Grassmannian. We construct embeddings of the graph closure to the projectivizations of certain cyclic representations of a degenerate special linear Lie algebra and study algebraic and combinatorial properties of these representations. In particular, we describe monomial bases, generalizing the FFLV bases. The proof relies on combinatorial properties of a new family of poset polytopes, which are of independent interest. As a consequence we obtain flat toric degenerations of the graph closure studied by Borovik, Sturmfels and Sverrisdóttir.

It is well-known that the generalized Auslander-Reiten condition (GARC) and the symmetric Auslander condition (SAC) are equivalent, and (GARC) implies that the Auslander-Reiten condition (ARC). In this paper we explore (SAC) along with the several canonical change of rings (R rightarrow S). First, we prove the equivalence of (SAC) for R and R/xR, where x is a non-zerodivisor on R, and the equivalence of (SAC) and (SACC) for rings with positive depth, where (SACC) is the symmetric Auslander condition for modules with constant rank. The latter assertion affirmatively answers a question posed by Celikbas and Takahashi. Secondly, for a ring homomorphism (R rightarrow S), we prove that if S satisfies (SAC) (resp. (ARC)), then R also satisfies (SAC) (resp. (ARC)) if the flat dimension of S over R is finite. We also prove that (SAC) holds for R implies that (SAC) holds for S when R is Gorenstein and (S=R/Q^ell ), where Q is generated by a regular sequence of R and the length of the sequence is at least (ell ). This is a consequence of more general results about Ulrich ideals proved in this paper. Applying these results to determinantal rings and numerical semigroup rings, we provide new classes of rings satisfying (SAC). A relation between (SAC) and an invariant related to the finitistic extension degree is also explored.

In this paper, we study simple weak and ordinary twisted modules of the extended Heisenberg-Virasoro vertex operator algebra (V_{tilde{mathcal {L}}_{F}}(ell _{123},0)). We first determine the full automorphism groups of (V_{tilde{mathcal {L}}_{F}}(ell _{123},0)) for all (ell _{1}, ell _{2},ell _{3},Fin {mathbb C}). They are isomorphic to certain subgroups of the general linear group (text {GL}_{2}({mathbb C})). Then for a family of finite order automorphisms (sigma _{r_{1},r_{2}}) of (V_{tilde{mathcal {L}}_{F}}(ell _{123},0)), we show that weak (sigma _{r_{1},r_{2}})-twisted (V_{tilde{mathcal {L}}_{F}}(ell _{123},0))-modules are in one-to-one correspondence with restricted modules of certain Lie algebras of level (ell _{123}), where (r_{1}, r_2in {mathbb N}). By this identification and vertex algebra theory, we give complete lists of simple ordinary (sigma _{r_{1},r_{2}})-twisted modules over (V_{tilde{mathcal {L}}_{F}}(ell _{123},0)). The results depend on whether F or (ell _{2}) is zero or not. Furthermore, simple weak (sigma _{r_{1},r_{2}})-twisted (V_{tilde{mathcal {L}}_{F}}(ell _{123},0))-modules are also investigated. For this, we introduce and study restricted modules (including Whittaker modules) of a new Lie algebra (mathcal {L}_{r_{1},r_{2}}) which is related to the mirror Heisenberg-Virasoro algebra.

We verify the Mukai conjecture for Fano quiver moduli spaces associated to dimension vectors in the interior of the fundamental domain.

We consider the operations of projection and lifting of Weyl chambers to and from a root subsystems of a finite roots system. Extending these operations to labeled galleries, we produce pairs of such galleries that satisfy some common wall crossing properties. These pairs give rise to certain morphisms in the category of Bott-Samelson varieties earlier considered by the author. We prove here that all these morphisms define embeddings of Bott-Samelson varieties (considered in the original interpretation based on compact Lie groups due to Raoul Bott and Hans Samelson) skew invariant with respect to the compact torus. We prove that those embeddings that come from projection and lifting preserve two natural orders on the set of the points fixed by the compact torus. We also consider the application of these embeddings to equivariant cohomology. The operations of projection and lifting can also be applied separately to each segment of a gallery. We describe conditions that allow us to glue together the galleries obtained this way.

Gabriel’s Theorem states that the quivers which have finitely many isomorphism classes of indecomposable representations are exactly those with underlying graph one of the ADE Dynkin diagrams and that the indecomposables are in bijection with the positive roots of this graph. When the underlying graph is (varvec{A_n}), these indecomposable representations are thin (either 0 or 1 dimensional at every vertex) and in bijection with the connected subquivers. Using linear algebraic methods we show that every (possibly infinite-dimensional) representation of a quiver with underlying graph (varvec{A_{infty , infty }}) is infinite Krull-Schmidt, i.e. a direct sum of indecomposables, as long as the arrows in the quiver eventually point outward. We furthermore prove that these indecomposable are again thin and in bijection with both the connected subquivers and the limits of the positive roots of (varvec{A_{infty , infty }}) with respect to a certain uniform topology on the root space. Finally we give an example of an (varvec{A_{infty , infty }}) quiver which is not infinite Krull-Schmidt and hence necessarily is not eventually-outward.

Let E and F be finite graphs with no sinks, and k any field. We show that shift equivalence of the adjacency matrices (A_E) and (A_F), together with an additional compatibility condition, implies that the Leavitt path algebras (L_k(E)) and (L_k(F)) are graded Morita equivalent. Along the way, we build a new type of (L_k(E))–(L_k(F))-bimodule (a bridging bimodule), which we use to establish the graded equivalence.

In this paper, we give a more down-to-earth introduction to the connection between Gelfand-Tsetlin modules over (mathfrak {gl}_n) and diagrammatic KLRW algebras and develop some of its consequences. In addition to a new proof of this description of the category Gelfand-Tsetlin modules appearing in earlier work, we show three new results of independent interest: (1) we show that every simple Gelfand-Tsetlin module is a canonical module in the sense of Early, Mazorchuk and Vishnyakova, and characterize when two maximal ideals have isomorphic canonical modules, (2) we show that the dimensions of Gelfand-Tsetlin weight spaces in simple modules can be computed using an appropriate modification of Leclerc’s algorithm for computing dual canonical bases, and (3) we construct a basis of the Verma modules of (mathfrak {sl}_n) which consists of generalized eigenvectors for the Gelfand-Tsetlin subalgebra. Furthermore, we present computations of multiplicities and Gelfand-Kirillov dimensions for all integral Gelfand-Tsetlin modules in ranks 3 and 4; unfortunately, for ranks (>4), our computers are not adequate to perform these computations.

We show that the small and large restricted injective dimensions coincide for Cohen-Macaulay rings of finite Krull dimension. Based on this, and inspired by the recent work of Sather-Wagstaff and Totushek, we suggest a new definition of Cohen-Macaulay Hom injective dimension. We show that the class of Cohen-Macaulay Hom injective modules is the right constituent of a perfect cotorsion pair. Our approach relies on tilting theory, and in particular, on the explicit construction of the tilting module inducing the minimal tilting class recently obtained in (Hrbek et al. 2022).