Algebras and Representation Theory最新文献

It is well-established that weak n-tilting modules serve as generalizations of both n-tilting and n-cotilting modules. The primary objective of this paper is to delineate the characterizations of weak n-silting modules and elaborate on their applications. Specifically, we aim to establish the "triangular relation" within the framework of silting theory in a module category, and provide novel characterizations of weak n-tilting modules. Furthermore, we delve into the properties of n-(co)silting modules and their interrelations with some other types of modules. Additionally, we explore the conditions under which a weak n-silting module can be classified as partial n-silting, weak n-tilting, or partial n-tilting. Notably, we establish and prove that Bazzoni’s renowned characterization of pure-injectivity for cotilting modules remains valid for weak n-silting modules with respect to (mathcal {F}_{mathbb {T}}). Lastly, we investigate weak n-silting and weak n-tilting objects in a morphism category.

Let G be a finite p-group with normal subgroup (varvec{N}) of order (varvec{p}). The first author and Zalesskii have previously shown that a (mathbb {Z}_p) (varvec{G})-lattice is a permutation module if, and only if, its (varvec{N})-invariants, its (varvec{N})-coinvariants, and a third module are all G/N permutation modules over (mathbb {Z}_p, mathbb {Z}_p) and (mathbb {Z}_p) respectively. The necessity of the first two conditions is easily shown but the necessity of the third was not known. We apply a correspondence due to Butler, which associates to a (mathbb {Z}_p) (varvec{G})-lattice for an abelian (varvec{p})-group a set of simple combinatorial data, to demonstrate the necessity of the conditions, using the correspondence to construct highly non-trivial counterexamples to the claim that if both the (varvec{N})-invariants and the (varvec{N})-coinvariants of a given lattice (varvec{U}) are permutation modules, then so is (varvec{U}). Our approach, which is new, is to translate the desired properties to the combinatorial side, find the counterexample there, and translate it back to a lattice.

Losev introduced the scheme X of almost commuting elements (i.e., elements commuting upto a rank one element) of (mathfrak {g}=mathfrak {sp}(V)) for a symplectic vector space V and discussed its algebro-geometric properties. We construct a Lagrangian subscheme (X^{nil}) of X and show that it is a complete intersection of dimension (text {dim}(mathfrak {g})+frac{1}{2}text {dim}(V)) and compute its irreducible onents. We also study the quantum Hamiltonian reduction of the algebra (mathcal {D}(mathfrak {g})) of differential operators on the Lie algebra (mathfrak {g}) tensored with the Weyl algebra with respect to the action of the symplectic group, and show that it is isomorphic to the spherical subalgebra of a certain rational Cherednik algebra of Type C. We contruct a category (mathcal {C}_c) of (mathcal {D})-modules whose characteristic variety is contained in (X^{nil}) and construct an exact functor from this category to the category (mathcal {O}) of the above rational Cherednik algebra. Simple objects of the category (mathcal {C}_c) are mirabolic analogs of Lusztig’s character sheaves.

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}).

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.