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In this paper, we first define a new family of Yoneda algebras, called F-Auslander-Yoneda algebras, in triangulated categories by introducing the notion of admissible sets F in N, which includes higher cohomologies indexed by F, and then present a general method to construct a family of new derived equivalences for these F-Auslander-Yoneda algebras (not necessarily Artin algebras), where the choices of the parameters F are rather abundant. Among applications of our method are the following results: (1) if A is a self-injective Artin algebra, then, for any A-module X and for any admissible set F in N, the F-Auslander-Yoneda algebras of A ⊕ X and A ⊕ OA(X) are derived equivalent, where O is the Heller loop operator. (2) Suppose that A and B are representation-finite self-injective algebras with additive generators AX and BY, respectively. If A and B are derived equivalent, then so are the F-Auslander-Yoneda algebras of X and Y for any admissible set F. In particular, the Auslander algebras of A and B are derived equivalent. The converse of this statement is open. Further, motivated by these derived equivalences between F-Auslander-Yoneda algebras, we show, among other results, that a derived equivalence between two basic self-injective algebras may transfer to a derived equivalence between their quotient algebras obtained by factoring out socles. [ABSTRACT FROM AUTHOR]

As an application of the double affine Hecke algebra with unequal parameters on Weyl orbits to representation theory of semisimple Lie algebras, we find the graded multiplicities of the trivial module and of the little adjoint module in the exterior algebra of the little adjoint module of a simple Lie algebra \mathfrak {g} with a non-simply laced Dynkin diagram. We prove that in type B, C or F these multiplicities can be expressed in terms of special exponents of positive long roots in the dual root system of \mathfrak {g}. [ABSTRACT FROM AUTHOR]

We provide an explicit construction for a class of commutative, non-associative algebras for each of the simple Chevalley groups of simply laced type. Moreover, we equip these algebras with an associating bilinear form, which turns them into Frobenius algebras. This class includes a 3876-dimensional algebra on which the Chevalley group of type E8 acts by automorphisms. We also prove that these algebras admit the structure of (axial) decomposition algebras. [ABSTRACT FROM AUTHOR]

A subcategory of an abelian category is wide if it is closed under sums, summands, kernels, cokernels, and extensions. Wide subcategories provide a significant interface between representation theory and combinatorics. If Φ is a finite dimensional algebra, then each functorially finite wide subcategory of mod(Φ) is of the form φ*(mod(Γ)) in an essentially unique way, where \Γ is a finite dimensional algebra and Φ φ → Γ is an algebra epimorphism satisfying TorΦ1(Γ,Γ) = 0. Let F ⊆ mod (Φ) be a d-cluster tilting subcategory as defined by Iyama. Then F is a d-abelian category as defined by Jasso, and we call a subcategory of F wide if it is closed under sums, summands, d-kernels, d-cokernels, and d-extensions. We generalise the above description of wide subcategories to this setting: Each functorially finite wide subcategory of F is of the form φ*(G) in an essentially unique way, where Φ φ → Γ is an algebra epimorphism satisfying TorΦd(Γ,Γ) = 0, and G ⊆ mod (\Gamma) is a d-cluster tilting subcategory. We illustrate the theory by computing the wide subcategories of some d-cluster tilting subcategories F ⊆ mod (Φ) over algebras of the form Φ = kAm/(rad kAm)l. [ABSTRACT FROM AUTHOR]

Twisted modules over vertex algebras formalize the relations among twisted vertex operators and have applications to conformal field theory and representation theory. A recent generalization, called a twisted logarithmic module, involves the logarithm of the formal variable and is related to logarithmic conformal field theory. We investigate twisted logarithmic modules of lattice vertex algebras, reducing their classification to the classification of modules over a certain group. This group is a semidirect product of a discrete Heisenberg group and a central extension of the additive group of the lattice. [ABSTRACT FROM AUTHOR]