Algebras and Representation Theory最新文献

The Graded Classification Conjecture states that the pointed (K_{0}^{operatorname {gr}})-group is a complete invariant of the Leavitt path algebras of finite graphs when these algebras are considered with their natural grading by (mathbb {Z}). The strong version of this conjecture states that the functor (K_{0}^{operatorname {gr}}) is full and faithful when considered on the category of Leavitt path algebras of finite graphs and their graded homomorphisms modulo conjugations by invertible elements of the zero components. We show that the functor (K_{0}^{operatorname {gr}}) is full for the unital Leavitt path algebras of countable graphs and that it is faithful (modulo specified conjugations) only in a certain weaker sense.

We show that for the family of complex reflection groups G = G(m, p,2) appearing in the Shephard–Todd classification, the endomorphism ring of the reduced hyperplane arrangement A(G) is a non-commutative resolution for the coordinate ring of the discriminant Δ of G. This furthers the work of Buchweitz, Faber and Ingalls who showed that this result holds for any true reflection group. In particular, we construct a matrix factorization for Δ from A(G) and decompose it using data from the irreducible representations of G. For G(m, p,2) we give a full decomposition of this matrix factorization, including for each irreducible representation a corresponding maximal Cohen–Macaulay module. The decomposition concludes that the endomorphism ring of the reduced hyperplane arrangement A(G) will be a non-commutative resolution. For the groups G(m,1,2), the coordinate rings of their respective discriminants are all isomorphic to each other. We also calculate and compare the Lusztig algebra for these groups.

Let k be an algebraically closed field of characteristic p > 0 and let G be a symplectic or general linear group over k. We consider induced modules for G under the assumption that p is bigger than the greatest hook length in the partitions involved. We give explicit constructions of left resolutions of induced modules by tilting modules. Furthermore, we give injective resolutions for induced modules in certain truncated categories. We show that the multiplicities of the indecomposable tilting and injective modules in these resolutions are the coefficients of certain Kazhdan-Lusztig polynomials. We also show that our truncated categories have a Kazhdan-Lusztig theory in the sense of Cline, Parshall and Scott. This builds further on work of Cox-De Visscher and Brundan-Stroppel.

Let G be a finite simple group and k be an algebraically closed field of prime characteristic dividing the order of G. We show that for all 2-cocycles α ∈ Z²(G;k^×), the first Hochschild cohomology group of the twisted group algebra HH¹(k_αG) is nonzero.

We demonstrate that for a fixed dominant integral weight and fixed defect d, there are only a finite number of Morita equivalence classes of blocks of cyclotomic Hecke algebras, by combining some combinatorics with the Chuang-Rouquier categorification of integrable highest weight modules over Kac-Moody algebras of affine type A. This is an extension of a proof for symmetric groups of a conjecture known as Donovan’s conjecture. We fix a dominant integral weight Λ. The blocks of cyclotomic Hecke algebras (H^{Lambda }_{n}) for the given Λ correspond to the weights P(Λ) of a highest weight representation with highest weight Λ. We connect these weights into a graph we call the reduced crystal (widehat {P}({Lambda })), in which vertices are connected by i-strings. We define the hub of a weight and show that a vertex is i-external for a residue i if the defect is less than the absolute value of the i-component of the hub. We demonstrate the existence of a bound on the degree after which all vertices of a given defect d are i-external in at least one i-string, lying at the high degree end of the i-string. For e = 2, we calculate an approximation to this bound.

We explain the construction of minimal tilting complexes for objects of highest weight categories and we study in detail the minimal tilting complexes for standard objects and simple objects. For certain categories of representations of complex simple Lie algebras, affine Kac-Moody algebras and quantum groups at roots of unity, we relate the multiplicities of indecomposable tilting objects appearing in the terms of these complexes to the coefficients of Kazhdan-Lusztig polynomials.

Let p be a an odd prime and let G be a finite p-group with cyclic commutator subgroup (G^{prime }). We prove that the exponent and the abelianization of the centralizer of (G^{prime }) in G are determined by the group algebra of G over any field of characteristic p. If, additionally, G is 2-generated then almost all the numerical invariants determining G up to isomorphism are determined by the same group algebras; as a consequence the isomorphism type of the centralizer of (G^{prime }) is determined. These claims are known to be false for p = 2.

We compute the homology of the complexes of finite Verma modules over the annihilation superalgebra (mathcal {A}({K}_{4}^{prime })), associated with the conformal superalgebra ({K}_{4}^{prime }), obtained in Bagnoli and Caselli (J. Math. Phys. 63, 091701, 2022). We use the computation of the homology in order to provide an explicit realization of all the irreducible quotients of finite Verma modules over (mathcal {A}({K}_{4}^{prime })).

In this paper, we study Weyl modules for a toroidal Lie algebra (mathcal {T}) with arbitrary n variables. Using the work of Rao (Pac. J. Math. 171(2), 511–528 1995), we prove that the level one global Weyl modules of (mathcal {T}) are isomorphic to suitable submodules of a Fock space representation of (mathcal {T}) up to a twist. As an application, we compute the graded character of the level one local Weyl module of (mathcal {T}), thereby generalising the work of Kodera (Lett. Math. Phys. 110(11) 3053–3080 2020).

We consider the action of a semisimple Hopf algebra H on an m-Koszul Artin–Schelter regular algebra A. Such an algebra A is a derivation-quotient algebra for some twisted superpotential w, and we show that the homological determinant of the action of H on A can be easily calculated using w. Using this, we show that the smash product A#H is also a derivation-quotient algebra, and use this to explicitly determine a quiver algebra Λ to which A#H is Morita equivalent, generalising a result of Bocklandt–Schedler–Wemyss. We also show how Λ can be used to determine whether the Auslander map is an isomorphism. We compute a number of examples, and show how several results for the quantum Kleinian singularities studied by Chan–Kirkman–Walton–Zhang follow using our techniques.