The Morita equivalences of classical Brauer algebras and classical Birman–Murakami–Wenzl (BMW) algebras have been well studied. Here, we study the Morita equivalence problems on these two kinds of algebras of simply-laced type, especially for them with the generic parameters. We show that Brauer algebras and BMW algebras of simply-laced type are Morita equivalent to the direct sums of some group algebras of Coxeter groups and some Hecke algebras of Coxeter groups, respectively.
Let be a simply connected and simple algebraic group defined and split over a finite prime field of elements. In this paper, using an -linear map splitting Frobenius endomorphism on a hyperalgebra relative to , we obtain some -linear isomorphisms induced by multiplication in the hyperalgebra.
Let be a positive integer and let be a finite field with elements, where is a prime power and . In this paper, we give the explicit factorization of over and count the number of its irreducible factors for the following conditions: are odd and . First, we present a method to obtain the set of all representatives of -cyclotomic cosets modulo , where . This set of representatives is then used to find the irreducible factors of and the cyclotomic polynomial over
For a semisimple quasi-triangular Hopf algebra over a field of characteristic zero, and a strongly separable quantum commutative -module algebra , we show that is a weak Hopf algebra, and it can be embedded into a weak Hopf algebra . With these structures, is the monoidal category introduced by Cohen and Westreich, and is tensor equivalent to . If is in the Müger center of , then the embedding is a quasi-triangular weak Hopf algebra morphism. This explains the presence of a subgroup inclusion in the characterization of irreducible Yetter–Drinfeld modules for a finite group algebra.
Rota–Baxter operators have been paid much attention in the last few decades as they have many applications in mathematics and physics. In this paper, our object of study is modified Rota–Baxter operators on Leibniz algebras. We investigate modified Rota–Baxter Leibniz algebras from the cohomological point of view. We study a one-parameter formal deformation theory of modified Rota–Baxter Leibniz algebras and define the associated deformation cohomology that controls the deformation. Finally, as an application, we characterize equivalence classes of abelian extensions in terms of second cohomology groups.
Let be a pair of finite subgroups of and a finite-dimensional fundamental -module. We study Kostant’s generating functions for the decomposition of the -module restricted to in connection with the McKay–Slodowy correspondence. In particular, the classical Kostant formula was generalized to a uniform version of the Poincaré series for the symmetric invariants in which the multiplicities of any individual module in the symmetric algebra are completely determined.
As a sequel to [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in Algebra Colloq.], in this article we first introduce a so-called duplex Hecke algebras of type which is a -algebra associated with the Weyl group of type , and symmetric groups for , satisfying some Hecke relations (see Definition 3.1). This notion originates from the degenerate duplex Hecke algebra arising from the course of study of a kind of Schur–Weyl duality of Levi-type (see [B. Shu and Y. Yao, On enhanced reductive groups (I): Enhanced Schur algebras and the dualities related to degenerate duplex Hecke algebras, with an appendix by B. Liu, submitted (2023)]), extending the duplex Hecke algebra of type arising from the related -Schur–Weyl duality of Levi-type (see [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in Algebra Colloq.]). A duplex Hecke algebra of type admits natural representations on certain tensor spaces. We then establish a Levi-type -Schur–Weyl duality of type , which reveals the double centralizer property between such duplex Hecke algebras and
Let be a finite group and be the set of irreducible characters of . The codegree of an irreducible character of the group is defined as . In this paper, we study two topics related to the character codegrees. The first result is related to the prime graph of character codegrees and we show that the codegree prime graphs of several classes of groups can be characterized only by graph theoretical terms. The second result is about the -parts of the codegrees and character degrees.
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