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The weight modules of the Lie algebra s l (2 , C) are well known. In the first part of this paper we deal with a realization of weight modules of s l (2 , C) in the space t ν C [ t , t − 1 ] , ν ∈ R , where C [ t , t − 1 ] is the algebra of Laurent polynomials. In the second part, we consider the Hom-Lie algebra s l q (2 , C) of Jackson where q ≠ 0 , 1 . The q-analogue of the above realization in the space t ν C (q) [ t , t − 1 ] , ν ∈ R is considered. We obtain two kinds of q-modules. The regular q-modules which have limits the modules obtained in the classical realization when q goes to 1. The other q-modules have no limits when q goes to 1 and they are called singular modules. [ABSTRACT FROM AUTHOR]

Let O be a discrete valuation ring with maximal ideal p and with finite residue field F q , the field with q elements where q is a power of a prime p. For r ≥ 1 , we write O r for the reduction of O modulo the ideal p r . An irreducible ordinary representation of the finite group GL N (O r) is called stable if its restriction to the principal congruence kernel K l = 1 + p l M N (O r) , where l = ⌈ r 2 ⌉ , consists of irreducible representations whose stabilizers modulo K l ′ , where l ′ = r − l , are centralizers of certain matrices in g l ′ = M N (O l ′) , called stable matrices. The study of stable representations is motivated by constructions of strongly semisimple representations, introduced by Hill, which is a special case of stable representations. In this paper, we explore the construction of stable irreducible representations of the finite group GL N (O r) for N ≥ 2 . Communicated by Scott Chapman [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show that there are Hom-Lie algebra structures on s l 2 (F) ⊕ F D , where D is a special type of generalized derivation of s l 2 (F) , and F is an algebraically closed field of characteristic zero. We study the representation theory of Hom-Lie algebras within the appropriate category and prove that any finite dimensional representation of a Hom-Lie algebra of the form s l 2 (F) ⊕ F D is completely reducible, analogously to the well-known theorem of Weyl from the classical Lie theory. We apply this result to characterize the non-solvable Lie algebras of this type having an invertible generalized derivation D. [ABSTRACT FROM AUTHOR]

First, we give a partial solution to the isomorphism problem for uniserial modules of finite length with the help of the morphisms between these modules. Later, under suitable assumptions on the lattice of the submodules, we give a method to partially solve the isomorphism problem for uniserial modules over an arbitrary ring. Particular attention is given to the natural class of uniserial modules defined over algebras given by quivers. [ABSTRACT FROM AUTHOR]

In this article, we demonstrate a technique for calculating the irreducible representations of the endomorphism algebra of the Gelfand–Graev representations of the general symplectic group where q is an odd prime power. [ABSTRACT FROM AUTHOR]

We define a new class of racks, called finitely stable racks, which, to some extent, share various flavors with Abelian groups. Characterization of finitely stable Alexander quandles is established. Further, we study twisted rack dynamical systems, construct their cross-products, and introduce representation theory of racks and quandles. We prove several results on the strong representations of finite connected involutive racks analogous to the properties of finite Abelian groups. Finally, we define the Pontryagin dual of a rack as an Abelian group which, in the finite involutive connected case, coincides with the set of its strong irreducible representations. [ABSTRACT FROM AUTHOR]

This paper is primarily concerned with δ-Hom-Jordan Lie superalgebras. We discuss the concepts of αk-derivations, representations and T*-extensions of δ-Hom-Jordan Lie superalgebras in detail, and some cohomological characterizations are established. [ABSTRACT FROM AUTHOR]

We prove that all irreducible representations of the bismash producthave Frobenius–Schur indicators +1 or 0 where 𝕜 is an algebraically closed field andis an exact factorization. Moreover, we have a description of those simple modules which are not self-dual. We prove some new results for bismash products in general and make conjectures about bismash products that arise from other exact factorizations ofSn. [ABSTRACT FROM AUTHOR]

The purpose of this article is to study representations andT*-extensions of δ-hom–Jordan–Lie algebras. In particular, adjoint representations, trivial representations, deformations, and many properties ofT*-extensions of δ-hom–Jordan–Lie algebras are studied in detail. Derivations and central extensions of δ-hom–Jordan–Lie algebras are also discussed as an application. [ABSTRACT FROM PUBLISHER]

We give a characterization of indecomposable exceptional modules over finite dimensional gentle algebras. As an application, we study gentle algebras arising from an unpunctured surface and show that a class of indecomposable modules related to curves without self-intersections, as exceptional modules, are uniquely determined by their dimension vectors. [ABSTRACT FROM AUTHOR]

To describe the representation theory of the quantum Weyl algebra at anlth primitive root γ of unity, Boyette, Leyk, Plunkett, Sipe, and Talley found all nonsingular irreducible matrix solutions to the equationyx − γxy = 1, assumingyx ≠ xy. In this note, we complete their result by finding and classifying, up to equivalence, all irreducible matrix solutions (X,Y), whereXis singular. [ABSTRACT FROM AUTHOR]

Letdbe a prehomogeneous dimension vector for a connected quiverQwith the property thatcℓdhas a negative entry for some ℓ ∈ ℕ wherecis the Coxeter transformation corresponding to an admissible numbering of the verticesQ0ofQ. Denote by rep(Q,d) the variety ofd-dimensional representations ofQand by Sl(d) the product of the special linear groups at all verticesQ0. We show how to find the irreducible components of the null cone of the algebraic quotient rep(Q,d)//Sl(d). [ABSTRACT FROM PUBLISHER]

We give a module-theoretic interpretation of Schiffler's expansion formula which is defined combinatorially in terms of complete (Γ, γ)-paths in order to get the expansion of the cluster variables in the cluster algebra of a marked surface (S, M). Based on the geometric description of the indecomposable objects of the cluster category of the marked surface (S, M), we show the coincidence of Schiffler–Thomas’ expansion formula and the cluster character defined by Palu. [ABSTRACT FROM PUBLISHER]

Let 𝕂 be an algebraically closed field. We classify the finite groups having exactly one irreducible 𝕂-representation of degree bigger than one. The case where the characteristic of 𝕂 is zero, was done by G. Seitz in 1968. [ABSTRACT FROM PUBLISHER]

Using a unified method, we determine the structure of automorphisms and representations of arbitrary polyadic groups. More precisely, for a polyadic group (G, f) = der θ, b (G, ·), we obtain a complete description of automorphisms and representations of (G, f) in terms of automorphisms and representations of the binary group (G, ·). [ABSTRACT FROM PUBLISHER]

We prove that there is a one-one correspondence between sets of irreducible representations of a polyadic group and its Post's cover. Using this correspondence, we generalize some well-known properties of irreducible characters in finite groups to the case of polyadic groups. [ABSTRACT FROM PUBLISHER]

We give a classification of the smooth (complex) representations of GL2(), where  is the ring of integers in a non-Archimedean local field. The approach is based on Clifford theory of finite groups and a corresponding study of orbits and stabilizers. In terms of this classification, we identify the representations which are geometrically or infinitesimally induced, respectively. [ABSTRACT FROM AUTHOR]

If G is a group of odd order and χ ∈ Irr(G) lifts an irreducible Brauer character ϕ, then we associate to χ a canonical pair (Q, δ) up to G-conjugacy, where Q is a vertex of ϕ and δ ∈ Irr(Q) is a linear character of Q. We show that (Q, δ) is a Navarro vertex for χ. We also discuss examples. [ABSTRACT FROM AUTHOR]

Given a discrete valuation ring R, a non-negative integer m, and a finite partially ordered set S, the representation type of a category rep(S*, R, m) of finite rank free R-module representations of S is characterized in terms of S and m. As an application, representation types of some categories of torsion-free abelian groups of finite rank are determined. [ABSTRACT FROM AUTHOR]

Modular irreducible p-restricted representations of algebraic groups of types En (n = 6, 7,8) and F4 are studied, The composition factors of the restrictions of such representations to naturally embedded Al-subgroups are found. Jordan block sizes of root unipotent elements of such groups in the representations with locally small highest weights with respect to p are determined. [ABSTRACT FROM AUTHOR]

Let Vn = 〈 e1, …, en+1) be the vector product n-Lie algebra with n-Lie commutator [e1, …, êi, … en+1] = (-1)iei over the field of complex numbers. Any finite-dimensional n-Lie Vn-module is completely reducible. Any finite-dimensional irreducible n-Lie Vn-module is isomorphic to an n-Lie extension of son+1-module with highest weight tπ1 for some nonnegative integer t. [ABSTRACT FROM AUTHOR]

In this note we show that certain propertie sof the quantum function algebra at roots of unity hold over any algebraically closed field of characteristic zero. [ABSTRACT FROM AUTHOR]

Restrictions of irreducible representations of classical algebraic groups to root A[SUB1]-subgroups, i.e., subgroups of type A[SUB1] generated by root subgroups associated with two opposite roots, are studied. Composition factors of such restrictions are found in the following cases: for groups of types A[SUBn] with n > 2 and D[SUBn], for groups of type B[SUBn], n > 2, and long root subgroups, for groups of type C[SUBn], n > 2, and short root subgroups, and for p-restricted representations of A[SUB2](K), C[SUB2](K) (recall that B[SUB2](K) ≅ C[SUB2](K)), and of B[SUBn](K), n > 2, and short root subgroups. Here we assume that p > 2 for G = B[SUBn](K) or C[SUBn](K). [ABSTRACT FROM AUTHOR]

We give a complete description of primitive linear representations of a p-group over a field F of characteristic ≠2, p, including whether they are of orthogonal or symplectic type. For p odd this is easy and essentially well-known. For p=2, although the proofs offer no special difficulty the discussion is quite involved, depending on the cyclotomy of F. When F is non-exceptional in the sense of Harris and Segal, primitive representations can be obtained from "generic" ones (Theorems 9.8 and 10.4): this may be considered as the main result of the paper. In passing, we improve an old result of O. Schilling on the division ring of endomorphisms of an irreducible representation of a p-group. [ABSTRACT FROM AUTHOR]