Representation theory of the algebra generated by a pair of complex structures

The objective of this paper is to determine the finite-dimensional, indecomposable representations of the algebra that is generated by two complex structures over the real numbers. Since the generators satisfy relations that are similar to those of the infinite dihedral group, we give the algebra the name [Formula: see text].

On the representation theory of the infinite Temperley–Lieb algebra

We begin the study of the representation theory of the infinite Temperley–Lieb algebra. We fully classify its finite-dimensional representations, then introduce infinite link state representations and classify when they are irreducible or indecomposable. We also define a construction of projective indecomposable representations for TL[Formula: see text] that generalizes to give extensions of TL[Formula: see text] representations. Finally, we define a generalization of the spin chain representation and conjecture a generalization of Schur–Weyl duality.

Diagrammatic morphisms between indecomposable modules of Ūq(𝔰𝔩2)

We give diagrammatic formulae for morphisms between indecomposable representations of [Formula: see text] appearing in the decomposition of [Formula: see text], including projections and second endomorphisms on projective indecomposable representations.

Sifat-Sifat Representasi Indekomposabel The Properties of Indecomposable Representations

A directed graph is also called as a quiver where is a finite set of vertices, is a set of arrows, and are two maps from to . A representation of a quiver is an assignment of a vector space to each vertex of and a linear mapping to each arrow. We denote by the direct sum of representasions and of a quiver . A representation is called indecomposable if is not ishomorphic to a direct sum of non-zero representations. This paper study about the properties of indecomposable representations. These properties will be used to investigate the necessary and sufficient condition of indecomposable representations.

Kac Polynomials for Canonical Algebras

We prove that the number of geometrically indecomposable representations of fixed dimension vector $\mathbf{d}$ of a canonical algebra $C$ defined over a finite field $\mathbb{F}_q$ is given by a polynomial in $q$ (depending on $C$ and $\mathbf{d}$). We prove a similar result for squid algebras. Finally, we express the volume of the moduli stacks of representations of these algebras of a fixed dimension vector in terms of the corresponding Kac polynomials.

Gröbner-Shirshov Basis of Twisted Generic Composition Algebras of Affine Type

In this paper, by using PBW bases for the twisted generic composition algebras of affine type, we prove that the set of the skew-commutator relations of the iso-classes of indecomposable representations forms a minimal Gröbner-Shirshov basis for the twisted generic composition algebras of affine type.