In this article we study the properties of preprojective algebras of representation finite species. To understand the structure of a preprojective algebra, one often studies its Nakayama automorphism. A complete description of the Nakayama automorphism is given by Brenner, Butler and King when the algebra is given by a path algebra. We partially generalize this result to the species case, i.e. we manage to describe the Nakayama automorphism up to an unknown constant. We show that the preprojective algebra of a representation finite species is an almost Koszul algebra. With this we know that almost Koszul complexes exist. It turns out that the almost Koszul complex for a representation finite species is given by a mapping cone of a certain chain map. We also study a higher dimensional analogue of representation finite hereditary algebras called d-representation finite algebras. One source of $d$-representation finite algebras comes from taking tensor products. By introducing a functor called the Segre product, we manage to give a complete description of the almost Koszul complex of the preprojective algebra of a tensor product of two species with relations with certain properties, in terms of the knowledge of the given species with relations. This allows us to compute the almost Koszul complex explicitly for certain species with relations more easily.

No description available.

No description available.

No description available.

No description available.

No description available.

No description available.

In this thesis we aim to study the Lefschetz properties ofmonomial algebras. First, we present the necessary concepts and resultsfrom commutative algebra, in particular we build up to the Hilbert-Serre theorem regarding the rationality of Hilbert series. We then reviewsome important results from the literature on the Lefschetz properties(whereof many provide drastic computational shortcuts under certainconditions) and provide some examples of these. The second half isdevoted to the study of Artinian quadratic monomial algebras of theform A(Δ) = K[x1, . . . , xn]/JΔ, where JΔ = (x21, . . . , x2n) + IΔ and IΔis the ideal obtained from some (abstract) simplicial complex Δ viathe Stanley-Reisner correspondence. In particular, we review a recentarticle [2] by H. Dao and R. Nair, provide examples and refine some ofthe formulations and results.

Let Q be a Dynkin quiver of type A and K be an algebraically closedfield. We start with the preprojective algebra Λ associated to Q andpresent a 2-cluster tilting module of Λ constructed by Geiss, Leclerc, andSchr¨oer.After that we move on to something more general. Let A be a drepresentationfinite algebra, its (d+1)-preprojective algebra has a (d+1)-cluster tilting module given by Iyama and Opperman, we shall comparethis with the case d = 1. Then we will compute an example where d = 2. In both cases we will investigate the endomorphism algebra of thecorresponding d-cluster tilting module.

Exceptional algebraic groups are divided into five types, namely G2, F4,E6, E7 and E8. In this thesis we discuss G2, F4 and E6. We discuss the exceptionalalgebraic groups via octonion algebras and Jordan algebras. We firstconsider the groups of type G2. Groups of type G2 are automorphism groupsof octonion algebras, a form of composition algebras. We take the algebra of Zorn vector matrices and find the possible values of automorphisms of thisalgebra with the help of U-operators. We also discuss the product of two andthree U-operators. Then we discuss Albert algebras, since groups of type E6and F4 are related to these algebras. The Albert algebras are a form of Jordanalgebras. We also study the U-operators in Albert algebras. In this thesis wework over algebraically closed fields of characteristic zero.