by Alvares, Edson R.; Coelho, Flávio U.

In this work we show that a directed translation quiver such that every path from a injective vertex to a projective vertex that has at most two hooks and in case two, they are consecutives, can be embedded in a quiver $\mathbb{Z}\Delta$. This generalizes a result by (Li, Comm. Algebra, (10),4635–4645, 2000).

DOI: 10.1007/s10468-006-9041-2
Online Date: 12/19/2006
Print publication date: 4/1/2007
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by Herzog, Ivo

Given an R-T-bimodule
_R

K

_T
and R-S-bimodule
_R

M

_S
, we study how properties of
_R

K

_T
affect the K-double dual M** = Hom
_T
[Hom
_R
(M, K), K] considered as a right S-module. If
_R

K is a cogenerator, then for every R-S-bimodule, the natural morphism Φ
_M
: M → M** is a pure-monomorphism of right S-modules. If
_R

K is the minimal (injective) cogenerator and K

_T
is quasi-injective, then M
^** is a pure-injective right S-module. If
_R

K is the minimal (injective) cogenerator, and T = End
_R
K it is shown that K

_T
is quasi-injective if and only if the K-topology on R is linearly compact. If the
_R

K-topology on R is of finite type, then the natural morphism Φ
_R
: R → R** is the pure-injective envelope of R

_R
as a right module over itself.

DOI: 10.1007/s10468-006-9039-9
Online Date: 12/19/2006
Print publication date: 4/1/2007
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by Zhang, Shouchuan; Zhang, Yao-Zhong

It is shown that an algebra Λ can be lifted with nilpotent Jacobson radical r = r(Λ) and has a generalized matrix unit {e

_ii
}
_I
with each ē

_ii
in the center of $\bar \Lambda = \Lambda/r$ if Λ is isomorphic to a generalized path algebra with weak relations. Representations of the generalized path algebras are given. As a corollary, Λ is a finite algebra with non-zero unity element over a perfect field k (e.g., a field with characteristic zero or a finite field) if Λ is isomorphic to a generalized path algebra k (D, Ω, ρ) of finite directed graph with weak relations and dim < ∞; Λ is a generalized elementary algebra which can be lifted with nilpotent Jacobson radical and has a complete set of pairwise orthogonal idempotents if Λ is isomorphic to a path algebra with relations.

DOI: 10.1007/s10468-006-9036-z
Online Date: 12/19/2006
Print publication date: 4/1/2007
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by Delvaux, Lydia; Van Daele, Alfons

Let G be any group and let K(G) denote the multiplier Hopf algebra of complex functions with finite support in G. The product in K(G) is pointwise. The comultiplication on K(G) is defined with values in the multiplier algebra M(K(G) ⊗K(G )) by the formula $(\Delta(f\,)) (p,q)\!\! =\!\! f(pq)$ for all $f\! \in\! K(G)$ and $p, q \!\in\! G$. In this paper we consider multiplier Hopf algebras B (over $\Bbb C$) such that there is an embedding I: K(G) →M(B). This embedding is a non-degenerate algebra homomorphism which respects the comultiplication and maps K(G) into the center of M(B). These multiplier Hopf algebras are called G-cograded multiplier Hopf algebras. They are a generalization of the Hopf group-coalgebras as studied by Turaev and Virelizier. In this paper, we also consider an admissible action π of the group G on a G-cograded multiplier Hopf algebra B. When B is paired with a multiplier Hopf algebra A, we construct the Drinfel’d double D

^π
where the coproduct and the product depend on the action π. We also treat the ^*-algebra case. If π is the trivial action, we recover the usual Drinfel’d double associated with the pair $\langle A, B \rangle$. On the other hand, also the Drinfel’d double, as constructed by Zunino for a finite-type Hopf group-coalgebra, is an example of the construction above. In this case, the action is non-trivial but related with the adjoint action of the group on itself. Now, the double is again a G-cograded multiplier Hopf algebra.

DOI: 10.1007/s10468-006-9042-1
Online Date: 12/13/2006
Print publication date: 6/1/2007
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by Heinschel, Nancy; Huisgen-Zimmermann, Birge

For any increasing function $f: \mathbb{N} \rightarrow \mathbb{N}_{\ge 2}$ which takes only finitely many distinct values, a connected finite dimensional algebra $\Lambda$ is constructed, with the property that ${\rm {fin\; dim}}_{n} \Lambda = f(n)$ for all $n$; here ${\rm {fin\; dim}}_{n} \Lambda$ is the $n$-generated finitistic dimension of $\Lambda$. The stacking technique developed for this construction of homological examples permits strong control over the higher syzygies of $\Lambda$-modules in terms of the algebras serving as layers.

DOI: 10.1007/s10468-006-9029-y
Online Date: 12/5/2006
Print publication date: 2/1/2007
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by Cimprič, Jakob; Kochetov, Mikhail; Marshall, Murray

Our aim is to construct new examples of totally ordered and ∗-ordered noncommutative integral domains. We will discuss the following classes of rings: enveloping algebras U(L), group rings $\Bbbk$
G and smash products U(L)$\# _{\varphi } \Bbbk$
G. All of them are examples of Hopf algebras. Characterizations of orderability for enveloping algebras and group rings and of ∗-orderability for enveloping algebras have been found before and will be recalled in the article. Our main results are: for $\Bbbk = \mathbb{R}$ and L finite–dimensional, we characterize the orderability of U(L)$\# _{\varphi } \Bbbk$
G; for $\Bbbk = \mathbb{C}$, we give a necessary and a sufficient condition for ∗-orderability of $\Bbbk$
G (G orderable, respectively, G residually ‘torsion-free nilpotent’). Moreover, for $\Bbbk = \mathbb{C}$ and L finite-dimensional, we reduce the problem of characterizing the ∗-orderability of U(L)$\# _{\varphi } \Bbbk$
G to the problem of characterizing the ∗-orderability of $\Bbbk$
G. The latter remains open.

DOI: 10.1007/s10468-006-9038-x
Online Date: 12/2/2006
Print publication date: 2/1/2007
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