by Proskurin, Daniil; Samoĩlenko, Yurii

The universal enveloping C
^*-algebra
_μ of twisted canonical commutation relations is considered. It is shown that, for any με(−1,1), the C
^*-algebra
_μ is isomorphic to the C
^*-algebra
₀ generated by partial isometries t

_i
,t

_i

^*,i=1,¨,d satisfying the relations
t

_i

^*
t

_j
=δ
_ij
(1−∑
_k<i

t

_k

t

_k

^*), t

_j

t

_i
=0, i≠j
and it is proved that the Fock representation of
_μ is faithful.

DOI: 10.1023/A:1020129429475
Print publication date: 10/1/2002
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by Kerner, Otto; Yamagata, Kunio

Let A be a locally finite Abelian R-category with Auslander–Reiten sequences and with Auslander–Reiten quiver Γ(A). We give a criterion for Auslander–Reiten components to contain a cone and apply this result to various categories.

DOI: 10.1023/A:1020143721750
Print publication date: 10/1/2002
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by Brzeziński, Tomasz

Given a ring A and an A-coring C, we study when the forgetful functor from the category of right C-comodules to the category of right A-modules and its right adjoint −⊗
_A

C are separable. We then proceed to study when the induction functor −⊗
_A

C is also the left adjoint of the forgetful functor. This question is closely related to the problem when A→
_A
Hom(C,A) is a Frobenius extension. We introduce the notion of a Galois coring and analyse when the tensor functor over the subring of A fixed under the coaction of C is an equivalence. We also comment on possible dualisation of the notion of a coring.

DOI: 10.1023/A:1020139620841
Print publication date: 10/1/2002
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by Smith, S. Paul; Zhang, James J.

Let R be a finitely generated commutative algebra over an algebraically closed field k and let A=R[t;σ,δ] be the Ore extension with respect to an automorphism σ and a σ-derivation δ. We view A as the coordinate ring of an affine noncommutative space X. The inclusion R→A gives an affine map ξ: X→SpecR, and X is a noncommutative analogue of A
¹×SpecR. We define the fiber X

_p
of ξ over a closed point pεSpecR as a certain full subcategory ModX

_p
of ModA. The category ModX

_p
has the following structure. If p has infinite σ-orbit, then ModX

_p
is equivalent to the category of graded modules over the polynomial ring k[x] with deg x=1. If p is not fixed by σ, but has finite σ-orbit, say of size n, then ModX

_p
is equivalent to the representations of the quiver Ã
_n−1 with the arrows all going in the same direction. If p is fixed by σ, then ModX

_p
is equivalent to either Modk or Modk[x]. It is also shown that X is the disjoint union of the fibers X

_p
in a certain sense.

DOI: 10.1023/A:1020183419932
Print publication date: 10/1/2002
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by Brenner, Sheila; Butler, Michael C. R.; King, Alastair D.

The preprojective algebra and the trivial extension algebra of a Dynkin quiver (in bipartite orientation) are very close to being a Koszul dual pair of algebras. In this case the usual duality theory may be adapted to show that each algebra has a periodic bimodule resolution built using the other algebra and some extra data: an algebra automorphism. A general theory of such ‘almost Koszul’ algebras is developed and other examples are given.

DOI: 10.1023/A:1020146502185
Print publication date: 10/1/2002
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