by Bobiński, Grzegorz; Skowroński, Andrzej

Let A be a tame concealed or tubular algebra and the dimension-vector of a periodic module with respect to the action of the Auslander–Reiten translation. We prove that the affine variety mod
_A
() of all A-modules of dimension-vector is a normal complete intersection. Moreover, we show that a module M in mod
_A
() is nonsingular if and only if Ext
_A

²(M,M)=0.

DOI: 10.1023/A:1015606729502
Print publication date: 5/1/2002
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by Lenzing, Helmut; Meltzer, Hagen

We investigate complete exceptional sequences E=(E
₁,¨,E

_n
) in the derived category D
^b
Λ of finite-dimensional modules over a canonical algebra, equivalently in the derived category D
^b

X of coherent sheaves on a weighted projective line, and the associated Cartan matrices C(E)=(〈 [E

_i
],[E

_j
]〉). As a consequence of the transitivity of the braid group action on such sequences we show that a given Cartan matrix has at most finitely many realizations by an exceptional sequence E, up to an automorphism and a multi-translation (E
₁,¨,E

_n
)↦(E
₁[i
₁],¨,E

_n
[i

_n
]) of D
^b
Λ. Moreover, we determine a bound on the number of such realizations. Our results imply that a derived canonical algebra A is determined by its Cartan matrix up to isomorphism if and only if the Hochschild cohomology of A vanishes in nonzero degree, a condition satisfied if A is representation-finite.

DOI: 10.1023/A:1015646412663
Print publication date: 5/1/2002
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by Nishida, Kenji

We provide a criterion for a Λ-bimodule ω to be a dualizing module, where Λ is an order over a commutative Gorenstein complete local domain of dim R=1. Using this criterion, we give examples of dualizing modules which are neither isomorphic to Λ nor a dual of Λ. Thus we can also give such examples over an Artin algebra by modulo a nonzerodivisor.

DOI: 10.1023/A:1015642326776
Print publication date: 5/1/2002
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by Mazorchuk, Volodymyr; Turowska, Lyudmyla

We study bounded and unbounded *-representations of Twisted Generalized Weyl Algebras and algebras similar to them for different choices of involutions.

DOI: 10.1023/A:1015669525867
Print publication date: 5/1/2002
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by Buchweitz, Ragnar-Olaf; Liu, Shiping

We construct families of Artin algebras over fields of arbitrary characteristic that contain a loop in their ordinary quiver but admit no nontrivial outer derivation. This refuses the long-held belief that such algebras should not exist.

DOI: 10.1023/A:1015648905949
Print publication date: 5/1/2002
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by Billig, Y.; Pianzola, A.

We compute the Lie algebras of continuous right invariant derivations of free Kac–Moody groups.

DOI: 10.1023/A:1015691401406
Print publication date: 5/1/2002
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