by Lakatos, P.

A new proof is given of Dlab’s theorem asserting that the left regular representation of an algebra is filtered by the standard modules if and only if the right regular representation of it is filtered by the proper standard modules.

DOI: 10.1023/A:1009922008930
Print publication date: 3/1/2000
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by Kirichenko, V. V.

The present paper is devoted to the study of semi-perfect semi-distributive rings (SPSD-rings). In particular, the concept of a prime quiver of a semi-perfect ring and a quiver of an SPSD-ring is widely used. The description of semi-hereditary SPSD-rings is reduced to the case of prime semi-hereditary serial rings and finite posets without rhombuses. For semi-hereditary, semi-perfect, semi-distributive rings we prove the existence of classical quotient rings.

DOI: 10.1023/A:1009969911772
Print publication date: 3/1/2000
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by Lin, Zongzhu; Nakano, Daniel K.

In this paper we prove that there are no self-extensions of simple modules over restricted Lie algebras of Cartan type. The proof given by Andersen for classical Lie algebras not only uses the representation theory of the Lie algebra, but also representations of the corresponding reductive algebraic group. The proof presented in the paper follows in the same spirit by using the construction of a infinite-dimensional Hopf algebra D(G) u(
$$\mathfrak{g}$$
) containing u(
$$\mathfrak{g}$$
) as a normal Hopf subalgebra, and the representation theory of this algebra developed in our previous work. Finite-dimensional hyperalgebra analogs D(G

_r
) u(
$$\mathfrak{g}$$
) have also been constructed, and the results are stated in this setting.

DOI: 10.1023/A:1009917927702
Print publication date: 3/1/2000
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by Caenepeel, S.; Ion, Bogdan; Militaru, G.; Zhu, Shenglin

We consider the factorization problem for bialgebras. Let L and H be algebras and coalgebras (but not necessarily bialgebras) and consider two maps R : H ⊗ L → L ⊗ H and W : L ⊗ H → H ⊗ L. We introduce a product K = L

_W
⋈
_R

H and we give necessary and sufficient conditions for K to be a bialgebra. Our construction generalizes products introduced by Majid and Radford. Also, some of the pointed Hopf algebras that were recently constructed by Beattie, Dăscălescu and Grünenfelder appear as special cases.

DOI: 10.1023/A:1009917210863
Print publication date: 3/1/2000
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by Cuadra, J.; García Rozas, J. R.; Torrecillas, B.

We introduce Shur and projective Schur subgroup of the Brauer group of a cocommutative coalgebra by means of twisted cogroup coalgebras and we study their properties. In particular we show that these subgroups are always torsion (in contrast with the whole Brauer group). Moreover, when C is coreflexive and irreducible both subgroups coincide with the coradical ones. We illustrate the theory with several examples.

DOI: 10.1023/A:1009913901047
Print publication date: 3/1/2000
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