Symmetric Hochschild extension algebras

Yosuke Ohnuki; Kaoru Takeda; Kunio Yamagata

Colloquium Mathematicae (1999)

  • Volume: 80, Issue: 2, page 155-174
  • ISSN: 0010-1354

Abstract

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By an extension algebra of a finite-dimensional K-algebra A we mean a Hochschild extension algebra of A by the dual A-bimodule H o m K ( A , K ) . We study the problem of when extension algebras of a K-algebra A are symmetric. (1) For an algebra A= KQ/I with an arbitrary finite quiver Q, we show a sufficient condition in terms of a 2-cocycle for an extension algebra to be symmetric. (2) Let L be a finite extension field of K. By using a given 2-cocycle of the K-algebra L, we construct a 2-cocycle of the K-algebra LQ for an arbitrary finite quiver Q without oriented cycles. Then we show a criterion on L for all those K-algebras LQ to have symmetric non-splittable extension algebras defined by the 2-cocycles.

How to cite

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Ohnuki, Yosuke, Takeda, Kaoru, and Yamagata, Kunio. "Symmetric Hochschild extension algebras." Colloquium Mathematicae 80.2 (1999): 155-174. <http://eudml.org/doc/210709>.

@article{Ohnuki1999,
abstract = {By an extension algebra of a finite-dimensional K-algebra A we mean a Hochschild extension algebra of A by the dual A-bimodule $Hom_K(A,K)$. We study the problem of when extension algebras of a K-algebra A are symmetric. (1) For an algebra A= KQ/I with an arbitrary finite quiver Q, we show a sufficient condition in terms of a 2-cocycle for an extension algebra to be symmetric. (2) Let L be a finite extension field of K. By using a given 2-cocycle of the K-algebra L, we construct a 2-cocycle of the K-algebra LQ for an arbitrary finite quiver Q without oriented cycles. Then we show a criterion on L for all those K-algebras LQ to have symmetric non-splittable extension algebras defined by the 2-cocycles.},
author = {Ohnuki, Yosuke, Takeda, Kaoru, Yamagata, Kunio},
journal = {Colloquium Mathematicae},
keywords = {Morita duality cocycles; symmetric algebras; finite-dimensional algebras; Hochschild extension algebras; cohomology groups; finite quivers; path algebras},
language = {eng},
number = {2},
pages = {155-174},
title = {Symmetric Hochschild extension algebras},
url = {http://eudml.org/doc/210709},
volume = {80},
year = {1999},
}

TY - JOUR
AU - Ohnuki, Yosuke
AU - Takeda, Kaoru
AU - Yamagata, Kunio
TI - Symmetric Hochschild extension algebras
JO - Colloquium Mathematicae
PY - 1999
VL - 80
IS - 2
SP - 155
EP - 174
AB - By an extension algebra of a finite-dimensional K-algebra A we mean a Hochschild extension algebra of A by the dual A-bimodule $Hom_K(A,K)$. We study the problem of when extension algebras of a K-algebra A are symmetric. (1) For an algebra A= KQ/I with an arbitrary finite quiver Q, we show a sufficient condition in terms of a 2-cocycle for an extension algebra to be symmetric. (2) Let L be a finite extension field of K. By using a given 2-cocycle of the K-algebra L, we construct a 2-cocycle of the K-algebra LQ for an arbitrary finite quiver Q without oriented cycles. Then we show a criterion on L for all those K-algebras LQ to have symmetric non-splittable extension algebras defined by the 2-cocycles.
LA - eng
KW - Morita duality cocycles; symmetric algebras; finite-dimensional algebras; Hochschild extension algebras; cohomology groups; finite quivers; path algebras
UR - http://eudml.org/doc/210709
ER -

References

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  9. [9] A. Skowroński and K. Yamagata, Socle deformations of self-injective algebras, Proc. London Math. Soc. 72 (1996), 545-566. Zbl0862.16001
  10. [10] A. Skowroński and K. Yamagata, Stable equivalence of selfinjective algebras of tilted type, Arch. Math. (Basel) 70 (1998), 341-350. Zbl0915.16005
  11. [11] K. Yamagata, Extensions over hereditary artinian rings with self-dualities, I, J. Algebra 73 (1981), 386-433. Zbl0471.16022
  12. [12] K. Yamagata, Representations of non-splittable extension algebras, J. Algebra 115 (1988), 32-45. Zbl0644.16020
  13. [13] K. Yamagata, Frobenius algebras, in: Handbook of Algebra, Vol. 1, Elsevier, 1996, 841-887. Zbl0879.16008

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