# Symmetric Hochschild extension algebras

Yosuke Ohnuki; Kaoru Takeda; Kunio Yamagata

Colloquium Mathematicae (1999)

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

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topOhnuki, 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 -

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