Vol. 71, No. 1, 1977

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Multiplication alteration and related rigidity properties of algebras

Dave Riffelmacher

Vol. 71 (1977), No. 1, 139–157

Given an algebra C over a commutative ring k and an element (called a C-two-cocycle) σ = iai bi ci in C kC kC satisfying certain relations, Sweedler defined a new multiplication on C by xy = iaixbiyci for all x, y in C and denoted C with this new multiplication by Cσ. This paper studies three rigidity properties which arise by asking whether:

  1. Cσ C as algebras;
  2. a certain functor from the category of C-bimodules to the category of Cσ-bimodules is an equivalence;
  3. a certain functor from the category of algebras over C to the category of algebras over Cσ is an equivalence.

For certain algebras over a field k (including finite dimensional algebras possessing a Wedderburn factor), these rigidity properties are shown to be equivalent to (respectively): (i) all k-separable subalgebras B of C are commutative and for a separability idempotent ixi yi of B,{c C| ixicyi = 0} is an ideal with square {0}; (ii) all k-separable subalgebras of C are central; (iii) k is the only k-separable subalgebra of C.

Mathematical Subject Classification 2000
Primary: 16A16, 16A16
Secondary: 12F15, 16A62
Received: 19 November 1976
Published: 1 July 1977
Dave Riffelmacher