When do the symmetric tensors of a commutative algebra form a Frobenius algebra?

Authors:
Annetta G. Aramova and Luchezar L. Avramov

Journal:
Proc. Amer. Math. Soc. **85** (1982), 299-304

MSC:
Primary 13E10

MathSciNet review:
656088

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Abstract: For a commutative -algebra , consider the subalgebra of the th tensor power of , formed by the tensors invariant under arbitrary permutations of the indices. Necessary and sufficient conditions are found for to be Frobenius. When , these say that is Frobenius and ! is invertible in , unless is separable. Some additional cases occur for two-dimensional algebras in positive characteristic, depending on the divisibility of .

**[1]**Gorô Azumaya,*Fixed subalgebra of a commutative Frobenius algebra*, Proc. Amer. Math. Soc.**81**(1981), no. 2, 213–216. MR**593459**, 10.1090/S0002-9939-1981-0593459-9**[2]**Keiichi Watanabe,*Certain invariant subrings are Gorenstein. I, II*, Osaka J. Math.**11**(1974), 1–8; ibid. 11 (1974), 379–388. MR**0354646****[3]**Kei-ichi Watanabe, Takeshi Ishikawa, Sadao Tachibana, and Kayo Otsuka,*On tensor products of Gorenstein rings*, J. Math. Kyoto Univ.**9**(1969), 413–423. MR**0257062**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1982-0656088-5

Keywords:
Frobenius algebra,
zero-dimensional Gorenstein ring,
symmetric tensors,
invariants of a finite group of automorphisms

Article copyright:
© Copyright 1982
American Mathematical Society