Abstract
Self-dual algebras are ones with an A bimodule isomorphism A → A ∨op, where A ∨ = Hom k (A, k) and A ∨op is the same underlying k-module as A ∨ but with left and right operations by A interchanged. These are in particular quasi self-dual algebras, i.e., ones with an isomorphism H*(A,A) ≌ H*(A,A ∨ op). For all such algebras H*(A,A) is a contravariant functor of A. Finite-dimensional associative self-dual algebras over a field are identical with symmetric Frobenius algebras; an example of deformation of one is given. (The monoidal category of commutative Frobenius algebras is known to be equivalent to that of 1+1 dimensional topological quantum field theories.) All finite poset algebras are quasi self-dual.
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References
L. Abrams, Two-dimensional quantum field theories and Frobenius algebras, Journal of Knot Theory and its Ramifications 5 (1996), 569–587.
M. Auslander and O. Goldman, The Brauer group of a commutative ring, Journal of Algebra 4 (1960), 220–272.
G. Azumaya, On maximally central algebras, Nagoya Mathematical Journal 2 (1951), 119–150.
F. DeMeyer and E. Ingraham, Separable Algebras over Commutative Rings, Lecture Notes in Mathematics, Vol. 181, Springer-Verlag, Berlin, 1971.
M. Gerstenhaber, The cohomology structure of an associative ring, Annals of Mathematics 78 (1963), 267–288.
M. Gerstenhaber, On the deformation of rings and algebras, Annals of Mathematics 79 (1964), 59–103.
M. Gerstenhaber, A. Giaquinto and M. E. Shaps, The Donald-Flanigan problem for finite reflection groups, Letters in Mathematical Physics 56 (2001), 41–72.
M. Gerstenhaber and S. D. Schack, Triangular algebras, in Deformation Theory of Algebras and Structures and Applications (Il Ciocco, 1986), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, Vol. 247, Kluwer Academic Publishing, Dordrecht, 1988, pp. 447–498.
M. Gerstenhaber and S. D. Schack, Simplicial cohomology is Hochschild cohomology, Journal of Pure and Applied Algebra 30 (1983), 143–156.
M. Gerstenhaber and M. Schaps, Finite posets and their representation algebras, International Journal of Algebra and Computation 20 (2010), 27–38.
N. Jacobson, Lie Algebras, Interscirnce Tracts in Pure and Applied Mathematics, Vol. 10, Interscience Publishers, New York-London, 1962.
T. Nakayama, On Frobeniusean algebras I, Annals of Mathematics 40 (1939), 611–633.
R. W. Richardson, Jr., On the rigidity of semi-direct products of Lie algebras, Pacific Journal of Mathematics 22 (1967), 339–344.
S. Sawin, Direct sum decompositions and indecomposable TQFTs, Journal of Mathematical Physics 36 (1995), 6673–6680.
S. Sawin, Links, quantum groups and TQFTs, Bulletin of the American Mathematical Society 33 (1996), 413–445.
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Gerstenhaber, M. Self-dual and quasi self-dual algebras. Isr. J. Math. 200, 193–211 (2014). https://doi.org/10.1007/s11856-014-0013-7
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DOI: https://doi.org/10.1007/s11856-014-0013-7