Abstract
It is well known that every Frobenius algebra is quasi-Frobenius, that is to say, if a finite dimensional algebra A over a field F has a nondegenerate bilinear form B such that B(xy, z) = B(x, yz) for all x, y, z ∊ A, then the maps L → Ann r (L) and R → Ann l (R) give inclusion preserving bijections between lattices of left and right ideals of A satisfying (a) Ann r (L 1 + L 2) = Ann r (L 1) ⋂ Ann r (L 2), Ann r (L 1 ⋂ L 2) = Ann r (L 1) + Ann r (L 2) (b) Ann l (R 1 + R 2) = Ann l (R 1) ⋂ Ann l (R 2), Ann l (R 1 ⋂ R 2) = Ann l (R 1) + Ann l (R 2) (c) Ann l(Ann r (L)) = LandAnn r (Ann l (R)) = R. (For example see [2].)
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References
I. Dibag, Duality for ideals in the Grassmann algebra, J. Algebra 183 (1996), 24–37.
G. Karpilovsky, Symmetric and G-algebras, Kluwer Academic Publishers, Dordrecht, Boston, London, 1990
S. Sternberg, Lectures on Differential Geometry, Prentice-Hall, New York, 1964
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Koc, C., Esin, S. (2002). Annihilators of Principal Ideals in the Grassmann Algebra. In: Dorst, L., Doran, C., Lasenby, J. (eds) Applications of Geometric Algebra in Computer Science and Engineering. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0089-5_18
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DOI: https://doi.org/10.1007/978-1-4612-0089-5_18
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