Terminology for (anti)symmetric tensors in characteristic 2

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Hurkyl
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Main Question or Discussion Point

When working over a field of characteristic not 2, or otherwise with modules over a ring where 2 is invertible, there is no ambiguity in what one means by symmetric or anti-symmetric rank 2 tensors. All of definitions of the anti-symmetric tensors
  • The module of anti-symmetric tensors is the quotient of [itex]M \otimes M[/itex] by the relations [itex]x \otimes y = -(y \otimes x)[/itex].
  • The module of anti-symmetric tensors is the quotient of [itex]M \otimes M[/itex] by the relations [itex]x \otimes x = 0[/itex].
  • The module of anti-symmetric tensors is the image of the anti-symmetrization operation [itex]x \otimes y \mapsto (1/2)(x \otimes y - y \otimes x)[/itex] on [itex]M \otimes M[/itex]
  • The module of anti-symmetric tensors is the kernel of the symmetrization operation [itex]x \otimes y \mapsto (1/2)(x \otimes y + y \otimes x)[/itex]
give the same module, and there is a standard notation for it: [itex]\wedge^2 M[/itex] or [itex]M \wedge M[/itex].

Similarly for symmetric tensors.



When 2 is not invertible, the different definitions can give different groups. Is there standard terminology and notation for the various possibilities? The only one I'm aware of is that the module [itex]\wedge^2 M[/itex] refers to the second in the list of definitions above. :frown:
 

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If the characteristic is ##2##, then you normally adjust the definition accordingly. E.g. instead of demanding anti-commutativity of a Lie algebra, ##[X,Y]=-[Y,X]## we demand ##[X,X]=0## instead. This is the more general case which covers all characteristics, since ##0=[X+Y,X+Y]=[X,Y]+[Y,X]## gets the usual definition from the general one.
 

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