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Can an exterior algebra $$ k\langle x_{1},\dots,x_{n} \rangle/(x_{1}x_{2}-x_{2}x_{1},\dots,x_{1}^{2},\dots) $$ can be seen as a skew group algebra?

A skew group ring is defined for example in the introduction of this paper. I read this fact(?) somewhere but I cannot find a group action $G \rightarrow \mathrm{Aut}(k)$ that cooks up the exterior algebra.

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    Should it not be $x_ix_j+x_jx_i$?2012-08-10
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    I would be interested in knowing what $G$ you had in mind.2012-08-10
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    I'm guessing there are two usages of "skew" here. Probably the exterior algebra is "skew" in the sense that it's not quite commutative.2012-08-10
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    "skew group algebra" tends to mean "cross product of the group and a field" with a nontrivial action.2012-08-10
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    @JimConant The standard group ring construction causes coefficients to commute with the group basis elements. The paper referenced by the OP uses skew to mean "has a rule which modifies commuting coefficients with elements of $G$". This is conventional in noncommutative algebra.2012-08-13
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    @rschwieb: So "skew" in both cases basically means "noncommutative."2012-08-13
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    @JimConant I still don't know what second case you are thinking of, but yes, skew constructions are usually noncommutative.2012-08-13
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    @rschwieb: Case 1 is the skew group ring and case 2 is the exterior algebra.2012-08-14
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    @JimConant Ah! "Skew" as in skew matrices and skew-symmetry! Natural thing I should have picked up on :) Yeah those two are completely different things in my mind... and they appear to be unrelated terms as far as I can tell.2012-08-14

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