Let $V$ be an $N$-dim vector space and denote its exterior algebra by $\Lambda^{\ast}(V)$. The algebra $\Lambda$ has an obvious Frobenius algebra structure $B(-,-)$ given by wedgeing and then projecting onto the one dim space of top forms $\Lambda^N(V)$. When $N$ is an odd number this Frobenius structure is symmetric, meaning that $(v,w) = (w,v), \forall v,w \in \Lambda^*(V)$. When $N$ is odd we get that $(v,w) = \pm (w,v)$, where the sign depends on the degree of $v$ or $w$. Does this phenomenon have a name? I would like to call such spaces graded anti-symmetric but perhaps this term does not exist.
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asked May 3, 2023 at 14:15
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1$\begingroup$ Surely in all cases the rule is just $(v,w)=(-1)^{|v||w|}(w,v)$. If $N$ is odd, $|v|$ and $|w|$ have opposite parity so the sign is $+1$ as you observe, while if $N$ is even, they have the same parity so you can use either in place of the product. But in any case, I'd just call this "graded symmetric" by analogy with "graded commutative"; or even just "symmetric" to avoid overloading the language. "Graded anti-symmetric" (or maybe "graded skew-symmetric") should mean $(v,w)=-(-1)^{|v||w|}(w,v)$. $\endgroup$– Dave BensonCommented May 3, 2023 at 17:18
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1$\begingroup$ @Dave: Thanks a lot for the suggestion! $\endgroup$– Jake WetlockCommented May 3, 2023 at 17:47
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