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Is there standard notation for

(1) exterior algebras

(2) free graded commutative algebras

(3) divided polynomial algebras ?

I've seen (and used) $\Lambda$, $\Gamma$, $\Delta$ etc. used for some or all of these things, and I have no idea if there is a consensus about which notation goes with which algebra.

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    $\begingroup$ The language of exterior algebra is old and often used, but the other algebras come up in more specialized contexts and are unlikely to have labels everyone recognizes. While LaTeX provides a handy symbol $\bigwedge$ (that is, \bigwedge) for exterior algebra as noted by Robin Chapman, it seems the use of $\Lambda$ (in written or verbal form) is just a corruption of the wedge symbol. There is also an older tradition of writing something like $E(V)$ for the exterior algebra of a vector space, but with the operation usually written as $v \wedge w$. The wedge has become standard $\endgroup$
    – Jim Humphreys
    Commented Jun 19, 2010 at 18:47

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It's pretty standard to use $\bigwedge(V)$ or $\Lambda(V)$ for the exterior algebra on a vector space $V$ and $\bigwedge^k(V)$ or $\Lambda^k(V)$ for the $k$-th graded part. For symmetric algebras $S(V)$ or $\mathrm{Sym}(V)$ etc. are frequent notations with again $S^k(V)$ or $\mathrm{Sym}^k(V)$ for the graded parts. I wouldn't say divided polynomial algebras come up often enough to have a standard notation; you could probably use the above notations for exterior or symmetric algebras without further comment and expect to be understood, but whatever notation you choose for divided polynomial algebras, you'd probably have to explain it.

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    $\begingroup$ In rational homotopy theory, $\Lambda(V)$ is universally used for free graded commutative algebras. $\endgroup$
    – Jeff Strom
    Commented Jun 19, 2010 at 17:53
  • $\begingroup$ But is "commutative" in RHT used for what other mathematicians would call anti-commutative or skew-commutative? $\endgroup$
    – Robin Chapman
    Commented Jun 19, 2010 at 18:04
  • $\begingroup$ I meant "graded-commutative" $\endgroup$
    – Jeff Strom
    Commented Jun 20, 2010 at 3:51
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    $\begingroup$ One has to keep in mind a very annoying an outdated (?) convention of denoting the symmetric algebra of a vector space $V$ (i.e. the quotient of the tensor algebra by commutativity relations) by $S(V^{*}).$ $\endgroup$
    – Victor Protsak
    Commented Jun 20, 2010 at 15:14
  • $\begingroup$ @VictorProtsak, I have never seen that notation! Do you just mean that people sometimes write $S(V^*)$ to mean the algebra of polynomial functions on $V$ (rather than the symmetric algebra of $V^*$), or do people really literally sometimes call $T(V)/(v \otimes w - w \otimes v)$ the symmetric algebra of $V^*$? $\endgroup$
    – LSpice
    Commented May 19, 2021 at 20:39

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