Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
Explained more generaly the construction of the symmetric algebra in a CMon-enriched symmetric monoidal category.
Added that permutations $\sigma:A^{\otimes n} \rightarrow A^{\otimes}$ are defined in the entry symmetric monoidal category
added pointer to
Should $\operatorname{S}^n V$ be defined as the cokernel of the symmetrisation map? That’s quite a strange definition which I don’t think it correct. Even in char $0$ and in the vector space case, when $n = 1$, the coequalizer is $0$ and when $n = 2$, the resulting elements are anti-symmetric. Am I missing something?
I think it’s correct to define it as the coequalizer of the $S_n$-action.
Thanks for the alert. The first “cokernel” here (which originates way back in revision 1) is probably a typo for “image”, meaning to refer to the standard construction laid out for instance here: pdf (p. 27).
I’ll make a quick fix to the entry now, but won’t be doing it justice (since I am still on family vacation and not allowed to be online :-).
changed “cokernel” to “image” and added pointer to:
Jean Gallier, Tensor Algebras, Symmetric Algebras and Exterior Algebras [pdf], section 22 in: Notes on Differential Geometry and Lie Groups (2011)
Jean Gallier, Jocelyn Quaintance, Tensor Algebras and Symmetric Algebras, Ch 2 in: Differential Geometry and Lie Groups – A second course, Geometry and Computing 13, Springer (2020) [doi:10.1007/978-3-030-46047-1, webpage]
added pointer to:
1 to 9 of 9