Talk:Divided power structure

From Wikipedia, the free encyclopedia

WikiProject Mathematics
Mathematics Portal
This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating: Start Class Low Priority  Field: Algebra

[edit] Formatting

Somebody more experienced than I am please check the formatting for the references section. Do I need to put a link to the reference somewhere near the beginning?

Schepler 22:48, 18 November 2006 (UTC)

[edit] Be more specific on the dual-to-symmetric-algebra example?

I'm wondering whether it would be worth it to indicate exactly what the PD structure on (S^\cdot M) \check{~} is, or whether it is a bit too complex and would obscure things.

If I included it, it would go something like:

Addition is just the normal pointwise addition of functions. For multiplication, given \phi, \psi : S^\cdot M \to A, their product \phi \psi : S^\cdot M \to A is defined so that for x_1, \ldots, x_n \in M,

(\phi \psi)(x_1 \cdots x_n) = \sum_{S \subseteq \{ 1, 2, \ldots, n \}} \phi \left(\prod_{i \in S} x_i \right) \psi \left(\prod_{j \notin S} x_j \right).

The set I of functions φ such that φ(1) = 0 can easily be seen to be an ideal with respect to this ring structure. Then defining \gamma_m \phi : S^\cdot M \to A such that

(\gamma_m \phi)(x_1 \cdots x_n) = \sum_{\pi \in P_{n,m}} \prod_{S\in \pi} \phi \left(\prod_{i \in S} x_i \right)

gives a divided power structure on I. Here Pn,m denotes the set of (unordered) partitions of \{ 1, 2, \ldots, n \} into m parts.

(Note that by definition, (\phi^m)(x_1 \cdots x_n) is equal to the corresponding sum where π ranges over ordered partitions of \{ 1, 2, \ldots, n \} into m parts, thus making the above definition of the PD structure a natural one.)

Daniel Schepler 15:20, 21 November 2006 (UTC)