# Knot at Lunch on December 12 2007

## Invitation

Dear Knot at Lunch People,

We will have our next fall lunch on Wednesday December 12, at the usual place, Bahen 6180, at 12 noon.

As always, please bring brown-bag lunch and fresh ideas. This is break time, and I expect relatively low attendance. So whatever we will do, it will be light and easy.

As always, if you know anyone I should add to this mailing list or if you wish to be removed from this mailing list please let me know. To prevent junk accumulation in mailboxes, I will actively remove inactive people unless they request otherwise.

Best,

Dror.

## Bone Soup

Let ${\displaystyle n}$ be a natural number and let ${\displaystyle R={\mathbb {Z} }[x_{1},\ldots ,x_{n}]}$ be the ring of polynomials in ${\displaystyle n}$ variables ${\displaystyle x_{1},\ldots ,x_{n}}$. The ${\displaystyle n}$'th bone soup module ${\displaystyle B_{n}}$ is the ${\displaystyle R}$-module generated by symbols ${\displaystyle B_{ij;kl}}$ (the "bones") where ${\displaystyle 1\leq i,j,k,l\leq n}$, subject to the relations:

• ${\displaystyle B_{ij;kl}=B_{kl;ij}=-B_{ji;kl}=-B_{ij;lk}}$.
• ${\displaystyle x_{m}B_{ij;kl}-x_{j}B_{im;kl}+x_{i}B_{jm;kl}=0}$.

Question 1. Can you find a simple basis for ${\displaystyle B_{n}}$?

Question 2. Is ${\displaystyle B_{n}}$ related to curvature tensors and Bianchi identities?