2024-11-08

This following is a very brief summary of what happened in class on 2024-11-08.

We recapped the constructions and key properties of the tensor algebra and symmetric algebra constructions. We then finished the construction of the exterior algebra and investigated some of its properties. We noted that in the exterior algebra $\bigwedge (M)$ we use the notation "wedge product" notation $z\wedge w$ for the product of two elements. The key feature of this new algebra is that $z\wedge z=0$ for every $z\in \bigwedge (M)$. We also saw that one consequence is that $m_1\wedge m_2=-(m_2\wedge m_1)$ for $m_1,m_2\in M$. (We also warned that we do not always have $z\wedge w=-(w\wedge z)$ for general elements $z,w\in \bigwedge (M)$.)

We ended by noting that there is a left action of the symmetric group $S_k$ on the $k^{\text{th}}$-tensor power ${\mathcal{T}}^k(M)$. This allowed us to define the notions of symmetric and alternating tensors, and the collections of such tensors form submodules of ${\mathcal{T}}^k(M)$. There are also symmetrization and skew-symmetrization morphisms that, at least when $k!$ is a unit in $R$, provide isomorphisms of those submodules with ${\mathcal{S}}^k(M)$ and $\stackrel{k}{\bigwedge }(M)$, respectively.

Next week: modules over a PID!

Concepts

• Tensor algebras
• Symmetric algebras
• Exterior algebras
• Symmetric and alternating tensors

References

• Dummit & Foote: Section 11.5