2025-11-03

This following is a very brief summary of what happened in class on 2025-11-03.

Inspired by the tensor algebra functor, we proceeded to ask for an analogous functor which produced commutative $R$ -algebras. This led us to the symmetric algebra functor. The construction was simple, namely quotienting the tensor algebra $T (M)$ by the (graded) ideal $C (M)$ generated by all tensors of the form $m \otimes m^{'} - m^{'} \otimes m$ . This created commutative graded $R$ -algebra $S (M)$ , called the symmetric algebra. We also introduced new notation, writing, for example, simply $m_{1} m_{2}$ for the coset $m_{1} \otimes m_{2} + C (M)$ .

We noted how to think about this construction in the vector space example.

We also recalled a bit about the classical determinant function and started the construction of our final algebra, namely the exterior algebra. We briefly outlined the construction of the exterior algebra and investigated some of its properties. We noted that in the exterior algebra $⋀ (M)$ we use the "wedge product" notation $z \land w$ for the product of two elements. The key feature of this new algebra is that $z \land z = 0$ for every $z \in ⋀ (M)$ . We also saw that one consequence is that $m_{1} \land m_{2} = - (m_{2} \land m_{1})$ for $m_{1}, m_{2} \in M$ .

Next time we'll recap the three constructions, as well as investigate the related notion of "symmetric" and "alternating" tensors.

Concepts

References

Dummit & Foote, Abstract Algebra: Section 11.5