2025-11-04

This following is a very brief summary of what happened in class on 2025-11-04.

We began by briefly summarizing our three algebra constructions:

The tensor algebra construction, $T (M)$
The symmetric algebra construction, $S (M)$
The exterior algebra construction, $⋀ (M)$

We then noted that each of these constructions produced a "special" type of $R$ -algebra.

The tensor algebra $T (M)$ is a graded $R$ -algebra
The symmetric algebra $S (M)$ is a commutative graded $R$ -algebra
The exterior algebra $⋀ (M)$ is a somewhat exotic type of algebra, sometimes called a graded-commutative super algebra (or something along those lines)

Each of our constructions is actually a functor to the appropriate category of "special" algebras, and is left adjoint to the forgetful functor back to $R - Mod$ .

In the case $R = k$ is a field and $M = V$ is a $k$ -vector space of dimension $n$ , we described bases for:

The $k$ -vector space $T^{i} (V)$ of $i$ -tensors
The $k$ -vector space $S^{i} (V)$ of $i$ -symmetric products
The $k$ -vector space $\overset{i}{⋀} (V)$ of $i$ -wedge products

We noted that $\overset{n}{⋀} (V) ≃ k$ (with single basis vector given by $v_{1} \land v_{2} \land \dots \land v_{n}$ ). We also noted that, if $T : V \to V$ is a $k$ -linear transformation, then by functoriality we have an induced $k$ -linear transformation

\overset{n}{⋀} (T) : \overset{n}{⋀} (V) \to \overset{n}{⋀} (V) .

This transformation is entirely determined by the image of the single basis vector $v_{1} \land v_{2} \land \dots \land v_{n}$ , which must be sent to some $k$ -multiple of itself. We denoted that constant by $D (T)$ and noted that this function, $D : T \to k$ , satisfies the three properties that characterize the determinant function. Thus, $D (T) = det (T)$ .

We ran out of time to discuss symmetric and alternating tensors, but that's okay. Next class we'll move on to studying modules over a PID.

Concepts

References

Dummit & Foote, Abstract Algebra: Section 11.5