## Tensor, symmetric and exterior algebras

Last update: 17 March 2012

## Definition of the algebras

Let $V$ be a vector space.

The tensor algebra $\left(T\left(V\right),\phi \right)$ is the pair with

1. $T\left(V\right)$ is an algebra and $\phi :V\to T\left(V\right)$ is a linear transformation, and
2. if $E$ is an algebra and $f:V\to E$ is a linear transformation then there exists a unique algebra homomorphism $g:T\left(V\right)\to E$ such that $f=g\circ \phi .$

The symmetric algebra $\left(S\left(V\right),\iota \right)$ is the pair with

1. $S\left(V\right)$ is an algebra and $\iota :V\to S\left(V\right)$ is a linear transformation such that
2. if $E$ is an algebra and $f:V\to E$ is a linear transformation such that then there exists a unique algebra homomorphism $g:S\left(V\right)\to E$ such that $f=g\circ \iota .$

The exterior algebra is the pair $\left(\Lambda \left(V\right),\iota \right)$ where

1. $\Lambda \left(V\right)$ is an algebra and $\iota :V\to \Lambda \left(V\right)$ is a linear transformation such that
2. if $E$ is an algebra and $f:V\to E$ is a linear transformation such that then there exists a unique homomorphism of algebras $g:\Lambda \left(V\right)\to E$ such that $f=g\circ \iota .$

## The functors

### The functor ${T}^{k}$

Define a functor by and if $u:V\to W$ is a linear transformation then $Tku: V⊗k→ W⊗k is given by (Tku) (bi1⊗⋯⊗bik) =u(bi1)⊗⋯⊗u(bik)$ with for ${c}_{1},{c}_{2}\in 𝔽$ and $l\in \left\{1,...,k\right\}.$

Then $T(V) = ⨁k∈ℤ≥0 V⊗k with V⊗0 = 𝔽,$ $(bi1⊗⋯⊗bik) (bj1⊗⋯⊗bjl) = bi1⊗⋯⊗bik⊗bj1⊗⋯⊗bjl$ and $ι: V → ⊕k∈ℤ≥0 V⊗k bi ↦ bi.$

### The functor ${S}^{k}$

Define a functor $Sk: { vector spaces } → { vector spaces } V ↦ Sk(V)$ by and if $u:V\to W$ is a linear transformation then $Sku: Sk(V) → Sk(W) is given by (Sku) (xi1⋯xik) = (uxi1) (uxi2) ⋯ (uxik)$ with for ${c}_{1},{c}_{2}\in 𝔽$ and $l\in \left\{1,...,k\right\}$ and

Then $S(V) = ⨁k∈ℤ≥0 Sk(V) with S0(V) = 𝔽,$ $(xi1⋯xik) (xj1⋯xjl) = xi1⋯xikxj1⋯xjl$ and $ι: V → S(V) xi ↦ xi.$

### The functor ${\Lambda }^{k}$

Define a functor $Λk: { vector spaces } → { vector spaces } V ↦ ΛkV$ by and if $u:V\to W$ is a linear transformation then $Λku: ΛkV→ ΛkW is given by (Λku) (ei1∧⋯∧eik) = (uei1) ∧⋯∧ (ueik)$ with for ${c}_{1},{c}_{2}\in 𝔽$ and $l\in \left\{1,...,k\right\}$ and

## Homework problems

HW: If $\mathrm{dim}\left(V\right)=n$ compute $dim(Tk(V)), dim(Sk(V)) and dim(Λk(V)).$

HW: Identifying a matrix $A\in {M}_{n×m}\left(𝔽\right)$ with a linear transformation $u:{𝔽}^{m}\to {𝔽}^{n}$ compute the matrix $TkA corresponding to Tku SkA corresponding to Sku ΛkA corresponding to Λku.$ First do this for $k=1$ and $2.$

HW: Let $\sigma \in {S}_{k}.$ Show that $det(σ) = (-1)l(σ) where l(σ) = Card(R(σ))$ and Interpret

HW: Let $A\in {M}_{n}\left(𝔽\right)$ be identified with a linear transformation $u:{𝔽}^{n}\to {𝔽}^{n}.$ Compute in terms of the matrix entries of $A.$

HW: Let $A\in {M}_{n}\left(𝔽\right)$ be identified with a linear transformation $u:{𝔽}^{n}\to {𝔽}^{n}.$ Compute in terms of the matrix entries of $A.$

HW: Show that if $\mathrm{dim}\left(V\right)=n$ then $\mathrm{dim}\left(T\left(V\right)\right)=\frac{1}{1-n}.$

## Notes and References

Where are these from?

References?