Tensor, symmetric and exterior algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 17 March 2012

Definition of the algebras

Let $V$ be a vector space.

The tensor algebra $(T (V), φ)$ is the pair with

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

The symmetric algebra $(S (V), ι)$ is the pair with

$S (V)$ is an algebra and $ι : V \to S (V)$ is a linear transformation such that $if v_{1}, v_{2} \in V then ι (v_{1}) ι (v_{2}) = ι (v_{2}) ι (v_{1}), and$
if $E$ is an algebra and $f : V \to E$ is a linear transformation such that $if v_{1}, v_{2} \in V then f (v_{1}) f (v_{2}) = f (v_{2}) f (v_{1}),$ then there exists a unique algebra homomorphism $g : S (V) \to E$ such that $f = g \circ ι .$

The exterior algebra is the pair $(Λ (V), ι)$ where

$Λ (V)$ is an algebra and $ι : V \to Λ (V)$ is a linear transformation such that $if v \in V then ι {(v)}^{2} = 0, and$
if $E$ is an algebra and $f : V \to E$ is a linear transformation such that $if v \in V then f {(v)}^{2} = 0,$ then there exists a unique homomorphism of algebras $g : Λ (V) \to E$ such that $f = g \circ ι .$

The functors $T^{k}, S^{k}, and Λ^{k}$

The functor $T^{k}$

Define a functor $\begin{array}{rrcl} T^{k} : & {\binom{vector}{spaces}} & \to & {\binom{vector}{spaces}} \\ V & \mapsto & V^{\otimes k} = \underset{k factors}{\underset{⏟}{V \otimes V \otimes \dots \otimes V}} \end{array}$ by $\begin{aligned} if & b_{1} ... b_{n} is a basis of V \\ then & {b_{i_{1}} \otimes \dots \otimes b_{i_{k}} | 1 \leq i_{1}, ..., i_{k} \leq n} is a basis of V^{\otimes k} \end{aligned}$ and if $u : V \to W$ is a linear transformation then $\begin{aligned} T^{k} u : V^{\otimes k} \to W^{\otimes k} & is given by \\ (T^{k} u) (b_{i_{1}} \otimes \dots \otimes b_{i_{k}}) & = u (b_{i_{1}}) \otimes \dots \otimes u (b_{i_{k}}) \end{aligned}$ with $\begin{aligned} b_{i_{1}} \otimes \dots \otimes & \overset{l^{th} factor}{(c_{1} b_{i} + c_{2} b_{j})} \otimes \dots \otimes b_{i_{k}} \\ = & c_{1} b_{i_{1}} \otimes \dots \overset{l^{th} factor}{\otimes b_{i} \otimes} \dots \otimes b_{i_{k}} + c_{2} b_{i_{1}} \otimes \dots \overset{l^{th} factor}{\otimes b_{j} \otimes} \dots \otimes b_{i_{k}} \end{aligned}$ for $c_{1}, c_{2} \in 𝔽$ and $l \in {1, ..., k} .$

Then $T (V) = ⨁_{k \in ℤ_{\geq 0}} V^{\otimes k} with V^{\otimes 0} = 𝔽,$ $(b_{i_{1}} \otimes \dots \otimes b_{i_{k}}) (b_{j_{1}} \otimes \dots \otimes b_{j_{l}}) = b_{i_{1}} \otimes \dots \otimes b_{i_{k}} \otimes b_{j_{1}} \otimes \dots \otimes b_{j_{l}}$ and $\begin{array}{rrcl} ι : & V & \to & \underset{k \in ℤ_{\geq 0}}{\oplus} V^{\otimes k} \\ b_{i} & \mapsto & b_{i} . \end{array}$

The functor $S^{k}$

Define a functor $\begin{array}{rrcl} S^{k} : & {\binom{vector}{spaces}} & \to & {\binom{vector}{spaces}} \\ V & \mapsto & S^{k} (V) \end{array}$ by $\begin{aligned} if & x_{1} ... x_{n} is a basis of V \\ then & {x_{i_{1}} \dots x_{i_{k}} | 1 \leq i_{1} \leq i_{2} \leq \dots \leq i_{k} \leq n} is a basis of S^{k} (V) \end{aligned}$ and if $u : V \to W$ is a linear transformation then $\begin{aligned} S^{k} u : S^{k} (V) \to S^{k} (W) & is given by \\ (S^{k} u) (x_{i_{1}} \dots x_{i_{k}}) & = (u x_{i_{1}}) (u x_{i_{2}}) \dots (u x_{i_{k}}) \end{aligned}$ with $\begin{aligned} x_{i_{1}} \dots & \overset{l^{th} factor}{(c_{1} x_{i} + c_{2} x_{j})} \dots x_{i_{k}} \\ = & c_{1} x_{i_{1}} \overset{l^{th} factor}{\dots x_{i} \dots} x_{i_{k}} + c_{2} x_{i_{1}} \overset{l^{th} factor}{\dots x_{j} \dots} x_{i_{k}} \end{aligned}$ for $c_{1}, c_{2} \in 𝔽$ and $l \in {1, ..., k}$ and $x_{i_{1}} \dots x_{i_{k}} = x_{i_{σ (1)}} \dots x_{i_{σ (k)}} for σ \in S_{k} .$

Then $S (V) = ⨁_{k \in ℤ_{\geq 0}} S^{k} (V) with S^{0} (V) = 𝔽,$ $(x_{i_{1}} \dots x_{i_{k}}) (x_{j_{1}} \dots x_{j_{l}}) = x_{i_{1}} \dots x_{i_{k}} x_{j_{1}} \dots x_{j_{l}}$ and $\begin{array}{rrcl} ι : & V & \to & S (V) \\ x_{i} & \mapsto & x_{i} . \end{array}$

The functor $Λ^{k}$

Define a functor $\begin{array}{rrcl} Λ^{k} : & {\binom{vector}{spaces}} & \to & {\binom{vector}{spaces}} \\ V & \mapsto & Λ^{k} V \end{array}$ by $\begin{aligned} if & e_{1} ... e_{n} is a basis of V \\ then & {e_{i_{1}} \land \dots \land e_{i_{k}} | 1 \leq i_{1} < i_{2} < \dots < i_{k} \leq n} is a basis of Λ^{k} V \end{aligned}$ and if $u : V \to W$ is a linear transformation then $\begin{aligned} Λ^{k} u : Λ^{k} V \to Λ^{k} W & is given by \\ (Λ^{k} u) (e_{i_{1}} \land \dots \land e_{i_{k}}) & = (u e_{i_{1}}) \land \dots \land (u e_{i_{k}}) \end{aligned}$ with $\begin{aligned} e_{i_{1}} \land \dots \land & \overset{l^{th} factor}{(c_{1} e_{i} + c_{2} e_{j})} \land \dots \land e_{i_{k}} \\ = & c_{1} e_{i_{1}} \land \dots \overset{l^{th} factor}{\land e_{i} \land} \dots \land e_{i_{k}} + c_{2} e_{i_{1}} \land \dots \overset{l^{th} factor}{\land e_{j} \land} \dots \land e_{i_{k}} \end{aligned}$ for $c_{1}, c_{2} \in 𝔽$ and $l \in {1, ..., k}$ and $e_{i_{1}} \land \dots \land e_{i_{k}} = \det (σ) e_{i_{σ (1)}} \land \dots \land e_{i_{σ (k)}}, for σ \in S_{k} .$

Homework problems

HW: If $\dim (V) = n$ compute $\dim (T^{k} (V)), \dim (S^{k} (V)) and \dim (Λ^{k} (V)) .$

HW: Identifying a matrix $A \in M_{n \times m} (𝔽)$ with a linear transformation $u : 𝔽^{m} \to 𝔽^{n}$ compute the matrix $\begin{array}{ccl} T^{k} A & corresponding to & T^{k} u \\ S^{k} A & corresponding to & S^{k} u \\ Λ^{k} A & corresponding to & Λ^{k} u . \end{array}$ First do this for $k = 1$ and $2 .$

HW: Let $σ \in S_{k} .$ Show that $\det (σ) = {(- 1)}^{l (σ)} where l (σ) = Card (R (σ))$ and $R (σ) = {(i, j) | i, j \in {1, 2, ..., k}, i < j and σ (i) > σ (j)} .$ Interpret $l (σ) as the 'number of crossings in σ'.$

HW: Let $A \in M_{n} (𝔽)$ be identified with a linear transformation $u : 𝔽^{n} \to 𝔽^{n} .$ Compute $\begin{aligned} Λ^{n} A, & the matrix corresponding to the linear \\ transformation Λ^{n} u, \end{aligned}$ in terms of the matrix entries of $A .$

HW: Let $A \in M_{n} (𝔽)$ be identified with a linear transformation $u : 𝔽^{n} \to 𝔽^{n} .$ Compute $\begin{aligned} Λ^{k} A, & the matrix corresponding to the linear \\ transformation Λ^{k} u, \end{aligned}$ in terms of the matrix entries of $A .$

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

Notes and References

Where are these from?

References

References?

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