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Difference between revisions of "Tensor algebra"

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A part of [[Tensor calculus|tensor calculus]] in which algebraic operations on tensors (cf. [[Tensor on a vector space|Tensor on a vector space]]) are studied.
 
A part of [[Tensor calculus|tensor calculus]] in which algebraic operations on tensors (cf. [[Tensor on a vector space|Tensor on a vector space]]) are studied.
  
The tensor algebra of a unitary module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t0923501.png" /> over a commutative associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t0923502.png" /> with unit is the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t0923503.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t0923504.png" /> whose underlying module has the form
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The tensor algebra of a unitary module $V$ over a commutative associative ring $A$ with unit is the algebra $T(V)$ over $A$ whose underlying module has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t0923505.png" /></td> </tr></table>
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$$ T(V) = \bigoplus_{p=0}^\infty T^{p, 0}(V) = \bigoplus_{p=0}^\infty \bigotimes^p V $$
  
 
and in which multiplication is defined with the help of tensor multiplication (cf. [[Tensor on a vector space|Tensor on a vector space]]). Besides the contravariant tensor algebra, the covariant tensor algebra
 
and in which multiplication is defined with the help of tensor multiplication (cf. [[Tensor on a vector space|Tensor on a vector space]]). Besides the contravariant tensor algebra, the covariant tensor algebra
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t0923506.png" /></td> </tr></table>
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$$ T(V^*) = \bigoplus_{p=0}^\infty T^{0, p}(V) $$
  
 
is also considered, as well as the mixed tensor algebra
 
is also considered, as well as the mixed tensor algebra
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t0923507.png" /></td> </tr></table>
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$$ \widehat T(V) = \bigoplus_{p, q = 0}^\infty T^{p, q}(V) . $$
  
If the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t0923508.png" /> is free and finitely generated, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t0923509.png" /> is naturally isomorphic to the algebra of all multilinear forms (cf. [[Multilinear form|Multilinear form]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t09235010.png" />. Any homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t09235011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t09235012.png" />-modules naturally defines a tensor algebra homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t09235013.png" />.
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If the module $V$ is free and finitely generated, then $T(V^*)$ is naturally isomorphic to the algebra of all multilinear forms (cf. [[Multilinear form|Multilinear form]]) on $V$. Any homomorphism $V \to W$ of $A$-modules naturally defines a tensor algebra homomorphism $T(V) \to T(W)$.
  
The tensor algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t09235014.png" /> is associative, but in general not commutative. Its unit is the unit of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t09235015.png" />. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t09235016.png" />-linear mapping of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t09235017.png" /> into an associative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t09235018.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t09235019.png" /> with a unit can be naturally extended to a homomorphism of algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t09235020.png" /> mapping the unit to the unit. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t09235021.png" /> is a free module with basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t09235022.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t09235023.png" /> is the [[Free associative algebra|free associative algebra]] with system of generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092350/t09235024.png" />.
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The tensor algebra $T(V)$ is associative, but in general not commutative. Its unit is the unit of the ring $A = T^0(V)$. Any $A$-linear mapping of the module $V$ into an associative $A$-algebra $B$ with a unit can be naturally extended to a homomorphism of algebras $T(V) \to B$ mapping the unit to the unit. If $V$ is a free module with basis $(v_i)_{i \in I}$, then $T(V)$ is the
 +
[[Free associative algebra|free associative algebra]] with system of generators $(v_i)_{i \in I}$.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley  (1974)  pp. Chapt.1;2  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Kostrikin,  Yu.I. Manin,  "Linear algebra and geometry" , Gordon &amp; Breach  (1989)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley  (1974)  pp. Chapt.1;2  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Kostrikin,  Yu.I. Manin,  "Linear algebra and geometry" , Gordon &amp; Breach  (1989)  (Translated from Russian)</TD></TR></table>

Revision as of 04:48, 23 July 2018

A part of tensor calculus in which algebraic operations on tensors (cf. Tensor on a vector space) are studied.

The tensor algebra of a unitary module $V$ over a commutative associative ring $A$ with unit is the algebra $T(V)$ over $A$ whose underlying module has the form

$$ T(V) = \bigoplus_{p=0}^\infty T^{p, 0}(V) = \bigoplus_{p=0}^\infty \bigotimes^p V $$

and in which multiplication is defined with the help of tensor multiplication (cf. Tensor on a vector space). Besides the contravariant tensor algebra, the covariant tensor algebra

$$ T(V^*) = \bigoplus_{p=0}^\infty T^{0, p}(V) $$

is also considered, as well as the mixed tensor algebra

$$ \widehat T(V) = \bigoplus_{p, q = 0}^\infty T^{p, q}(V) . $$

If the module $V$ is free and finitely generated, then $T(V^*)$ is naturally isomorphic to the algebra of all multilinear forms (cf. Multilinear form) on $V$. Any homomorphism $V \to W$ of $A$-modules naturally defines a tensor algebra homomorphism $T(V) \to T(W)$.

The tensor algebra $T(V)$ is associative, but in general not commutative. Its unit is the unit of the ring $A = T^0(V)$. Any $A$-linear mapping of the module $V$ into an associative $A$-algebra $B$ with a unit can be naturally extended to a homomorphism of algebras $T(V) \to B$ mapping the unit to the unit. If $V$ is a free module with basis $(v_i)_{i \in I}$, then $T(V)$ is the free associative algebra with system of generators $(v_i)_{i \in I}$.

References

[1] N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French)
[2] A.I. Kostrikin, Yu.I. Manin, "Linear algebra and geometry" , Gordon & Breach (1989) (Translated from Russian)
How to Cite This Entry:
Tensor algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tensor_algebra&oldid=19203
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article