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====Tensor product of two unitary modules====
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The tensor product of two unitary modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t0924101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t0924102.png" /> over an associative commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t0924103.png" /> with a unit is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t0924104.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t0924105.png" /> together with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t0924106.png" />-bilinear mapping
 
The tensor product of two unitary modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t0924101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t0924102.png" /> over an associative commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t0924103.png" /> with a unit is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t0924104.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t0924105.png" /> together with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t0924106.png" />-bilinear mapping
  
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which is an isomorphism if all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241075.png" /> are free and finitely generated.
 
which is an isomorphism if all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241075.png" /> are free and finitely generated.
  
====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">  F. Kasch,  "Modules and rings" , Acad. Press  (1982)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Kostrikin,  Yu.I. Manin,  "Linear algebra and geometry" , Gordon &amp; Breach  (1989)  (Translated from Russian)</TD></TR></table>
 
  
  
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=====Comments=====
  
====Comments====
 
 
An important interpretation of the tensor product in (theoretical) physics is as follows. Often the states of an object, say, a particle, are defined as the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241076.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241077.png" /> of all complex linear combinations of a set of pure states <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241079.png" />. Let the pure states of a second similar object be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241081.png" />, yielding a second vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241082.png" />. Then the pure states of the ordered pair of objects are all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241083.png" /> and the space of states of this ordered pair is the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241084.png" />.
 
An important interpretation of the tensor product in (theoretical) physics is as follows. Often the states of an object, say, a particle, are defined as the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241076.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241077.png" /> of all complex linear combinations of a set of pure states <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241079.png" />. Let the pure states of a second similar object be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241081.png" />, yielding a second vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241082.png" />. Then the pure states of the ordered pair of objects are all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241083.png" /> and the space of states of this ordered pair is the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241084.png" />.
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====Tensor product of two algebras====
  
 
The tensor product of two algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241086.png" /> over an associative commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241087.png" /> with a unit is the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241088.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241089.png" /> which is obtained by introducing on the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241090.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241091.png" />-modules a multiplication according to the formula
 
The tensor product of two algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241086.png" /> over an associative commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241087.png" /> with a unit is the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241088.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241089.png" /> which is obtained by introducing on the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241090.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241091.png" />-modules a multiplication according to the formula
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This definition can be extended to the case of an arbitrary family of factors. The tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241093.png" /> is associative and commutative and contains a unit if both algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241094.png" /> have a unit. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241096.png" /> are algebras with a unit over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241097.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241099.png" /> are subalgebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410100.png" /> which are isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410102.png" /> and commute elementwise. Conversely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410103.png" /> be an algebra with a unit over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410104.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410106.png" /> be subalgebras of it containing its unit and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410107.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410108.png" />. Then there is a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410109.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410110.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410111.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410112.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410113.png" /> to be an isomorphism it is necessary and sufficient that there is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410114.png" /> a basis over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410115.png" /> which is also a basis of the right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410116.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410117.png" />.
 
This definition can be extended to the case of an arbitrary family of factors. The tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241093.png" /> is associative and commutative and contains a unit if both algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241094.png" /> have a unit. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241096.png" /> are algebras with a unit over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241097.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t09241099.png" /> are subalgebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410100.png" /> which are isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410102.png" /> and commute elementwise. Conversely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410103.png" /> be an algebra with a unit over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410104.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410106.png" /> be subalgebras of it containing its unit and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410107.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410108.png" />. Then there is a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410109.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410110.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410111.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410112.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410113.png" /> to be an isomorphism it is necessary and sufficient that there is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410114.png" /> a basis over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410115.png" /> which is also a basis of the right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410116.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410117.png" />.
  
====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></table>
 
  
  
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====Tensor product, or Kronecker product, of two matrices (by D.A. Suprunenko)====
  
 
The tensor product, or Kronecker product, of two matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410118.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410119.png" /> is the matrix
 
The tensor product, or Kronecker product, of two matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410118.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410119.png" /> is the matrix
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410145.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410146.png" /> are homomorphisms of unitary free finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410147.png" />-modules and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410148.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410149.png" /> are their matrices in certain bases, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410150.png" /> is the matrix of the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410151.png" /> in the basis consisting of the tensor products of the basis vectors.
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410145.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410146.png" /> are homomorphisms of unitary free finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410147.png" />-modules and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410148.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410149.png" /> are their matrices in certain bases, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410150.png" /> is the matrix of the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410151.png" /> in the basis consisting of the tensor products of the basis vectors.
  
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.R. Halmos,  "Finite-dimensional vector spaces" , v. Nostrand  (1958)</TD></TR><TR><TD valign="top">[2]</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></table>
 
  
''D.A. Suprunenko''
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====Tensor product of two representations (by A.I. Shtern)====
  
 
The tensor product of two representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410152.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410153.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410154.png" /> in vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410155.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410156.png" />, respectively, is the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410157.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410158.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410159.png" /> uniquely defined by the condition
 
The tensor product of two representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410152.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410153.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410154.png" /> in vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410155.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410156.png" />, respectively, is the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410157.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410158.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410159.png" /> uniquely defined by the condition
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for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410161.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410162.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410163.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410164.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410165.png" /> are continuous unitary representations of a topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410166.png" /> in Hilbert spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410167.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410168.png" />, respectively, then the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410169.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410170.png" />, in the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410171.png" /> admit a unique extension by continuity to continuous linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410172.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410173.png" />, in the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410174.png" /> (being the completion of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410175.png" /> with respect to the scalar product defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410176.png" />) and the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410177.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410178.png" />, is a continuous [[Unitary representation|unitary representation]] of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410179.png" /> in the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410180.png" />, called the tensor product of the unitary representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410181.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410182.png" />. The representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410183.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410184.png" /> are equivalent (unitarily equivalent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410185.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410186.png" /> are unitary). The operation of tensor multiplication can be defined also for continuous representations of a topological group in topological vector spaces of a general form.
 
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410161.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410162.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410163.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410164.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410165.png" /> are continuous unitary representations of a topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410166.png" /> in Hilbert spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410167.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410168.png" />, respectively, then the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410169.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410170.png" />, in the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410171.png" /> admit a unique extension by continuity to continuous linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410172.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410173.png" />, in the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410174.png" /> (being the completion of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410175.png" /> with respect to the scalar product defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410176.png" />) and the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410177.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410178.png" />, is a continuous [[Unitary representation|unitary representation]] of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410179.png" /> in the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410180.png" />, called the tensor product of the unitary representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410181.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410182.png" />. The representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410183.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410184.png" /> are equivalent (unitarily equivalent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410185.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410186.png" /> are unitary). The operation of tensor multiplication can be defined also for continuous representations of a topological group in topological vector spaces of a general form.
  
''A.I. Shtern''
 
  
====Comments====
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 +
=====Comments=====
 +
 
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410187.png" /> is a representation of an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410188.png" /> in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410189.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410190.png" />, one defines the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410191.png" />, which is a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410192.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410193.png" />, by
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410187.png" /> is a representation of an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410188.png" /> in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410189.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410190.png" />, one defines the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410191.png" />, which is a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410192.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410193.png" />, by
  
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<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/t092410/t092410209.png" /></td> </tr></table>
 
<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/t092410/t092410209.png" /></td> </tr></table>
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====Tensor product of two vector bundles====
  
 
The tensor product of two vector bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410210.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410211.png" /> over a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410212.png" /> is the vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410213.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410214.png" /> whose fibre at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410215.png" /> is the tensor product of the fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410216.png" />. The tensor product can be defined as the bundle whose transfer function is the tensor product of the transfer functions of the bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410217.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410218.png" /> in the same trivializing covering (see Tensor product of matrices, above).
 
The tensor product of two vector bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410210.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410211.png" /> over a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410212.png" /> is the vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410213.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410214.png" /> whose fibre at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410215.png" /> is the tensor product of the fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410216.png" />. The tensor product can be defined as the bundle whose transfer function is the tensor product of the transfer functions of the bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410217.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410218.png" /> in the same trivializing covering (see Tensor product of matrices, above).
  
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.F. Atiyah,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410219.png" />-theory: lectures" , Benjamin  (1967)</TD></TR></table>
 
  
  
 +
=====Comments=====
  
====Comments====
 
 
For a vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410220.png" /> over a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410221.png" /> and a vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410222.png" /> over a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410223.png" /> one defines the vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410224.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410225.png" /> (sometimes written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410226.png" />) as the vector bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410227.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410228.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410229.png" />. Pulling back this bundle by the diagonal mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410230.png" /> defines the tensor product defined above.
 
For a vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410220.png" /> over a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410221.png" /> and a vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410222.png" /> over a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410223.png" /> one defines the vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410224.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410225.png" /> (sometimes written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410226.png" />) as the vector bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410227.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410228.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410229.png" />. Pulling back this bundle by the diagonal mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092410/t092410230.png" /> defines the tensor product defined above.
 +
 +
 +
 +
====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">  F. Kasch,  "Modules and rings" , Acad. Press  (1982)  (Translated from German)</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Kostrikin,  Yu.I. Manin,  "Linear algebra and geometry" , Gordon &amp; Breach  (1989)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top">  P.R. Halmos,  "Finite-dimensional vector spaces" , v. Nostrand  (1958)</TD></TR>
 +
<TR><TD valign="top">[5]</TD> <TD valign="top">  M.F. Atiyah,  "$K$-theory: lectures" , Benjamin  (1967)</TD></TR>
 +
</table>

Revision as of 11:58, 21 June 2016


Tensor product of two unitary modules

The tensor product of two unitary modules and over an associative commutative ring with a unit is the -module together with an -bilinear mapping

which is universal in the following sense: For any -bilinear mapping , where is an arbitrary -module, there is a unique -linear mapping such that

The tensor product is uniquely defined up to a natural isomorphism. It always exists and can be constructed as the quotient module of the free -module generated by the set modulo the -submodule generated by the elements of the form

then . If one gives up the requirement of commutativity of , a construction close to the one described above allows one to form from a right -module and a left -module an Abelian group , also called the tensor product of these modules [1]. In what follows will be assumed to be commutative.

The tensor product has the following properties:

for any -modules , and .

If and are bases of the free -modules and , then is a basis of the module . In particular,

if the are free finitely-generated modules (for instance, finite-dimensional vector spaces over a field ). The tensor product of cyclic -modules is computed by the formula

where and are ideals in .

One also defines the tensor product of arbitrary (not necessarily finite) families of -modules. The tensor product

is called the -th tensor power of the -module ; its elements are the contravariant tensors (cf. Tensor on a vector space) of degree on .

To any pair of homomorphisms of -modules , , corresponds their tensor product , which is a homomorphism of -modules and is defined by the formula

This operation can also be extended to arbitrary families of homomorphisms and has functorial properties (see Module). It defines a homomorphism of -modules

which is an isomorphism if all the and are free and finitely generated.


Comments

An important interpretation of the tensor product in (theoretical) physics is as follows. Often the states of an object, say, a particle, are defined as the vector space over of all complex linear combinations of a set of pure states , . Let the pure states of a second similar object be , , yielding a second vector space . Then the pure states of the ordered pair of objects are all pairs and the space of states of this ordered pair is the tensor product .


Tensor product of two algebras

The tensor product of two algebras and over an associative commutative ring with a unit is the algebra over which is obtained by introducing on the tensor product of -modules a multiplication according to the formula

This definition can be extended to the case of an arbitrary family of factors. The tensor product is associative and commutative and contains a unit if both algebras have a unit. If and are algebras with a unit over the field , then and are subalgebras of which are isomorphic to and and commute elementwise. Conversely, let be an algebra with a unit over the field , and let and be subalgebras of it containing its unit and such that for any . Then there is a homomorphism of -algebras such that , . For to be an isomorphism it is necessary and sufficient that there is in a basis over which is also a basis of the right -module .


Tensor product, or Kronecker product, of two matrices (by D.A. Suprunenko)

The tensor product, or Kronecker product, of two matrices and is the matrix

Here, is an -matrix, is a -matrix and is an -matrix over an associative commutative ring with a unit.

Properties of the tensor product of matrices are:

where ,

If and , then

Let be a field, and . Then is similar to , and , where is the unit matrix, coincides with the resultant of the characteristic polynomials of and .

If and are homomorphisms of unitary free finitely-generated -modules and and are their matrices in certain bases, then is the matrix of the homomorphism in the basis consisting of the tensor products of the basis vectors.


Tensor product of two representations (by A.I. Shtern)

The tensor product of two representations and of a group in vector spaces and , respectively, is the representation of in uniquely defined by the condition

(*)

for all , and . If and are continuous unitary representations of a topological group in Hilbert spaces and , respectively, then the operators , , in the vector space admit a unique extension by continuity to continuous linear operators , , in the Hilbert space (being the completion of the space with respect to the scalar product defined by the formula ) and the mapping , , is a continuous unitary representation of the group in the Hilbert space , called the tensor product of the unitary representations and . The representations and are equivalent (unitarily equivalent if and are unitary). The operation of tensor multiplication can be defined also for continuous representations of a topological group in topological vector spaces of a general form.


Comments

If is a representation of an algebra in a vector space , , one defines the tensor product , which is a representation of in , by

In case is a bi-algebra (cf. Hopf algebra), composition of this representation with the comultiplication (which is an algebra homomorphism) yields a new representation of , (also) called the tensor product.

In case is a group, a representation of is the same as a representation of the group algebra of , which is a bi-algebra, so that the previous construction applies, giving the same definition as (*) above. (The comultiplication on is given by .)

In case is a Lie algebra, a representation of is the same as a representation of its universal enveloping algebra, , which is also a bi-algebra (with comultiplication defined by , ). This permits one to define the tensor product of two representations of a Lie algebra:


Tensor product of two vector bundles

The tensor product of two vector bundles and over a topological space is the vector bundle over whose fibre at a point is the tensor product of the fibres . The tensor product can be defined as the bundle whose transfer function is the tensor product of the transfer functions of the bundles and in the same trivializing covering (see Tensor product of matrices, above).


Comments

For a vector bundle over a space and a vector bundle over a space one defines the vector bundle over (sometimes written ) as the vector bundle over with fibre over . Pulling back this bundle by the diagonal mapping defines the tensor product defined above.


References

[1] N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French)
[2] F. Kasch, "Modules and rings" , Acad. Press (1982) (Translated from German)
[3] A.I. Kostrikin, Yu.I. Manin, "Linear algebra and geometry" , Gordon & Breach (1989) (Translated from Russian)
[4] P.R. Halmos, "Finite-dimensional vector spaces" , v. Nostrand (1958)
[5] M.F. Atiyah, "$K$-theory: lectures" , Benjamin (1967)
How to Cite This Entry:
Tensor product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tensor_product&oldid=12437
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article