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This operation can also be extended to arbitrary families of homomorphisms and has functorial properties (see
 
This operation can also be extended to arbitrary families of homomorphisms and has functorial properties (see
 
[[Module|Module]]). It defines a homomorphism of $A$-modules
 
[[Module|Module]]). It defines a homomorphism of $A$-modules
 
+
$$\Hom_A(V_1, W_1) \tensor_A \Hom_A(V_2, W_2) \to \Hom_A(V_1 \tensor V_2, W_1 \tensor W_2),$$
$$\Hom_A(V_1, W_1) \tensor_A \Hom_A(V_2, W_2) \to$$
 
 
 
$$\to \Hom_A(V_1 \tensor V_2, W_1 \tensor W_2),$$
 
 
which is an isomorphism if all the $V_i$ and $W_i$ are free and finitely generated.
 
which is an isomorphism if all the $V_i$ and $W_i$ are free and finitely generated.
  
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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 $V$ over $\C$ of all complex linear combinations of a set of pure states $e_i$, $i \in I$. Let the pure states of a second similar object be $f_j$, $j \in J$, yielding a second vector space $W$. Then the pure states of the ordered pair of objects are all pairs $(e_i, f_j)$ and the space of states of this ordered pair is the tensor product $V\tensor_\C W$.
 
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 $V$ over $\C$ of all complex linear combinations of a set of pure states $e_i$, $i \in I$. Let the pure states of a second similar object be $f_j$, $j \in J$, yielding a second vector space $W$. Then the pure states of the ordered pair of objects are all pairs $(e_i, f_j)$ and the space of states of this ordered pair is the tensor product $V\tensor_\C W$.
 
 
  
 
====Tensor product of two algebras====
 
====Tensor product of two algebras====

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Tensor product of two unitary modules

The tensor product of two unitary modules $V_1$ and $V_2$ over an associative commutative ring $A$ with a unit is the $A$-module $V_1 \tensor_A V_2$ together with an $A$-bilinear mapping

$$(x_1, x_2) \mapsto x_1 \tensor x_2 \in V_1 \tensor_A V_2$$ which is universal in the following sense: For any $A$-bilinear mapping $\beta: V_1 \times V_2 \to W$, where $W$ is an arbitrary $A$-module, there is a unique $A$-linear mapping $b : V_1 \tensor_A V_2 \to W$ such that

$$\beta(x_1, x_2) = b(x_1 \tensor x_2), \qquad x_1 \in V_1, \qquad x_2 \in V_2.$$ 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 $A$-module $F$ generated by the set $V_1 \times V_2$ modulo the $A$-submodule $R$ generated by the elements of the form

$$(x_1 + y, x_2) - (x_1, x_2) - (y, x_2),$$

$$(x_1, x_2 + z) - (x_1, x_2) - (x_1, z),$$

$$(cx_1, x_2) - c(x_1, x_2),$$

$$(x_1, cx_2) - c(x_1, x_2),$$

$$x_1, y \in V_1, \qquad x_2, z \in V_2, \qquad c \in A;$$ then $x_1 \tensor x_2 = (x_1, x_2) + R$. If one gives up the requirement of commutativity of $A$, a construction close to the one described above allows one to form from a right $A$-module $V_1$ and a left $A$-module $V_2$ an Abelian group $V_1 \tensor_A V_2$, also called the tensor product of these modules [1]. In what follows $A$ will be assumed to be commutative.

The tensor product has the following properties:

$$A \tensor_A V \iso V,$$

$$V_1 \tensor_A V_2 \iso V_2 \tensor_A V_1,$$

$$(V_1 \tensor_A V_2) \tensor V_3 \iso V_1 \tensor_A (V_2 \tensor_A V_3),$$

$$\left( \bigoplus_{i \in I} V_i \right) \tensor_A W \iso \bigoplus_{i \in I} (V_i \tensor_A W)$$ for any $A$-modules $V$, $V_i$ and $W$.

If $(x_i)_{i \in I}$ and $(y_j)_{j \in J}$ are bases of the free $A$-modules $V_1$ and $V_2$, then $(x_i \tensor y_j)_{(i,j) \in I\times J}$ is a basis of the module $V_1 \tensor_A V_2$. In particular,

$$\dim(V_1 \tensor_A V_2) = \dim V_1 \cdot \dim V_2$$ if the $V_i$ are free finitely-generated modules (for instance, finite-dimensional vector spaces over a field $A$). The tensor product of cyclic $A$-modules is computed by the formula

$$(A/I) \tensor_A (A/J) \iso A/(I+J)$$ where $I$ and $J$ are ideals in $A$.

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

$$\bigotimes^p V = V \tensor_A \cdots \tensor_A V \qquad (p \text{ factors})$$ is called the $p$-th tensor power of the $A$-module $V$; its elements are the contravariant tensors (cf. Tensor on a vector space) of degree $p$ on $V$.

To any pair of homomorphisms of $A$-modules $\alpha_i : V_i \to W_i$, $i=1,2$, corresponds their tensor product $\alpha_1 \tensor \alpha_2$, which is a homomorphism of $A$-modules $V_1 \tensor_A V_2 \to W_1 \tensor_A W_2$ and is defined by the formula

$$(\alpha_1 \tensor \alpha_2) (x_1 \tensor x_2) = \alpha(x_1)\tensor \alpha_2(x_2), \qquad x_i \in V_i.$$ This operation can also be extended to arbitrary families of homomorphisms and has functorial properties (see Module). It defines a homomorphism of $A$-modules $$\Hom_A(V_1, W_1) \tensor_A \Hom_A(V_2, W_2) \to \Hom_A(V_1 \tensor V_2, W_1 \tensor W_2),$$ which is an isomorphism if all the $V_i$ and $W_i$ 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 $V$ over $\C$ of all complex linear combinations of a set of pure states $e_i$, $i \in I$. Let the pure states of a second similar object be $f_j$, $j \in J$, yielding a second vector space $W$. Then the pure states of the ordered pair of objects are all pairs $(e_i, f_j)$ and the space of states of this ordered pair is the tensor product $V\tensor_\C W$.

Tensor product of two algebras

The tensor product of two algebras $C_1$ and $C_2$ over an associative commutative ring $A$ with a unit is the algebra $C_1 \tensor_A C_2$ over $A$ which is obtained by introducing on the tensor product $C_1 \tensor_A C_2$ of $A$-modules a multiplication according to the formula

$$(x_1 \tensor x_2)(y_1 \tensor y_2) = (x_1 y_1) \tensor (x_2 y_2), \qquad x_i, y_i \in C_i.$$ This definition can be extended to the case of an arbitrary family of factors. The tensor product $C_1 \tensor_A C_2$ is associative and commutative and contains a unit if both algebras $C_i$ have a unit. If $C_1$ and $C_2$ are algebras with a unit over the field $A$, then $\widetilde C_1 = C_1 \tensor \mathbf{1}$ and $\widetilde C_2 = \mathbf{1} \tensor C_2$ are subalgebras of $C_1 \tensor_A C_2$ which are isomorphic to $C_1$ and $C_2$ and commute elementwise. Conversely, let $C$ be an algebra with a unit over the field $A$, and let $C_1$ and $C_2$ be subalgebras of it containing its unit and such that $x_1 x_2 = x_2 x_1$ for any $x_i \in C_i$. Then there is a homomorphism of $A$-algebras $\phi : C_1 \tensor_A C_2 \to C$ such that $\phi(x_1 \tensor x_2) = x_1 x_2$, $x_i \in C_i$. For $\phi$ to be an isomorphism it is necessary and sufficient that there is in $C_1$ a basis over $A$ which is also a basis of the right $C_2$-module $C$.

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

The tensor product, or Kronecker product (cf. Matrix multiplication), of two matrices $A = [ \alpha_{ij} ]$ and $B$ is the matrix

$$A \tensor B = \begin{bmatrix} \alpha_{11} B & \cdots & \alpha_{1n} B \\ \vdots & \ddots & \vdots \\ \alpha_{m1} B & \cdots & \alpha_{mn} B \end{bmatrix}.$$ Here, $A$ is an $(m\times n)$-matrix, $B$ is a $(p \times q)$-matrix and $A \tensor B$ is an $(mp \times nq)$-matrix over an associative commutative ring $k$ with a unit.

Properties of the tensor product of matrices are:

$$(A_1 + A_2) \tensor B = A_1 \tensor B + A_2 \tensor B,$$

$$A \tensor (B_1 + B_2) = A \tensor B_1 + A\tensor B_2,$$

$$\alpha(A \tensor B) = \alpha A \tensor B = A \tensor \alpha B,$$ where $\alpha \in k$,

$$(A \tensor B)(C \tensor D) = AC \tensor BD).$$ If $m=n$ and $p=q$, then

$$\det(A \tensor B) = (\det A)^p (\det B)^n.$$ Let $k$ be a field, $m=n$ and $p=q$. Then $A\tensor B$ is similar to $B \tensor A$, and $\det(A \tensor E_p - E_n \tensor B)$, where $E_s$ is the unit matrix, coincides with the resultant of the characteristic polynomials of $A$ and $B$.

If $\alpha : V \to V'$ and $\beta : W \to W'$ are homomorphisms of unitary free finitely-generated $k$-modules and $A$ and $B$ are their matrices in certain bases, then $A \tensor B$ is the matrix of the homomorphism $\alpha \tensor \beta : V \tensor W \to V' \tensor W'$ 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 $\pi_1$ and $\pi_2$ of a group $G$ in vector spaces $E_1$ and $E_2$, respectively, is the representation $\pi_1 \tensor \pi_2$ of $G$ in $E_1 \tensor E_2$ uniquely defined by the condition

$$(\pi_1 \tensor \pi_2) (g) (\xi_1 \tensor \xi_2) = \pi_1(g) \xi_1 \tensor \pi_2(g) \xi_2 \tag{*}$$ for all $\xi_1 \in E_1$, $\xi_2 \in E_2$ and $g \in G$. If $\pi_1$ and $\pi_2$ are continuous unitary representations of a topological group $G$ in Hilbert spaces $E_1$ and $E_2$, respectively, then the operators $(\pi_1 \tensor \pi_2)(g)$, $g \in G$, in the vector space $E_1 \tensor E_2$ admit a unique extension by continuity to continuous linear operators $(\pi_1 \tensor -\pi_2)g$, $g\in G$, in the Hilbert space $E_1 \tensor -E_2$ (being the completion of the space $E_1 \tensor E_2$ with respect to the scalar product defined by the formula $(\xi_1 \tensor \xi_2, \eta_1 \tensor \eta_2) = (\xi_1, \eta_1)(\xi_2, \eta_2)$) and the mapping $\pi_1 \tensor \pi_2 : g \to (\pi_1 \tensor -\pi_2)g$, $g \in G$, is a continuous unitary representation of the group $G$ in the Hilbert space $E_1 \tensor -E_2$, called the tensor product of the unitary representations $\pi_1$ and $\pi_2$. The representations $\pi_1 \tensor \pi_2$ and $\pi_2 \tensor \pi_1$ are equivalent (unitarily equivalent if $\pi_1$ and $\pi_2$ 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 $\pi_i$ is a representation of an algebra $A_i$ in a vector space $E_i$, $i=1,2$, one defines the tensor product $\pi_1 \tensor \pi_2$, which is a representation of $A_1\tensor A_2$ in $E_1\tensor E_2$, by

$$(\pi_1 \tensor \pi) (a_1 \tensor a_2) = \pi_1(a_1) \tensor \pi_2(a_2).$$ In case $A = A_1 = A_2$ is a bi-algebra (cf. Hopf algebra), composition of this representation with the comultiplication $A \to A \tensor A$ (which is an algebra homomorphism) yields a new representation of $A$, (also) called the tensor product.

In case $G$ is a group, a representation of $G$ is the same as a representation of the group algebra $k[G]$ of $G$, which is a bi-algebra, so that the previous construction applies, giving the same definition as (*) above. (The comultiplication on $k[G]$ is given by $g\mapsto g \tensor g$.)

In case $\lieg$ is a Lie algebra, a representation of $\lieg$ is the same as a representation of its universal enveloping algebra, $U_\lieg$, which is also a bi-algebra (with comultiplication defined by $x\mapsto 1 \tensor x + x \tensor 1$, $x \in \lieg$). This permits one to define the tensor product of two representations of a Lie algebra:

$$(\pi_1 \tensor \pi_2)(x) = 1 \tensor \pi_2(x) + \pi_1(x) \tensor 1.$$

Tensor product of two vector bundles

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


Comments

For a vector bundle $E$ over a space $X$ and a vector bundle $F$ over a space $Y$ one defines the vector bundle $E \times F$ over $X \times Y$ (sometimes written $E \tensor F$) as the vector bundle over $X \times Y$ with fibre $E_x \tensor F_y$ over $(x, y)$. Pulling back this bundle by the diagonal mapping $x \mapsto (x, x)$ 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:
Artemisfowl3rd/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Artemisfowl3rd/sandbox&oldid=43378