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{{TEX|part}}
 
{{TEX|part}}
  
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$\DeclareMathOperator{\PSL}{PSL}$
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====Tensor product of two unitary modules====
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A subgroup $\Gamma$ of a topological group $G$ (in particular, a subgroup of a Lie group) which is a discrete subset of the topological space $G$. In locally compact topological groups (in particular, in Lie groups) one distinguishes lattices — i.e. discrete subgroups for which the quotient space $\Gamma\setminus G$ has finite volume in the sense of the measure induced by the left-invariant
 
−
[[Haar measure|Haar measure]] on the group $G$. The concept of lattices includes that of uniform discrete subgroups, for which the quotient space $\Gamma\setminus G$ is compact.
 
  
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If $K$ is a compact subgroup of a locally compact topological group $G$, a subgroup $\Gamma \subset G$ is discrete if and only if it is a
+
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
−
[[Discrete group of transformations|discrete group of transformations]] of the space $X=G/K$ (in the sense of the action induced by the natural action of the group $G$ on $X$). Here, $\Gamma$ is a lattice (a uniform discrete subgroup) if and only if the quotient space $\Gamma \setminus X$ has finite volume (is compact) in the sense of the measure induced by the $G$-invariant measure on $X$. This makes it possible to utilize geometric methods when studying discrete subgroups of Lie groups.
 
  
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One of the principal problems in the theory of discrete subgroups of Lie groups is the classification of such subgroups up to commensurability. Two subgroups $\Gamma_1$ and $\Gamma_2$ are said to be commensurable if $\Gamma_1 \cap \Gamma_2$ has finite index both in $\Gamma_1$ and in $\Gamma_2$. If one of two commensurable subgroups of a locally compact topological group is a discrete subgroup (or a lattice, or a uniform discrete subgroup), so is the other.
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$$(x_1, x_2) \mapsto x_1 \tensor x_2 \in V_1 \tensor_A V_2$$
 +
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
  
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Up to the middle of the 20th century one basically studied individual classes of discrete subgroups of Lie groups occurring in arithmetic, function theory and physics. Historically, the first non-trivial discrete subgroup — the subgroup $\SL_2(\mathbf{Z})$ of the group $\SL_2(\mathbf{R})$, subsequently named the Kleinian
+
$$\beta(x_1, x_2) = b(x_1 \tensor x_2), \qquad x_1 \in V_1, \qquad x_2 \in V_2.$$
−
[[Modular group|modular group]] — was in fact studied by J.L. Lagrange and C.F. Gauss in the context of the arithmetic of quadratic forms in two variables. The subgroup $\SL_n(\mathbf{Z})$ of $\SL_n(\mathbf{R})$ is its natural generalization. The study of this group as a discrete group of transformations of the space of positive-definite quadratic forms in $n$ variables formed the subject of reduction theory, developed by A.N. Korkin, E.I. Zolotarev, Ch. Hermite, H. Minkowski, and others in the second half of the nineteenth and in the beginning of the 20th century. A series of arithmetically definable discrete subgroups of classical Lie groups — groups of units of quadratic forms with rational coefficients, groups of units of simple algebras over $\mathbf{Q}$, groups of integral symplectic matrices — were studied by C.L. Siegel in the 1940s. He proved, in particular, that all these groups are lattices in the respective Lie groups.
+
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
  
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In the theory of functions of a complex variable the integration of algebraic functions and, more generally, the solution of differential equations with algebraic coefficients, resulted in the study of certain special functions (subsequently named automorphic functions, cf.
+
$$(x_1 + y, x_2) - (x_1, x_2) - (y, x_2),$$
−
[[Automorphic function|Automorphic function]]) which are invariant with respect to various discrete groups consisting of transformations of the form
 
  
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$$z \mapsto \frac{az+b}{cz+d}, \qquad z \in \mathbf{C}, \qquad \begin{bmatrix} a & b \\ c & d \end{bmatrix} \in \SL_2(\mathbf{R}).$$
+
$$(x_1, x_2 + z) - (x_1, x_2) - (x_1, z),$$
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Certain discrete subgroups of $\SL_2(\mathbf{R})$ were studied in the mid-19th century by Hermite, R. Dedekind and I.L. Fuchs. They also included the group $\SL_2(\mathbf{Z})$ (though represented differently from the presentation used by Lagrange and Gauss). A wide class of such groups, including the group $\SL_2(\mathbf{Z})$ and certain subgroups of $\SL_2(\mathbf{R})$ commensurable with it, were studied by F. Klein. Almost simultaneously (1881–1882) H. Poincaré gave a geometric description of all discrete groups consisting of transformations of the form (1). He named these groups Fuchsian groups (cf.
 
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[[Fuchsian group|Fuchsian group]]).
 
  
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In the first half of the 20th century studies were made of individual classes of automorphic functions in several variables. These functions were connected with certain arithmetically definable discrete subgroups of the group $\left(\SL_2(\mathbf{R})\right)^k$ (Hilbert's modular functions), $\Sp_{2n}(\mathbf{R})$ (Siegel's modular functions) and other semi-simple Lie groups.
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$$(cx_1, x_2) - c(x_1, x_2),$$
  
−
Since the late 19th century, crystallographic studies have centred on the symmetry groups of crystallographic lattices, which are identical with uniform discrete subgroups of the group of motions of three-dimensional Euclidean space. These, together with the related groups of motions of $n$-dimensional Euclidean space (the so-called crystallographic groups, cf.
+
$$(x_1, cx_2) - c(x_1, x_2),$$
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[[Crystallographic group|Crystallographic group]]) were studied in 1911 by L. Bieberbach from the algebraic point of view. He demonstrated, in particular, the theorem according to which any crystallographic group contains a uniform discrete subgroup of parallel translations.
 
  
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All these studies provided the initial material for the general theory of discrete subgroups of Lie groups, the foundations of which were laid in the 1950s and 1960s.
+
$$x_1, y \in V_1, \qquad x_2, z \in V_2, \qquad c \in A;$$
 +
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
 +
[[#References|[1]]]. In what follows $A$ will be assumed to be commutative.
  
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An exhaustive theory of discrete subgroups of nilpotent Lie groups has been constructed
+
The tensor product has the following properties:
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[[#References|[9]]]. Its main statements are listed below:
 
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# If $H$ is a unipotent algebraic group defined over $\mathbf{Q}$, then the group $H_\mathbf{Z}$ of its integer points is a uniform discrete subgroup in the group $H_\mathbf{R}$ of its real points. (Here $H_\mathbf{R}$ is a simply-connected nilpotent Lie group.)
 
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# Any uniform discrete subgroup $\Gamma$ of a simply-connected nilpotent Lie group $G$ is arithmetic in the sense that there exist a unipotent algebraic group $H$ defined over $\mathbf{Q}$ and an isomorphism $\phi: H_\mathbf{R} \to G$ such that the subgroup $\Gamma$ is commensurable with $\phi(H_\mathbf{Z})$.
 
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# If $\Gamma_1$, $\Gamma_2$ are uniform discrete subgroups of simply-connected nilpotent Lie groups $G_1$ and $G_2$ respectively, then any isomorphism $\Gamma_1 \to \Gamma_2$ can be uniquely extended to an isomorphism $G_1 \to G_2$.
 
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# An abstract group $\Gamma$ is imbeddable as a uniform discrete subgroup in a simply-connected nilpotent Lie group if and only if $\Gamma$ is a finitely-generated torsion-free nilpotent group.
 
  
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Discrete subgroups of solvable Lie groups have been fairly thoroughly studied, but the results are less complete than those obtained for nilpotent groups. Any lattice in a solvable Lie group is a uniform discrete subgroup. If $\Gamma$ is a lattice in a simply-connected solvable Lie group $G$, then $G$ has a faithful matrix representation in which the elements of $\Gamma$ are represented by integer matrices
+
$$A \tensor_A V \iso V,$$
−
[[#References|[13]]]. This statement may be regarded as a generalization of Mal'tsev's theorem 2) above. The following theorem is the analogue of theorem 4). Any lattice in a simply-connected solvable Lie group is a strictly
 
−
[[Polycyclic group|polycyclic group]]; conversely, any strictly polycyclic group has a subgroup of finite index which is isomorphic to a lattice in a simply-connected solvable Lie group.
 
  
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The most precise results in the theory of discrete subgroups of Lie groups concern discrete subgroups of non-solvable and, in particular, semi-simple Lie groups. In
+
$$V_1 \tensor_A V_2 \iso V_2 \tensor_A V_1,$$
−
[[#References|[4]]] the following theorem was demonstrated, which includes, as special cases, Mal'tsev's theorem 1), the
 
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[[Dirichlet theorem|Dirichlet theorem]] on the units of an algebraic number field and Siegel's results (see above) on certain arithmetic discrete subgroups of semi-simple Lie groups. Let $H$ be a linear algebraic group defined over $\mathbf{Q}$. For the subgroup $H_\mathbf{Z}$ to be a lattice in $H_\mathbf{R}$ it is necessary and sufficient for $H$ not to permit rational homomorphisms into the group $\mathbf{C}^*$, defined over $\mathbf{Q}$ (this condition is satisfied, for example, if $H$ is semi-simple or unipotent). For the subgroup $H_\mathbf{Z}$ to be a uniform discrete subgroup in $H_\mathbf{R}$ it is necessary and sufficient, in addition, that all unipotent elements of the group $H_\mathbf{Q}$ lie in $U_\mathbf{Q}$, where $U$ is the unipotent radical of $H$.
 
  
−
The arithmeticity theorem
+
$$(V_1 \tensor_A V_2) \tensor V_3 \iso V_1 \tensor_A (V_2 \tensor_A V_3),$$
−
[[#References|[11]]] which follows is the analogue of theorem 2) for discrete subgroups of semi-simple Lie groups. Let $\Gamma$ be a lattice in a connected semi-simple Lie group $G$ without compact factors, and let (for the sake of convenience in formulation) the centre of $G$ be trivial. Moreover, let the lattice $\Gamma$ be irreducible in the sense that $G$ cannot be non-trivially decomposed into a direct product $G_1 \times G_2$ so that $\Gamma$ is commensurable with a subgroup of the form $\Gamma_1 \times \Gamma_2$ where $\Gamma_1 \subset G_1$ and $\Gamma_2 \subset G_2$. Then, if the real rank of $G$ exceeds one, the group $\Gamma$ is arithmetic in the sense that there exist a semi-simple algebraic group $H$, defined over $\mathbf{Q}$, and a homomorphism $\phi: H_\mathbf{R}^0 \to G$ (where $H_\mathbf{R}^0$ is the connected component of the unit of the group $H_\mathbf{R}$) such that the kernel of the homomorphism $\phi$ is compact and the subgroup $\Gamma$ is commensurable with $\phi(H_\mathbf{Z})$. The assumption that the real rank of $G$ exceeds one is essential. It is known that the theorem is invalid for the group $\PSL_2(\mathbf{R})$ (the group of motions of the Lobachevskii plane), which on the whole plays an important role in the theory of discrete subgroups of Lie groups, and also for the groups of motions of the three-, four- and five-dimensional Lobachevskii spaces
 
−
[[#References|[6]]],
 
−
[[#References|[8]]].
 
  
−
The strong rigidity theorem which follows is the analogue of theorem 3) for discrete subgroups of semi-simple Lie groups. Let $\Gamma_1$, $\Gamma_2$ be irreducible lattices in connected semi-simple Lie groups $G_1$, $G_2$ without compact factors, and let the centres of $G_1$, $G_2$ be trivial. Then, if $G_1$ and $G_2$ are not isomorphic to $\PSL_2(\mathbf{R})$, any isomorphism $\Gamma_1 \to \Gamma_2$ can be uniquely extended to an isomorphism $G_1 \to G_2$
+
$$\left( \bigoplus_{i \in I} V_i \right) \tensor_A W \iso \bigoplus_{i \in I} (V_i \tensor_A W)$$
−
[[#References|[10]]],
+
for any $A$-modules $V$, $V_i$ and $W$.
−
[[#References|[14]]]. Historically, the proof of this theorem was preceded by the proof of the weak rigidity theorem
 
  
−
on the extension of isomorphisms which are sufficiently close to the identity (if $G_1 = G_2$). One consequence of the weak rigidity theorem is the existence of a basis in which the elements of a discrete subgroup are written in the form of algebraic numbers. This fact played an important role in the development of the theory of discrete subgroups of semi-simple Lie groups.
+
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,
  
−
Regarding discrete subgroups of the group $\PSL_2(\mathbf{R})$ see
+
$$\dim(V_1 \tensor_A V_2) = \dim V_1 \cdot \dim V_2$$
−
[[Fuchsian group|Fuchsian group]].
+
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
  
−
Of the other general theorems about discrete subgroups of semi-simple Lie groups one may mention Borel's density theorem and Wang's maximality theorem. Let $\Gamma$ be a lattice in a connected semi-simple Lie group $G$ which has no compact factors. Then $\Gamma$ is dense in $G$ in the Zariski topology
+
$$(A/I) \tensor_A (A/J) \iso A/(I+J)$$
−
[[#References|[3]]], and is contained in only a finite number of lattices in $G$
+
where $I$ and $J$ are ideals in $A$.
−
[[#References|[17]]].
 
  
−
The description of lattices in arbitrary Lie groups can be reduced, to some extent, to the description of lattices in semi-simple Lie groups, in view of theorems analogous to the Bieberbach theorem on crystallographic groups mentioned above. One says that a normal subgroup $N$ of a Lie group $G$ has the Bieberbach property if for any lattice $\Gamma$ in $G$ the subgroup $N\Gamma$ is closed (and, in such a case, $N \cap \Gamma$ is automatically a lattice in $N$, while $\Gamma / N\cap \Gamma$ is a lattice in $G/N$). Bieberbach's theorem says that, in the group of motions of Euclidean space, the subgroup of parallel translations has the Bieberbach property. There exists a generalization of this theorem to Lie groups which are extensions of a simply-connected nilpotent Lie group by a compact group
+
One also defines the tensor product of arbitrary (not necessarily finite) families of $A$-modules. The tensor product
−
[[#References|[1]]]. Another theorem of such a type is the following. Let $G$ be a connected Lie group, let $R$ be its radical, let $S$ be a maximal connected semi-simple subgroup, and let $C$ be a maximal connected compact normal subgroup of $S$. Then the subgroup $RC$ has the Bieberbach property in $G$
 
−
[[#References|[2]]]. It is also known that the Bieberbach property is displayed by the nilpotent radical of a connected solvable Lie group
 
−
[[#References|[12]]] and by the commutator subgroup of a simply-connected nilpotent Lie group
 
−
[[#References|[9]]].
 
  
−
Topological methods (cf.
+
$$\bigotimes^p V = V \tensor_A \cdots \tensor_A V \qquad (p \text{ factors})$$
−
[[Discrete group of transformations|Discrete group of transformations]]) can be used to prove that any uniform discrete subgroup of a connected Lie group is a finitely-presentable group . In fact, any lattice in a connected Lie group is finitely presentable
+
is called the $p$-th tensor power of the $A$-module $V$; its elements are the contravariant tensors (cf.
−
[[#References|[17]]],
+
[[Tensor on a vector space|Tensor on a vector space]]) of degree $p$ on $V$.
−
[[#References|[18]]].
 
  
−
====References====
+
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
−
<table><TR><TD valign="top">[1]</TD>
+
 
−
<TD valign="top"> L. Auslander, "Bieberbach's theorem on space groups and discrete uniform subgroups of Lie groups" ''Amer. J. Math.'' , '''83''' (1961) pp. 276–280 {{MR|123637}} {{ZBL|}} </TD>
+
$$(\alpha_1 \tensor \alpha_2) (x_1 \tensor x_2) = \alpha(x_1)\tensor \alpha_2(x_2), \qquad x_i \in V_i.$$
−
</TR><TR><TD valign="top">[2]</TD>
+
This operation can also be extended to arbitrary families of homomorphisms and has functorial properties (see
−
<TD valign="top"> L. Auslander, "On radicals of discrete subgroups of Lie groups" ''Amer. J. Math.'' , '''85''' (1963) pp. 145–150 {{MR|0152607}} {{ZBL|0217.37002}} </TD>
+
[[Module|Module]]). It defines a homomorphism of $A$-modules
−
</TR><TR><TD valign="top">[3]</TD>
+
 
−
<TD valign="top"> A. Borel, "Density properties for certain subgroups of semi-simple groups without compact components" ''Ann. of Math.'' , '''72''' (1960) pp. 179–188 {{MR|0123639}} {{ZBL|0094.24901}} </TD>
+
$$\Hom_A(V_1, W_1) \tensor_A \Hom_A(V_2, W_2) \to$$
−
</TR><TR><TD valign="top">[4]</TD>
+
 
−
<TD valign="top"> A. Borel, Harish-Chandra, "Arithmetic subgroups of algebraic groups" ''Ann. of Math.'' , '''75''' (1962) pp. 485–535 {{MR|0147566}} {{ZBL|0107.14804}} </TD>
+
$$\to \Hom_A(V_1 \tensor V_2, W_1 \tensor W_2),$$
−
</TR><TR><TD valign="top">[5a]</TD>
+
which is an isomorphism if all the $V_i$ and $W_i$ are free and finitely generated.
−
<TD valign="top"> A. Weil, "Discrete subgroups of Lie groups I" ''Ann. Math.'' , '''72''' (1960) pp. 369–384 {{MR|137792}} {{ZBL|0131.26602}} </TD>
+
 
−
</TR><TR><TD valign="top">[5b]</TD>
+
 
−
<TD valign="top"> A. Weil, "Discrete subgroups of Lie groups II" ''Ann. Math.'' , '''75''' (1962) pp. 578–602 {{MR|0137793}} {{ZBL|0131.26602}} </TD>
+
 
−
</TR><TR><TD valign="top">[6]</TD>
+
=====Comments=====
−
<TD valign="top"> E.B. Vinberg, "Discrete groups generated by reflections in Lobachevskii spaces" ''Math. USSR-Sb.'' , '''1''' : 3 (1967) pp. 429–444 ''Mat. Sb.'' , '''72''' : 3 (1967) pp. 471–488 {{MR|}} {{ZBL|}} </TD>
+
 
−
</TR><TR><TD valign="top">[7]</TD>
+
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 $V\tensor_\C W$.
−
<TD valign="top"> H. Garland, M.S. Raghunathan, "Fundamental domains for lattices in ($\mathbf{R}$-) rank 1 semisimple Lie groups" ''Ann. of Math.'' , '''92''' (1970) pp. 279–326 {{MR|267041}} {{ZBL|}} </TD>
+
 
−
</TR><TR><TD valign="top">[8]</TD>
+
 
−
<TD valign="top"> V.S. Makarov, "A certain class of discrete Lobachevskii space groups with an infinite fundamental region of finite measure" ''Soviet Math.-Dokl.'' , '''7''' (1966) pp. 328–331 ''Dokl. Akad. Nauk. SSSR'' , '''167''' : 1 (1966) pp. 30–33 {{MR|}} {{ZBL|}} </TD>
+
 
−
</TR><TR><TD valign="top">[9]</TD>
+
====Tensor product of two algebras====
−
<TD valign="top"> A.I. Mal'tsev, "On a class of homogeneous spaces" ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''13''' : 1 (1949) pp. 9–32 (In Russian) {{MR|}} {{ZBL|0034.01701}} </TD>
+
 
−
</TR><TR><TD valign="top">[10]</TD>
+
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
−
<TD valign="top"> G.A. Margulis, "Arithmetic properties of discrete subgroups" ''Russian Math. Surveys'' , '''29''' : 1 (1974) pp. 107–156 ''Uspekhi Mat. Nauk'' , '''29''' : 1 (1974) pp. 49–98 {{MR|0463353}} {{MR|0463354}} {{ZBL|0299.22010}} </TD>
+
 
−
</TR><TR><TD valign="top">[11]</TD>
+
$$(x_1 \tensor x_2)(y_1 \tensor y_2) = (x_1 y_1) \tensor (x_2 y_2), \qquad x_i, y_i \in C_i.$$
−
<TD valign="top"> G.A. Margulis, "Discrete groups of motions of manifolds of non-positive curvature" R. James (ed.) , ''Proc. Internat. Congress Mathematicians (Vancouver, 1974)'' , '''2''' , Canad. Math. Congress (1975) pp. 21–34 (In Russian) {{MR|492072}} {{ZBL|0336.57037}} </TD>
+
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 $\tilde C_1 = C_1 \tensor \mathbf{1}$ and $\tilde C_2 = \mathbf{1} \tensor C_2$ 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 $C$.
−
</TR><TR><TD valign="top">[12]</TD>
+
 
−
<TD valign="top"> G.D. Mostow, "Factor spaces of solvable groups" ''Ann. of Math.'' , '''60''' (1954) pp. 1–27 {{MR|0061611}} {{ZBL|0057.26103}} </TD>
+
 
−
</TR><TR><TD valign="top">[13]</TD>
+
 
−
<TD valign="top"> G.D. Mostov, "Representative functions on discrete groups and solvable arithmetic subgroups" ''Amer. J. Math.'' , '''92''' (1970) pp. 1–32 {{MR|}} {{ZBL|}} </TD>
+
====Tensor product of two matrices (by D.A. Suprunenko)====
−
</TR><TR><TD valign="top">[14]</TD>
+
 
−
<TD valign="top"> G.D. Mostow, "Strong rigidity of locally symmetric spaces" , Princeton Univ. Press (1973) {{MR|0385004}} {{ZBL|0265.53039}} </TD>
+
The tensor product, or
−
</TR><TR><TD valign="top">[15]</TD>
+
[[Kronecker product]] (cf.
−
<TD valign="top"> M.S. Raghunathan, "Discrete subgroups of Lie groups" , Springer (1972) {{MR|0507234}} {{MR|0507236}} {{ZBL|0254.22005}} </TD>
+
[[Matrix multiplication]]), of two matrices $A = \| \alpha_{ij} \|$ and $B$ is the matrix
−
</TR><TR><TD valign="top">[16]</TD>
+
 
−
<TD valign="top"> A. Selberg, "On discontinuous groups in higher-dimensional symmetric spaces" , ''Internat. Coll. function theory'' , Tata Inst. (1960) pp. 147–164 {{MR|0130324}} {{ZBL|0201.36603}} </TD>
+
$$A \tensor B = \begin{Vmatrix} \alpha_{11} B & \cdots & \alpha_{1n} B \\ \vdots & \ddots & \vdots \\ \alpha_{m1} B & \cdots & \alpha_{mn} B \end{Vmatrix}.$$
−
</TR><TR><TD valign="top">[17]</TD>
+
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.
−
<TD valign="top"> H.-C. Wang, "On a maximality property of subgroups with fundamental domain of finite measure" ''Amer. J. Math.'' , '''89''' (1967) pp. 124–132 {{MR|}} {{ZBL|0152.01002}} </TD>
+
 
−
</TR><TR><TD valign="top">[18]</TD>
+
Properties of the tensor product of matrices are:
−
<TD valign="top"> H.-C. Wang, "Topics on totally discontinuous groups" , ''Symmetric spaces'' , M. Dekker (1972) pp. 459–487 {{MR|0414787}} {{ZBL|0232.22018}} </TD>
+
 
−
</TR></table>
+
$$(A_1 + A_2) \tensor B = A_1 \tensor B + A_2 \tensor B,$$
 +
 
 +
$$A \tensor (B_1 + B_2) = A \tensor B_1 + A\tensor B_2,$$
 +
 
 +
$$\alpha(A \tensor B) = \alpha A \tensor B = A \tensor \alpha B,$$
 +
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|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|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|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|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.
  
  
−
====Comments====
 
−
The arithmeticity theorem, mentioned in the main article and saying that an irreducible lattice $\Gamma$ in a connected semi-simple Lie group $G$ without compact factors (and with trivial centre) is an
 
−
[[Arithmetic group|arithmetic group]] if the real rank of $G$ exceeds one, was conjectured by A. Selberg (for uniform discrete subgroups) and by I.I. Pyatetskii-Shapiro (general case), see also
 
−
[[#References|[a1]]]. A first important step to the understanding of non-compact subgroups $\Gamma$ of finite co-volume, i.e. $G/\Gamma$ of finite volume, was the proof by D.A. Kazhdan and G.A. Margulis of the existence of non-trivial unipotent elements in $\Gamma$; this is a related, more special, conjecture of Selberg, cf.
 
−
[[#References|[a5]]]. In
 
−
[[#References|[a2]]] it is proved that this theorem does not hold for the groups $\SU(n, 1)$, $n \le 3$.
 
−
[[Ergodic theory|Ergodic theory]] plays an important role in proving some of the arithmeticity results mentioned in the main article, cf. also
 
−
[[#References|[a3]]]. One result in the proof of which ergodic arguments play an important role (the multiplicative ergodic theorem) is Margulis' superrigidity theorem, which for groups of real rank $\ge 2$ generalizes the A. Weil and G.D. Mostow rigidity theorems. It states the following. Let $G$ be a simply-connected Lie group of real points of a real simply-connected algebraic group $\mathcal{G} \subset \GL_n(\mathbf{R})$ and let $G$ have no compact factors. Assume that the real rank of $G$ is $\ge 2$. Let $F$ be a locally compact non-discrete field and $\rho : \Gamma \to \GL_n(F)$ a linear representation such that $\rho(\Gamma)$ is not relatively compact and such that its Zariski closure is connected. Then $F = \mathbf{R}$ or $\mathbf{C}$ and $\rho$ extends to a rational representation of $\mathcal{G}$, cf.
 
−
[[#References|[a6]]] for a detailed discussion of these results and related matters; cf. also the discussion on strong rigidity in the main article above.
 
  
 
====References====
 
====References====
−
<table><TR><TD valign="top">[a1]</TD>
+
 
−
  <TD valign="top"> G.A. Margulus, "Arithmeticity of irreducible lattices in semi-simple groups of rank exceeding 1" , MIR (1977) (In Russian) (Appendix to the Russian translation of: M.S. Raghunathan: "On the congruence subgroup problem" Publ. Math. IHES '''46''' (1976), 107–161) {{MR|}} {{ZBL|}} </TD>
+
<table> <TR><TD valign="top">[1]</TD>
−
</TR><TR><TD valign="top">[a2]</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>
−
  <TD valign="top"> G.D. Mostow, "Existence of nonarithmetic monodromy groups" ''Proc. Nat. Acad. Sc. U.S.A.'' , '''78''' (1981) pp. 5948–5950 {{MR|0773821}} {{ZBL|0551.32024}} </TD>
+
</TR> <TR><TD valign="top">[2]</TD>
−
</TR><TR><TD valign="top">[a3]</TD>
+
  <TD valign="top"> F. Kasch,   "Modules and rings" , Acad. Press  (1982)  (Translated from German)</TD>
−
  <TD valign="top"> R.J. Zimmer, "Ergodic theory and semisimple groups" , Birkhäuser (1984) {{MR|0776417}} {{ZBL|0571.58015}} </TD>
+
</TR> <TR><TD valign="top">[3]</TD>
−
</TR><TR><TD valign="top">[a4]</TD>
+
  <TD valign="top"> A.I. Kostrikin,  Yu.I. Manin,   "Linear algebra and geometry" , Gordon &amp; Breach  (1989)  (Translated from Russian)</TD>
−
  <TD valign="top"> J.E. Humphreys, "Arithmetic groups" , ''Topics in the theory of arithmetic groups'' , Notre Dame Univ. (1982) pp. 73–97 {{MR|0698787}} {{ZBL|0504.22010}} </TD>
+
</TR> <TR><TD valign="top">[4]</TD>
−
</TR><TR><TD valign="top">[a5]</TD>
+
  <TD valign="top"> P.R. Halmos,   "Finite-dimensional vector spaces" , v. Nostrand  (1958)</TD>
−
  <TD valign="top"> D.A. Kazhdan, G.A. Margulis, "A proof of Selberg's conjecture" ''Math. USSR-Sb.'' , '''4''' : 1 (1968) pp. 147–152 ''Mat. Sb.'' , '''75''' (1968) pp. 163–168 {{MR|}} {{ZBL|}} </TD>
+
</TR> <TR><TD valign="top">[5]</TD>
−
</TR><TR><TD valign="top">[a6]</TD>
+
  <TD valign="top"> M.F. Atiyah,   "$K$-theory: lectures" , Benjamin  (1967)</TD>
−
  <TD valign="top"> J. Tits, "Travaux de Margulis sur les sous-groupes discrets de groupes de Lie" , ''Sem. Bourbaki 1975/1976'' , '''Exp. 482''' , Springer (1977) pp. 174–190 {{MR|0492073}} {{ZBL|0346.22011}} </TD>
+
</TR> </table>
−
</TR></table>
 

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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

$$\Hom_A(V_1, W_1) \tensor_A \Hom_A(V_2, W_2) \to$$

$$\to \Hom_A(V_1 \tensor V_2, W_1 \tensor W_2),$$ 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 $\tilde C_1 = C_1 \tensor \mathbf{1}$ and $\tilde C_2 = \mathbf{1} \tensor C_2$ 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

$$A \tensor B = \begin{Vmatrix} \alpha_{11} B & \cdots & \alpha_{1n} B \\ \vdots & \ddots & \vdots \\ \alpha_{m1} B & \cdots & \alpha_{mn} B \end{Vmatrix}.$$ 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)
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