1.8 Tensor products

Tensor products are a powerful tool in noncommutative geometry, allowing us to combine linear spaces and modules into larger structures. They're essential for studying multilinear maps and interactions between algebraic structures, forming the backbone of many advanced mathematical concepts.

In this topic, we'll explore the definition, construction, and properties of tensor products. We'll see how they apply to vector spaces, modules, and C*-algebras, and discover their role in quantum mechanics, representation theory, and algebraic topology.

Definition of tensor products

• Tensor products provide a way to combine linear spaces or modules to create a larger space or module that captures the structure of the original objects
• The tensor product is a fundamental construction in mathematics, particularly in algebra, geometry, and topology
• Tensor products allow for the study of multilinear maps and the interaction between different algebraic structures

Tensor products of vector spaces

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• Given two vector spaces $V$ and $W$ over a field $k$, their tensor product is a vector space $V \otimes_k W$ equipped with a bilinear map $\otimes: V \times W \to V \otimes_k W$
• Elements of $V \otimes_k W$ are finite sums of the form $\sum_{i} v_i \otimes w_i$, where $v_i \in V$ and $w_i \in W$
• The tensor product satisfies the following relations for all $v, v' \in V$, $w, w' \in W$, and $\alpha \in k$:
  • $(v + v') \otimes w = v \otimes w + v' \otimes w$
  • $v \otimes (w + w') = v \otimes w + v \otimes w'$
  • $(\alpha v) \otimes w = v \otimes (\alpha w) = \alpha (v \otimes w)$

Tensor products of modules

• The tensor product can be defined for modules over a ring $R$ in a similar manner to vector spaces
• Given two $R$-modules $M$ and $N$, their tensor product is an $R$-module $M \otimes_R N$ together with an $R$-bilinear map $\otimes: M \times N \to M \otimes_R N$
• The tensor product of modules satisfies analogous relations to those of vector spaces, with scalars replaced by elements of the ring $R$

Universal property of tensor products

• The tensor product is characterized by a universal property, which states that for any vector spaces (or modules) $V$ and $W$, and any bilinear map $f: V \times W \to U$, there exists a unique linear map $\tilde{f}: V \otimes W \to U$ such that $f = \tilde{f} \circ \otimes$
• This universal property determines the tensor product up to isomorphism and allows for the extension of bilinear maps to linear maps on the tensor product

Construction of tensor products

• There are several equivalent ways to construct the tensor product of vector spaces or modules, each highlighting different aspects of the construction

Tensor products as quotient spaces

• The tensor product $V \otimes_k W$ can be constructed as a quotient space of the free vector space generated by the Cartesian product $V \times W$
• The subspace quotiented out is generated by elements of the form $(v + v', w) - (v, w) - (v', w)$, $(v, w + w') - (v, w) - (v, w')$, and $(\alpha v, w) - (v, \alpha w)$ for all $v, v' \in V$, $w, w' \in W$, and $\alpha \in k$
• The quotient space ensures that the relations defining the tensor product hold in the resulting space

Tensor products via universal property

• An alternative construction of the tensor product uses the universal property as a defining characteristic
• The tensor product $V \otimes_k W$ is defined as a vector space equipped with a bilinear map $\otimes: V \times W \to V \otimes_k W$ satisfying the universal property
• The existence and uniqueness of the tensor product can be proven using the concept of free objects and universal properties

Examples of tensor product constructions

• The tensor product of two vector spaces $\mathbb{R}^n$ and $\mathbb{R}^m$ over the real numbers is isomorphic to the space of $n \times m$ matrices, $\mathbb{R}^{n \times m}$
• For any vector space $V$, the tensor product $V \otimes_k k$ is isomorphic to $V$ itself, where $k$ is viewed as a one-dimensional vector space over $k$
• The tensor product of two polynomial rings $k[x]$ and $k[y]$ over a field $k$ is isomorphic to the polynomial ring in two variables, $k[x, y]$

Properties of tensor products

• Tensor products possess several important properties that make them useful in various areas of mathematics

Associativity and commutativity

• The tensor product is associative up to isomorphism: $(U \otimes_k V) \otimes_k W \cong U \otimes_k (V \otimes_k W)$ for any vector spaces $U$, $V$, and $W$ over a field $k$
• The tensor product is commutative up to isomorphism: $V \otimes_k W \cong W \otimes_k V$ for any vector spaces $V$ and $W$ over a field $k$
• These properties allow for the unambiguous definition of the tensor product of multiple spaces and the reordering of factors

Distributivity over direct sums

• The tensor product is distributive over direct sums: $(U \oplus V) \otimes_k W \cong (U \otimes_k W) \oplus (V \otimes_k W)$ for any vector spaces $U$, $V$, and $W$ over a field $k$
• This property allows for the decomposition of tensor products into simpler components and the study of the interaction between tensor products and direct sum decompositions

Tensor products and bases

• If $\{e_i\}$ and $\{f_j\}$ are bases for vector spaces $V$ and $W$ respectively, then $\{e_i \otimes f_j\}$ forms a basis for the tensor product $V \otimes_k W$
• This property allows for the computation of the dimension of tensor products and the representation of elements in the tensor product using basis vectors

Tensor products and linear maps

• Given linear maps $f: V \to V'$ and $g: W \to W'$ between vector spaces, there is an induced linear map $f \otimes g: V \otimes_k W \to V' \otimes_k W'$ defined by $(f \otimes g)(v \otimes w) = f(v) \otimes g(w)$
• This property allows for the study of the functorial behavior of tensor products and the interaction between tensor products and linear maps

Tensor products in noncommutative geometry

• Tensor products play a crucial role in noncommutative geometry, where they are used to construct and study noncommutative analogs of classical geometric objects

Tensor products of C*-algebras

• C*-algebras are noncommutative analogs of topological spaces, and their tensor products provide a way to construct new C*-algebras from existing ones
• The tensor product of two C*-algebras $A$ and $B$ is a C*-algebra $A \otimes B$ equipped with a *-homomorphism $\otimes: A \times B \to A \otimes B$ satisfying a universal property
• The tensor product of C*-algebras is used to define noncommutative analogs of topological products and to study the structure of C*-algebras

Tensor products of Hilbert spaces

• Hilbert spaces are fundamental objects in noncommutative geometry, and their tensor products allow for the construction of larger Hilbert spaces from smaller ones
• Given two Hilbert spaces $H$ and $K$, their tensor product is a Hilbert space $H \otimes K$ equipped with an inner product satisfying $\langle x_1 \otimes y_1, x_2 \otimes y_2 \rangle = \langle x_1, x_2 \rangle \langle y_1, y_2 \rangle$ for all $x_1, x_2 \in H$ and $y_1, y_2 \in K$
• The tensor product of Hilbert spaces is used to construct representations of C*-algebras and to study the geometry of noncommutative spaces

Tensor products and representations

• Representations of C*-algebras and other algebraic objects play a central role in noncommutative geometry, and tensor products provide a way to construct new representations from existing ones
• Given representations $\pi: A \to B(H)$ and $\rho: A \to B(K)$ of a C*-algebra $A$, their tensor product is a representation $\pi \otimes \rho: A \to B(H \otimes K)$ defined by $(\pi \otimes \rho)(a) = \pi(a) \otimes \rho(a)$ for all $a \in A$
• Tensor products of representations are used to study the structure of C*-algebras and to construct invariants of noncommutative spaces

Applications of tensor products

• Tensor products have numerous applications in various areas of mathematics and physics, where they are used to model complex systems and interactions

Tensor products in quantum mechanics

• In quantum mechanics, the state space of a composite system is described by the tensor product of the state spaces of its constituent parts
• For example, the state space of a system consisting of two particles is the tensor product of the state spaces of each particle
• Tensor products are used to model entanglement, a key feature of quantum systems, and to study the behavior of composite quantum systems

Tensor products in representation theory

• Representation theory studies the ways in which abstract algebraic objects, such as groups and algebras, can be represented as linear transformations on vector spaces
• Tensor products of representations provide a way to construct new representations from existing ones and to study the structure of the underlying algebraic objects
• For example, the tensor product of two irreducible representations of a group is often decomposable into a direct sum of irreducible representations, and the multiplicities of these representations provide important information about the group

Tensor products in algebraic topology

• In algebraic topology, tensor products are used to construct algebraic invariants of topological spaces, such as homology and cohomology groups
• The tensor product of abelian groups is a key ingredient in the definition of the tensor product of chain complexes, which is used to study the algebraic properties of topological spaces
• Tensor products also appear in the definition of the cup product, a fundamental operation in cohomology theory

Tensor products in homological algebra

• Homological algebra is the study of algebraic structures, such as modules and chain complexes, using the tools of category theory and homological methods
• Tensor products of modules and chain complexes play a central role in homological algebra, where they are used to construct derived functors and to study the homological properties of algebraic objects
• For example, the tensor product is used to define the torsion product of modules, which measures the failure of the tensor product to be exact, and the derived tensor product, which is a homological invariant of modules

Computations with tensor products

• Computational aspects of tensor products are important in many applications, where explicit calculations and representations of tensor products are required

Tensor products of matrices

• The tensor product of two matrices $A$ and $B$ of sizes $n \times m$ and $p \times q$, respectively, is a matrix $A \otimes B$ of size $np \times mq$ defined by $(A \otimes B)_{(i-1)p+k, (j-1)q+l} = a_{ij}b_{kl}$
• Tensor products of matrices are used to represent linear maps between tensor product spaces and to perform computations in various applications, such as quantum information theory

Tensor products and Kronecker products

• The Kronecker product of two matrices $A$ and $B$ is closely related to their tensor product and is defined by $(A \otimes B)_{(i-1)p+k, (j-1)q+l} = a_{ij}b_{kl}$, where $A$ is of size $n \times m$ and $B$ is of size $p \times q$
• Kronecker products are used in various applications, such as signal processing and control theory, where they provide a convenient way to represent certain linear transformations

Tensor products and multilinear maps

• Tensor products are intimately connected with the concept of multilinear maps, which are functions that are linear in each of their arguments separately
• Given vector spaces $V_1, \ldots, V_n$ and $W$, a multilinear map $f: V_1 \times \cdots \times V_n \to W$ induces a unique linear map $\tilde{f}: V_1 \otimes \cdots \otimes V_n \to W$ satisfying $\tilde{f}(v_1 \otimes \cdots \otimes v_n) = f(v_1, \ldots, v_n)$
• This correspondence between multilinear maps and linear maps on tensor products is a key feature of the tensor product construction and is used in various applications, such as the study of differential forms and the representation theory of Lie algebras

Generalizations of tensor products

• The concept of tensor products can be generalized in various ways to accommodate different algebraic structures and settings

Tensor products of sheaves

• Sheaves are mathematical objects that capture the idea of local-to-global passage and are used in algebraic geometry and topology to study the properties of spaces
• The tensor product of sheaves is a sheaf-theoretic analog of the tensor product of modules and is used to construct new sheaves from existing ones
• Tensor products of sheaves are used in the study of coherent sheaves, which are a key tool in algebraic geometry, and in the definition of the derived category of sheaves, which is a powerful invariant of spaces

Tensor products of operator algebras

• Operator algebras are algebras of bounded linear operators on Hilbert spaces and are used in the study of noncommutative geometry and quantum field theory
• The tensor product of operator algebras is a generalization of the tensor product of C*-algebras and is used to construct new operator algebras from existing ones
• Tensor products of operator algebras are used in the study of quantum groups, which are noncommutative analogs of groups, and in the construction of quantum field theories on noncommutative spaces

Tensor products in monoidal categories

• Monoidal categories are a general framework for studying algebraic structures with a tensor product-like operation
• In a monoidal category, objects are equipped with a tensor product operation, and morphisms are required to be compatible with this operation in a certain sense
• Examples of monoidal categories include the category of vector spaces with the usual tensor product, the category of sets with the Cartesian product, and the category of endofunctors of a category with composition as the tensor product
• The study of tensor products in monoidal categories provides a unified perspective on various algebraic constructions and is used in areas such as quantum algebra and topological quantum field theory