Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Abstract Algebra → Modules ↓


Tensor Products


Tensor products are a fundamental concept in abstract algebra and linear algebra. They allow the construction of new modules or vector spaces from existing modules or vector spaces. Understanding tensor products is essential in many areas of mathematics, including geometry, topology, and more. To fully understand the concept, it is important to understand the algebraic structures involved and how the tensor product works on them. This explanation will go deep into the definitions, properties, and examples of tensor products in the context of modules.

What is a module?

Before diving into tensor products, let's first understand what a module is. A module is like a generalization of a vector space. However, instead of a field of scalars (like the real numbers in a vector space), a module uses a ring as the set of scalars. Here is the formal definition:

A set M is a left R module if there is an operation R × M → M such that:
1. r · (m + n) = r · m + r · n for all r ∈ R and m, n ∈ M
2. (r + s) · m = r · m + s · m for all r, s ∈ R and m ∈ M
3. (r · s) · m = r · (s · m) for all r, s ∈ R and m ∈ M
4. 1_R · m = m for all m ∈ M where 1_R is a multiplicative identity in R

Definition of tensor products

Now, on to the tensor product of modules. The tensor product allows us to "multiply" two modules over the same ring to form another module. This operation extends the notion of multiplying numbers or vectors to the more general context of modules.

If M and N are modules over a ring R, then the tensor product M ⊗_R N is an R module, constructed such that for any R module P and bilinear map f: M × N → P, there exists a unique linear map g: M ⊗_R N → P such that transforms the following diagram:

m × n --f--> p
 ,
 |G∘⊗ |
M ⊗_R N -----G--→ P

The operation ⊗ maps the pair (m, n) to the elements in the tensor product M ⊗_R N.

Construction of the tensor product

The construction of the tensor product can get quite technical, but it essentially involves starting with a free module generated by the pairs (m, n) and then quotienting by certain relations to ensure bilinearity. Specifically:

To construct M ⊗_R N, consider the free module generated by the symbols [m, n] for m ∈ M and n ∈ N Then, apply the following relation:
1. [m + m', n] = [m, n] + [m', n]
2. [m, n + n'] = [m, n] + [m, n']
3. [rm, n] = r[m, n] and [m, rn] = r[m, n].

These relations ensure that the tensor product is bilinear.

Properties of tensor products

  • Distributivity: (M ⊕ M') ⊗_R N ≅ (M ⊗_R N) ⊕ (M' ⊗_R N)
  • Associativity: (M ⊗_R N) ⊗_R P ≅ M ⊗_R (N ⊗_R P)
  • Commutativity: M ⊗_R N ≅ N ⊗_R M (only if R is commutative)

Here, ⊕ denotes the direct sum of modules, and ≅ denotes isomorphism.

Examples of tensor products

Let's walk through some examples of tensor products in action.

Example 1: Tensor product of vector spaces

Let V and W be vector spaces over a field K The tensor product V ⊗_K W is a vector space whose dimension is the product of the dimensions of V and W

If {v_1, v_2, ..., v_m} is a basis for V and {w_1, w_2, ..., w_n} is a basis for W, 
So {v_i ⊗ w_j | i = 1, ..., m; j = 1, ..., n} forms a basis for V ⊗_K W.
v_1 w_1 v_2 w_2

Example 2: Tensor product with Z-modules

Consider integers modulo 2, ℤ/2ℤ, and integers modulo 3, ℤ/3ℤ. Compute their tensor product on ℤ.

Since ℤ/2ℤ is similar to ℤ[x]/(x²-x) and ℤ/3ℤ is similar to ℤ[y]/(y²-y), 
The tensor product (ℤ/2ℤ) ⊗_ℤ (ℤ/3ℤ) is trivial.
Z/2Z Z/3Z

Applications of tensor products

Tensor products are widely used in various mathematical fields. For example, in algebraic geometry, tensor products allow the construction of sheaves and are essential in defining schemes. In physics, especially quantum mechanics, tensor products are used to describe systems with many particles.

In the context of quantum mechanics, if H_1 and H_2 are Hilbert spaces, 
The tensor product H_1 ⊗ H_2 represents the combined state space of the two-part system.

Conclusion

Tensor products are a powerful and versatile mathematical concept. They provide a way to combine modules and vector spaces in a rigorous algebraic framework. Understanding tensor products enhances our ability to work with complex algebraic structures and to see connections between different mathematical fields.

In short, tensor products extend fundamental operations in mathematics to a broader context, giving us the opportunity to explore new and exciting areas of mathematical theory and application.


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