Modern Mathematics

Table of Contents

1. Algebraic Structures

1.1. Group-like

Magma_to_group4.svg

1.1.1. Monoid

  • Group without the inverse elements.

1.1.2. Group

1.2. Ring-like

1.2.1. Ring

1.2.2. Field

1.3. Module-like

1.3.1. Vector Space

See vector space for the definition.

1.3.1.1. Symplectic Vector Space
  • \((V, \omega)\)
1.3.1.1.1. Definition
1.3.1.1.2. Standard Symplectic Space
  • \(\mathbb{R}^{2n}\) with the symplectic form given by a nonsingular, skew-symmetric matrix, typically chosen to be: \[ \omega = \begin{bmatrix} 0 & I_n \\ - I_n & 0 \end{bmatrix}. \]
1.3.1.1.3. Symplectic Map
  • Linear map \(f\colon V\to W\) between symplectic vector spaces \((V, \omega), (W, \rho)\), that the preserves the symplectic form: \[ f^*\rho = \omega. \]
1.3.1.1.4. Symplectic Group
  • Symplectic map \(f\colon V\to V\) is called a linear symplectic transformation of \(V\). It preserves the symplectic form \(\omega(f(u), f(v)) = \omega(u,v)\).
  • The set of symplectic transformations forms a Lie group called the symplectic group, denoted \(\mathrm{Sp}(V)\) or \(\mathrm{Sp}(V, \omega)\).
  • Matrix form of symplectic transformations are symplectic matrices.
  • The
1.3.1.1.5. Symplectic Complement
  • For a linear subspace \(W \subset V\), the symplectic complement of \(W\) is: \[ W^\bot := \{v\in V\mid \forall w\in W, \omega(v,w) = 0\}. \]
1.3.1.1.6. Properties
  • \((W^\bot)^\bot = W\)
  • \(\dim W + \dim W^\bot = \dim V\)
  • Unlike orthogonal complements, \(W^\bot\cap W\) need not be \(\{0\}\), distinguished four cases are:
    • \(W\ \text{symplectic}\): \(W^\bot\cap W = \{0\}\)
      • If and only if \(\omega\) restricts to a nondegenerate form on \(W\)
      • Symplectic subspace with the restricted form is a symplectic vector space.
    • \(W\ \text{isotropic}\): \(W\subseteq W^\bot\)
      • If and only if \(\omega\) restricts to 0 on \(W\)
      • Any one-dimensional subspace is isotropic.
    • \(W\ \text{coisotropic}\): \(W^\bot \subseteq W\)
      • If and only if \(\omega\) descends to a nondegenerate form on the quotient space \(W/W^\bot\)
      • Equivalently, \(W\) is coisotropic if and only if \(W^\bot\) is isotropic.
      • Any codimension-one subspace is coisotropic
    • \(W\ \text{Lagrangian}\): \(W = W^\bot\)
      • If and only if it is both isotropic and coisotropic
      • In a finite-dimensional vector space, a Lagrangian subspace is an isotropic one whose dimension is half that of \(V\)
      • Every isotropic subspace can be extended to a Lagrangian one.

1.3.2. Module

1.3.2.1. Definition

A module is a vector space over a ring.

Every axiom of vector spaces applies, including the linearity of scalar multiplication in both arguments.

1.3.2.2. Simple Module

Module that contains no non-trivial submodules.

1.3.2.3. Semi-Simple Module

\( A \)-Module \( M \) isomorphic to a direct sum of simple \( A \)-modules.

1.3.2.4. Representation

A group representation \( (V, \rho) \) of a group \( G \) on a vector space \( V \) over a field \( K \) can be linearly extended into an algebra representation \( (V, \tilde{\rho}) \) of group algebra \( K[G] \), and form a \( K[G] \)-module (often, \( G \)-module). The converse is also true; a \( K[G] \)-module uniquely determines a representation.

The linear extension into an algebra homomorphism is given by: \[ \tilde{\rho}\colon K[G] \to \operatorname{Hom}(V,V)\colon a_ig_i \to a_i\rho(g_i). \] This map induces the algebra action: \[ \cdot \colon (g, v) \mapsto \tilde{\rho}(g)v. \] Notice the scalar multiplication by \( K \) can be fully replaced by the algebra action whenever the algebra is unital: \[ av = \rho(ae)v. \]

In fact, any algebra \( A \) can form a \( A \)-module on top of a vector space \( V \) over a field \( K \), given its represetation \( \tilde{\rho}\colon A\to \operatorname{Hom}(V, V) \).

1.4. Algebra-like

1.4.1. Algebra

Vector space with bilinear, associative, unital multiplication.

1.4.1.1. Simple Algebra
1.4.1.2. Semi-Simple Algebra

An algebra \( A \) whose \( A \)-modules are all semi-simple.

2. Topological Structures

2.1. Topological Space

See topological space

2.2. Metric Space

  • \((M, d)\)
  • Set with a notion of distance between its elements

2.2.1. Definition

  • Ordered pair \((M, d)\) where \(M\) is a set, and \(d : M\times M \to \mathbb{R}\) is a metric on \(M\) with:
    • \(d(x, x) = 0\)
    • Positivity: \(x\neq y \implies d(x, y) > 0\)
    • Symmetry: \(d(x, y) = d(y, x)\)
    • Triangle Inequality: \( d(x,z) \le d(x,y) + d(y,z) \)

2.2.2. Properties

2.2.3. Metrizable Space

  • Topological Space that is homeomorphic to a metric space.
  • There exists a metric \(d\) that induces the topology \(\tau\).

2.2.4. Closeness

  • Arbitrarily near
2.2.4.1. Definition
  • In a metric space \((X, d)\), a point \(p\) is close or near to a set \(A\), if: \[ d(p, A) := \inf_{a\in A} d(p, a) = 0 \] where \(\inf\) is the infimum.
  • Similarly, a set \(B\) is close to a set \(A\), if: \[ d(B, A) := \inf_{b\in B} d(b, A) = 0. \]

2.3. Normed Vector Space

  • Normed Space
  • \((V, \Vert\cdot\Vert)\)

2.3.1. Definition

  • Vector space \(V\) over \(K\) on which a norm \(\Vert\cdot\Vert\) is defined, such that:
    • Non-Negativity: \(\forall x\in V, \Vert x\Vert \ge 0\)
    • Positive Definiteness: \(\forall x\in V, (\Vert x\Vert = 0 \iff x = \mathbf{0})\)
    • Absolute Homogeneity: \(\forall \lambda \in K, \forall x\in V, \Vert\lambda x\Vert = |\lambda|\Vert x\Vert\)
    • Triangle Inequality: \( \forall x, y\in V, \Vert x+y\Vert \le \Vert x\Vert + \Vert y\Vert \)

2.3.2. Properties

  • It is also a metric space with the metric \(d\) induced by the norm: \[ d(x, y) = \Vert y - x\Vert. \]
  • If the norm satisfies the polarization identity then the inner product can be induced.

2.3.3. Polarization Identity

  • Any formula that expresses the inner product in terms of the norm.
  • Every inner product satisfies: \[ \Vert x+ y \Vert^2 = \Vert x\Vert^2 + \Vert y\Vert^2 + 2\mathfrak{R}\langle x, y\rangle \] for the induced norm.
2.3.3.1. Theorem
  • Norm satisfies the parallelogram law, if and only if, there exists an inner product \(\langle \cdot, \cdot \rangle\) such that \(\Vert x\Vert^2 = \langle x, x\rangle\) for all \(x\).
  • The map between the norm and the inner product is bijective.
2.3.3.2. For Real Vector Space
\begin{align*} \langle x, y \rangle &= \frac{1}{4} \left(\|x+y\|^2 - \|x-y\|^2\right) \\ &= \frac{1}{2} \left(\|x+y\|^2 - \|x\|^2 - \|y\|^2\right) \\ &= \frac{1}{2} \left(\|x\|^2 + \|y\|^2 - \|x-y\|^2\right) \\ \end{align*}
  • These forms are related by the Parallelogram law.
2.3.3.3. For Complex Vector Space

By stipulating the properties of the inner product:

\begin{align*} \langle x \,|\, y \rangle &= \frac{1}{4} \left(\|x+y\|^2 - \|x-y\|^2 - i\|x + iy\|^2 + i\|x - iy\|^2\right) \\ &= R(x, y) - i R(x, iy) \\ &= R(x, y) + i R(ix, y) \\ \end{align*}

if antilinear in the first argument,

\begin{align*} \langle x,\, y \rangle &= \frac{1}{4} \left(\|x+y\|^2 - \|x-y\|^2 + i\|x + iy\|^2 - i\|x - iy\|^2\right) \\ &= R(x, y) + i R(x, iy) \\ &= R(x, y) - i R(ix, y) \\ \end{align*}

if antilinear in the second argument.

2.3.4. Examples

2.3.4.1. Lᵖ Space
  • \(L^p\) Spaces, Lebesque Spaces
2.3.4.1.1. Definition
  • Function spaces defined using a generalization of the \(p\)-norm---\(L^p\)-norm.
  • It is the space of measurable functions for which the \(L^p\)-norm is defined---\(|f|^p\) is Lebesgue integrable, modulo the equivalence relation \(f\sim g :\!\!\iff \| f - g\|_p = 0\).
2.3.4.1.1.1. p-Norm
  • For a real number \(p\le 1\): \[ \|x\|_p = \left(|x_1|^p + |x_2|^p + \dotsb + |x_n|^p\right)^{1/p}. \]
2.3.4.1.1.2. Lᵖ-Norm
  • \[ \| f\|_p = \left(\int_S |f|^p\, \mathrm{d}\mu\right)^{1/p} \]
2.3.4.1.2. Uniform Norm
  • Sup Norm, Supremum Norm, Chebyshev Norm, Infinity Norm, Max Norm, Maximum Norm
2.3.4.1.2.1. Definition
  • The uniform norm of a real- or complex-valued bounded functions \(f\) defined on a set \(S\) is \[ \| f\|_\infty := \sup\{|f(x)|: x\in S\}. \]
2.3.4.2. Sobolev Space
2.3.4.2.1. Definition
  • \(W^{k,p}(\mathbb{F})\)
  • It is the subset of \(L^p(\mathbb{F})\)
  • A normed vector space of functions equipped with a norm that is a combination of Lᵖ-norms of a funtion and its derivatives up to a given order.
2.3.4.2.2. Norm
  • \[ \|f\|_{k,p} = \left(\sum_{i=0}^k\left\|f^{(i)}\right\|_p^p\right)^{\frac{1}{p}}. \]

2.4. Banach Space

  • Complete normed vector space
  • The norm of Banach space is called complete norm, and the canonical metric is called the complete metric.

2.5. Inner Product Space

  • \((V, \langle\cdot, \cdot\rangle)\)

2.5.1. Definition

  • Vector space \(V\) over the field \(F\)—which can be either \(\mathbb{R}\) or \(\mathbb{C}\)—with an inner product \(\langle\cdot,\cdot\rangle : V\times V \to F\) satisfying:
    • Conjugate Symmetry: \(\langle x,y\rangle = \overline{\langle y, x\rangle}\)
    • Linearity in the First Argument: \(\langle ax+by, z\rangle = a\langle x, z\rangle + b\langle y,z\rangle\)
    • Positive Definiteness: \(x\neq 0 \implies \langle x,x\rangle > 0\)
  • It immediately follows from the definition that the inner product is antilinear in the second argument.

2.5.2. Properties

  • It is also a normed vector space with the norm induced by the inner product: \[ \Vert x\Vert = \sqrt{\langle x, x\rangle}. \]

2.5.3. Vectors

2.5.3.1. Null Vector
  • \(\langle x, x\rangle = 0\)
2.5.3.2. Degenerate Vector
  • \(\forall y \in V, \langle x, y\rangle = 0\)

2.6. Hilbert Space

2.6.1. Definition

2.6.1.1. Complete Metric Space
  • If a series converges absolutely then the series converges. \[ \sum_{n=0}^{\infty}\Vert \mathbf{x}_n\Vert = L \implies \sum_{n=0}^{\infty}\mathbf{x}_n = \mathbf{L} \]
  • This is making sure that the limit exists.

2.6.2. Examples

2.6.2.1. L² Space
  • Function space from measure space \((X, \Sigma, \mu)\) to either \(\mathbb{R}\) or \(\mathbb{C}\), equipped with the inner product: \[ \langle f, g\rangle = \int_X f(x)\overline{g(x)}\,\mathrm{d}\mu(x)\ \ \text{or}\ \int_X \overline{f(x)}g(x)\,\mathrm{d}\mu(x). \]
  • This is the only Hilbert space among Lᵖ spaces.
  • Hilbert space - Wikipedia

2.7. Affine Space

  • \((A, \vec{A}, +)\)
  • Informally, a vector space whose origin is forgotten by adding translations to the linear maps.

2.7.1. Definition

  • A set \(A\)—points—with an associated vector space \(\vec{A}\)—isplacement vectors— and a transitive free action of the additive group of \(\vec{A}\) on the set \(A\), \(+\).
  • The action \(+: A\times \vec{A}\to A\) satisfies:
    • Right Identity: \[ \forall a \in A, a + 0 = a. \]
    • Associativity: \[ \forall v, w\in \vec{A}, \forall a \in A, (a+v)+w = a+(v+w). \]
    • Free and Transitive action: \[ \forall a\in A, \text{the mapping } \vec{A}\to A: v\mapsto a+v \text{ is a bijection}. \]

2.7.2. Affine Combination

  • Linear combination in which the coefficients sums to 1.
  • Affine combination of the points makes sense, which in turn can constitute a normalized barycentric coordinates.

2.7.3. Affine Structure

  • Defined by the values of affine combination

2.8. Euclidean Space

  • Basically, \(\mathbb{R}^n\) with all the nice structures.

2.8.1. Definition

2.8.1.1. Euclidean Vector Space
  • \((\mathbb{R}^n, \cdot)\)
  • A Euclidean vector space is a finite-dimensional inner product space over the real numbers.
2.8.1.2. Euclidean Space
  • Euclidean space is an affine space over the reals such that the associated vector space is a Euclidean vector space.

2.8.2. Euclidean Distance

  • The Euclidean distance between two points in Euclidean space is the length of the line segment between them.

2.8.3. Properties

  • Every Euclidean space of each dimension are all isomorphic to each other. One of them is represented using Cartesian coordinate system as the real \(n\)-space \(\mathbb{R}^n\) such that the associated vector space is equipped with the standard dot product .

3. Measure Structures

3.0.1. Measure Space

See measure space

4. Order Structures

4.1. Preorder

  • Quasiorder
  • \(\lesssim\), \(\gtrsim\)

4.1.1. Definition

  • For elements that are comparable:
    • Reflexive
    • Transitive
  • A set equipped with a preorder is called preordered set, proset.

4.1.2. Properties

  • Antisymmetric preorder is a partial order, and symmetric preorder is an equivalence relation.
  • Preorder induces an equivalence relation \(\sim\) with symmetry axiom: \[ a\sim b \iff a \lesssim b \land a \gtrsim b. \]
  • Similarly it induces an strict partial order \(<\) with anti-symmetry axiom: \[ a < b \iff a \lesssim b \land \neg(a \gtrsim b). \]

4.2. Partial Order

  • Reflexive, Weak, or Non-Strict Partial Order
  • The set equipped with partial order is called partially ordered set, poset.
  • As the name suggests not every pair needs to be comparable.

4.2.1. Definition

  • For the pairs that the partial order is defined:
    • Reflexivity
    • Antisymmetry: \(a\le b \land b\le a \implies a = b.\)
      • Therefore, the order relation on \(\mathbb{C}\) is not partial order, rather, preorder.
    • Transitivity
  • It is an antisymmetric preorder.

4.2.2. Strict Partial Order

  • Irreflexive, Strong, or Strict Partial Order
4.2.2.1. Definition
  • Irreflexivity
  • Asymmetry
  • Transitivity

4.2.3. Reflexive Closure and Irreflexive Kernel

  • Closure of the irreflexive relation is the union with the reflection.
  • Kernel of the reflexive relation is the subtraction of the reflection.

4.3. Total Order

  • Simple Order, Connex Order, Full Order

4.3.1. Definition

  • Binary relation \(\le\) on \(X\), that satisfies:
    • Reflexivity: \(a\le a.\)
    • Transitivity: \(a\le b \land b\le c \implies a\le c.\)
    • Antisymmetry: \(a\le b \land b\le a \implies a = b.\)
    • Strongly Connected(Formerly, Total): \(a\le b \lor b\le a.\)
  • The set \((X, \le)\) is called totally ordered set, simply ordered set, linearly ordered set, loset.
  • Chain is generally the totally ordered subset of a partially ordered set.

4.3.2. Strict Total Order

  • Binary relation \(<\) on \(X\)

4.3.3. Definition

  • Irreflexivity: \(\neg( a < a).\)
  • Asymmetry: \(a < b \implies \neg (b< a).\)
  • Transitivity: \(a< b \land b< c\implies a < c.\)
  • Connected: \(a\neq b \implies a < b \lor b < a.\)

4.4. Directed Set

4.4.1. Definition

  • Set with a preorder \(\lesssim\), where every pair of elements has an upper bound.
  • The preorder of a directed set is called a direction.

4.4.2. Upward and Downward Directed Sets

  • Upward directed set requires the existance of the common upper bound, and Downward directed set requires that of the common lower bound.

4.5. Hasse Diagram

  • Mathematical diagram used to represent a partially ordered set.
  • Larger objects on top, and smaller objects on the bottom, with the containment relationships indicated with lines.

5. Composite Structures

5.0.1. Vector Bundle

5.0.2. Fiber Bundle

6. Operations

6.1. Element-wise

6.1.1. Formal Product

  • \(*\)
  • Form of a product with no additional structure.

6.1.2. Tensor Product

  • \(\otimes\)
  • The tensor product \(v\otimes w\) is then defined as \(v*w+I\), where \(I\) is the coset.

6.1.3. Wedge Product

  • \(\wedge\)
  • See

6.1.4. Dot Product

6.1.5. Pseudo-Inner Product

6.1.5.1. Definition
  • Bilinear (or sesquilinear)
  • Symmetric
  • Non-Degenerate
    • See degeneracy
    • Weakening of the positive-definiteness.
    • \[ (\forall w \in V, \langle v, w\rangle = 0)\implies v = 0. \]

6.1.6. Inner Product

6.1.6.1. Definition

6.1.7. Geometric Product

6.2. Structure-wise

6.2.1. Formal Product

  • \(*\)
  • Given two vector spaces \(V\) and \(W\), \(V*W = \operatorname{span}_\mathbb{R}\{a*b\mid a\in V, b\in W\}\) forms a vector space of dimension \(|V|\cdot|W|\).

6.2.2. Tensor Product

  • \(\otimes\)
  • \(V^{\otimes n}\): \(V\) tensor producted with itself \(n\) times.
  • Tensor product of two vector spaces \(V\oplus W\) is the quotient vector space \(V*W/I\) with respect to the subspace \(I\): \[ I = \operatorname{span}_\mathbb{R}\left\lbrace\begin{array}{l|} (cv)*w - c(v*w), \\ v*(cw) - c(v*w), \\ (v_1 + v_2)*w - (v_1*w+v_2*w), \\ v*(w_1+w_2) - (v*w_1 + v*w_2) \end{array}\,\ c\in \mathbb{R},\ v\in V,\ w\in W\right\rbrace \]
6.2.2.1. Properties
  • \(\dim (V\otimes W) = \dim V\cdot \dim{W}\)
  • \(a\otimes b\in \mathbb{R}\otimes \mathbb{R}\) then \((ca)\otimes b = c(a\otimes b)\)

6.2.3. Direct Product

  • \(\times\)
  • This is what is meant by \(\mathbb{R}^n\).
6.2.3.1. Properties
  • \(\dim(V\times W) = \dim V + \dim W\)
  • \((a, b)\in \mathbb{R}\times \mathbb{R}\) then \((ca, cb) = c(a, b)\)

6.2.4. Direct Sum

  • \(\oplus\)
  • The two sets become just one set.
  • Direct Product with inclusions defined.
6.2.4.1. Properties
  • \(\dim(V\oplus W) = \dim V + \dim W\)
  • It is the coproduct

7. Substructures

Subset that respect the structure.

7.1. Submodule

A subspace that is also a module with respect to the same ring.

7.2. Subalgebra

8. Quotient

  • It is the general idea of forming a space of equivalence classes generated by dividing the collection with a equivalence relation.

8.1. Quotient Group

8.2. Quotient Ring

8.3. Quotient Space

  • Quotient of a Vector Space
  • The quotient space \(V/N\) consists of equivalence classes defined by the equivalence relation \(\sim\): \[ x\sim y \iff x-y \in N \] where \(N\) is a linear subspace of \(V\).
    • The equivalence class (or the coset) of \(x\) is denoted: \[ [x] = x+N = \lbrace x+n: n\in N\rbrace. \]
    • The scalar multiplication and addition are defined as:
      • \(\alpha [x] = [\alpha x],\)
      • \([x]+[y] = [x+y].\)

8.4. Quotient Space

  • Quotient Space of a Topological Space
  • Topological of equivalence classes generated by a equivalence relation.

9. Filtration

9.1. Defintion

Filtration of a structure \(X\) is the totally ordered collection of substructures \( (\mathcal{F}_i)_{i\in I}\), such that \(i\le j\implies \mathcal{F}_i \subseteq \mathcal{F}_j \subseteq X\).

10. Form

A function from a vector space to the underlying field. It may have a corresponding representation in the matrix form.

10.1. Bilinear Form

10.1.1. Definition

A bilinear form \( B \) is a bilinear map on a vector space \( V \) over a field \( K \): \[ B: V\times V \to K \] such that it's linear in both arguments.

10.1.1.1. Symmetric

\( B(v,u) = B(u,v) \)

10.1.1.2. Skew-Symmetric

\( B(v,u) = - B(u,v) \)

10.1.1.3. Altenating

\( B(v,v) = 0 \)

10.2. Sesquilinear Form

  • sesqui-, one and a half.

10.2.1. Definition

  • One of its arguments is semilinear: antilinear, and others.

10.2.2. Antilinearity

  • \( \varphi(ax) = a^*\varphi(x) \) , where \( ^* \) denotes the complex conjugate.
  • I just want to call it semilinearity, but there so much other things that are called semilinear.

10.3. Symplectic Form

10.3.1. Symplectic Bilinear Form

  • Mapping \(\omega: V\times V \to F\) that has following properties:
    • Bilinear
    • Alternating: \(\forall v\in V, \omega(v, v) = 0\)
    • Non-degenerate: \((\forall v\in V, \omega(v, u) = 0) \implies u = 0\)
  • If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry: \(\omega(v,u) = -\omega(u,v)\).
  • If the characteristic is 2, alternation implies skew-symmetry but not in the other direction.

10.3.2. Example

  • 2-Form

10.4. Quadratic Form

10.4.1. Definition

  • Homogeneous polynomial of degree two. ("form" is another name for a homogeneous polynomial1)
  • Quadratic form on a vector space \(V\) over a field \(K\) is \[ q: V\to K \] such that \(q(a\mathbf{v}) = a^2q(\mathbf{v})\).

10.4.2. Properties

  • Every quadratic form has an associated symmetric matrix: \[ q(\mathbf{x}) = \mathbf{x}^{\rm T}A\mathbf{x}. \]
  • That is every quadratic form can be orthogonally diagonalized using the spectral decomposition: \[ \mathbf{x}^{\rm T}\mathbf{A}\mathbf{x} = \mathbf{x}^{\rm T} Q\Lambda Q^{\rm T}\mathbf{x} = \tilde{\mathbf{x}}^{\rm T}\Lambda \tilde{\mathbf{x}} \]

11. Isomorphism

11.1. Definition

  • Structure-preserving mapping between two structures that can be reversed by an inverse mapping.

11.2. Canonical Isomorphism

  • The most trivial isomorphism between two structures.

11.3. Special Isomorphisms

11.3.1. Automorphism

Isomorphism to itself.

11.3.2. Isometry

Isomorphism between metric spaces. See isometry.

11.3.3. Homeomorphism

Isomorphism between topological spaces. See homeomorphism.

11.3.4. Diffeomorphism

Isomorphism between differential manifolds. See .

11.3.5. Symplectomorphism

Isomorphism between symplectic manifolds.

11.3.6. Permutation

Automorphism of a Set

11.3.7. Transformation

Automorphism of a geometric space (metric, affine, projective)

11.4. Isomorphism Theorems for Groups

11.4.1. First

  • For a group homomorphism \(\varphi\colon G\to H\),
  • \[ G/\ker \varphi \cong H \]

11.4.2. Second

  • Let \(H\) and \(N\) be a two normal subgroups of a group \(G\),
  • \[ G/H \cong (G/N)/(H/N) \]

11.4.3. Third

  • For a subgroup \(H \le G\), and a normal subgroup \(N\triangleleft G\),
  • \[ H/(H\cap N) \cong (HN)/N \]

11.5. Isomorphism Theorems for Rings

11.5.1. First

  • For a ring homomorphism \(\varphi\colon R\to S\), there exists a unique isomorphism \(\psi\colon R/\operatorname{ker}\varphi \to \operatorname{im}\varphi\) such that \(\psi(r+\operatorname{ker}\varphi) = \varphi(r)\).

12. Tensor

12.1. Intuitions

  • Tensor is multidimensional array
  • Tensor is multilinear map
  • Tensor is tensor product of vectors and covectors
  • Tensor (from physics) is an object that transforms like a tensor

12.2. Multilinear Map

\(k\)-linear map is a function \(f\) of vector spaces \(V_1,\dots,V_n, W\) (or modules over a commutative ring) \[ f: V_1\times \cdots \times V_n \to W \] that is linear in each argument when other variables are held constant.

  • Multilinear map from \(V^q\) to \(V^p\), corresponds to the (p,q)-tensor.

12.3. Definition

  • Given a finite set of vector spaces \(\{V_i\}_{i=1}^n\) over a common field \(F\), the element of their tensor product is a tensor \(T\): \[ T \in \bigotimes_{i=1}^nV_i. \]

12.3.1. On a Vector Space

  • Tensor \(T\) of type \((p,q)\) on a vector space \(V\) is defined as: \[ T \in T_q^p(V) := V^{\otimes p}\otimes (V^*)^{\otimes q}. \]

12.4. Type

  • Order, Rank, Valence, Degree
  • Pair of orders of contravariance and covariance.
  • \((p, q)\). \(p\) contravariant components (vectors), \(q\) covariant components (covectors).

12.5. Order

  • Degree, Rank
  • The dimension of a tensor
  • \(p+q\).

12.6. Rank

  • The minimum number of simple tensors that sum to the tensor.
  • Often used to mean the order of a tensor.

12.7. Universal Property

  • From the universal characterization of the tensor product, the space of (p,q)-tensors admits a natural isomorphism: \[ T_q^p(V) \cong L(\underbrace{V^*\otimes \cdots \otimes V^*}_{p}\otimes \underbrace{V\otimes \cdots \otimes V}_{q}; F) \cong L^{p+q}(\underbrace{V^*,\dots, V^*}_{p}, \underbrace{V,\dots,V}_q; F), \] when \(V\) is finite dimensional.
    • \(L^n(V_1, \dots, V_n;W)\) is denoting the space of n-linear maps from \(V_1\times \cdots \times V_n\) to \(W\).

12.8. (1, 1)-Tensor

12.8.1. Geometric Interpretation

12.8.2. Examples

12.9. Tensor Contraction

  • It is the generalization of
    • For a tensor of type \((p,q)\), contraction is a linear map from type \((p,q)\)-tensor to type \((p-1, q-1)\)-tensor defined by the canonical pairing of \(k\)th vector space and \(l\)th dual vector space: \[ C: \bigotimes_{i=1}^p v_i \otimes \bigotimes_{j=1}^q \alpha_j \mapsto \alpha_k(v_l)\bigotimes_{i=1, i\neq k}^p v_i \otimes \bigotimes_{j=1, j\neq l}^q \alpha_j \]
    • The application of tensor is also a contraction: \[ T(V_1, \dots, V_n) = C^n(T\otimes V_1\otimes \cdots \otimes V_n). \]

12.10. Tensor Field

12.10.1. Definition

  • A tensor field \(T\) of type \((p,q)\) on a manifold \(M\) is: \[ T \in \Gamma(M, V^{\otimes p}\otimes (V^*)^{\otimes q}), \]
  • where \(V\) is a vector bundle on \(M\).

13. Tensor Algebra

13.1. Definition

  • Tensor algebra \((T(V), \otimes)\) of a vector space \(V\) over af field \(K\) is: \[ TV = \bigoplus_{k=0}^\infty T^kV = \bigoplus_{k=0}^\infty V^{\otimes k}. \]

13.1.1. Tensor Power

  • \(T^kV\) is called the \(k\)th tensor power of \(V\) with \(T^0V := K\), and the multiplication \(\otimes\) is the linear extension of canonical isomorphism: \[ T^k V\otimes T^\ell V \to T^{k+\ell} V. \]
    • Note that \(\otimes\) of \(T(V)\) is not a tensor product.

13.2. Properties

  • It is the functor form the category of vector spaces \(\mathbf{Vect}\) to the category of algebras \(\mathbf{Alg}\)
    • \[ T: \mathbf{Vect}_K \to \mathbf{Alg}. \]

14. Exterior Algebra

14.1. Definiton

  • The quotient algebra of the tensor algebra \(T(V)\) with the ideal \(I = \{ x\otimes x : x\in V\}\): \[ \bigwedge(V) := T(V)/I. \]

14.2. Exterior Power

  • \(k\)th exterior power of \(V\) is the vector subspace of the exterior algebra with grade \(k\) is \[ \bigwedge\nolimits^k V. \]

14.3. Exterior Product

14.3.1. Of Tensors

  • For \(\alpha, \beta \in \bigwedge (V)\), \[ \alpha \wedge \beta = [\alpha \otimes \beta] \]
  • where \([\ \cdot\ ]\) denotes the equivalence classes with respect to \(I\).

14.3.2. Of Differential Forms

14.4. Grassmann Algebra

  • \[ \bigwedge V^* \]
  • \(\mathrm{Gr}(M)\) can be thought of as the space of all on \(M\),
    • \[ \Omega(M) := \bigoplus_{k = 0}^{\dim M} \Omega^k(M) \]
  • equipped with addition \(+\), the direct sum on elements, and the scalar multiplication \(\cdot\), and the bilinear map \(\wedge\) that is a linear continuation of : \(\mathrm{Gr}(M) = (\Omega(M), +, \cdot, \wedge)\).

15. Symmetric Algebra

15.1. Symmetric Power

Quotient of the \(n\)-fold Cartesian product \(X^n\) by the permutation action of the \(S_n\).

15.2. Definition

Direct sum of symmetric powers \(S^n(V)\) of a vector space \(V\): \[ S(V) := \bigoplus_{n=0}^\infty S^n(V). \]

16. Clifford Algebra

16.1. Definition

\[ \mathrm{Cl}(V, Q) := T(V)/\langle v\otimes v - Q(v)1\rangle \] where \( T(V) \) is the tensor algebra on a vector space \( V \) over a field \( K \), and \( Q \) is a quadratic form, \( 1 \in T(V) \) is the multiplicative identity. If \( Q \) is nondegenerate, it can be identified as \( \mathrm{Cl}_{p,q}(\mathbb{R}) \) with \( p \) elements that squares to \( 1 \), and \( q \) elements that squares to \( -1 \), for an orthogonal basis of \( V \).

16.2. Bilinear Form

If 2 is invertible in \(K\), the fundamental identity yields: \[ uv + vu = Q(u+v) - Q(u) - Q(v) =: 2\langle u, v\rangle 1 \] forming a symmetric bilinear form associated with \(Q\), via the polarization identity.

16.3. Exterior Algebra

If \(K\) does not have characteristic 2, then there is a natural isomorphism between \(\bigwedge V\) and \(\mathrm{Cl}(V,Q)\), which is not an algebra isomorphism unless \(Q\equiv 0\).

16.4. Antiautomorphisms

Grade Reflection Transposition Clifford Conjugation
0 +    
1 -    
2 +    
3 -    

16.4.1. Reflection

  • Grade Involution

The ideal is invariant under the linear map on \(V\) defined by \(v\mapsto v\), therefore it extends to an algebra automorphism \(\alpha\). It changes signs based on the grade.

16.4.2. Transpose

  • Reversal

The antiautomorphism of the tensor algebra: \[ v_1\otimes v_2 \otimes \cdots \otimes v_k \mapsto v_k\otimes \cdots \otimes v_2\otimes v_1 \] descends to an antiautomorphism of Clifford algebra called the transpose operation, denoted by \(x^\mathrm{t}\).

16.4.3. Clifford Conjugation

  • Composition of two involutions.
  • \(\bar{x} := \alpha(x^\mathrm{t}) = \alpha(x)^\mathrm{t}\)

16.5. Clifford Scalar Product

If the characteristic of \(K\) is not 2, the quadratic form \(Q\) on \(V\) can be extended to a quadratic form on all of \(\mathrm{Cl}(V, Q)\): \[ Q(x) := \langle x^\mathrm{t}x\rangle_0. \] It satisfies \[ Q(v_1v_2\cdots v_k) = Q(v_1)Q(v_2)\cdots Q(v_k). \] Note that this identity is not true for arbitrary elements, except \(v_i \in V\).

The associated symmetric bilinear form on \(\mathrm{Cl}(V, Q)\) is given by \[ \langle x, y\rangle = \langle x^\mathrm{t}y\rangle_0. \]

16.6. Twisted Conjugation

\[ \alpha(U)VU^{-1} \] where \(\alpha\) is the grade involution.

17. Exact Sequence

17.1. Definition

  • Sequence of morphisms between objects
    • \[ A_1 \xrightarrow{f_1} A_2 \xrightarrow{f_2} \cdots \xrightarrow{f_n} A_n \]
  • such that it is exact at each \(A_i\), that is, \(\operatorname{im}(f_i) = \ker(f_{i+1})\).

17.2. Properties

  • For the sequence of group homomorphisms,
    • \(0\to A\stackrel{f}{\to} B\) is exact, if and only if \(f\) is a monomorphism.
    • \(B\stackrel{g}{\to} C\to 0\) is exact, if and only if \(g\) is a epimorphism.

17.3. Short Exact Sequence

  • Exact sequence of the form
  • \[ 0 \longrightarrow A\xrightarrow{f} B\xrightarrow{g} C \longrightarrow 0. \]

17.3.1. Properties

  • Intuitively: \(A\) is a subobject of \(B\) and \(C\) is the correspoding factor object \(B/A\)
    • \[ C \cong B/\operatorname{im}(f) = B/\ker(g) \]

17.3.2. Split Exact Sequence

  • A short exact sequence is called split if there exists a homomorphism \(h\colon C\to B\) such that \(g\circ h = \mathrm{id}_C\).
  • If the objects are abelian groups, \(B \cong A\oplus C\).

17.4. Splitting Lemma

  • In any abelian category, for a short exact sequence \[ 0 \longrightarrow A\xrightarrow{q} B\xrightarrow{r} C \longrightarrow 0, \]
  • the following statements are equivalent:
    1. Left Split: \(\exists t\colon B\to A, t\circ q = \mathrm{id}_A\)
    2. Right Split: \(\exists u\colon C\to B, r\circ u = \mathrm{id}_C\)
    3. Direct Sum: logseq.order-list-type:: number
      • \(\exists h\colon B \stackrel{\sim}{\to} A\oplus C, (h\circ q: A \hookrightarrow_\natural A\oplus C) \land (r\circ h^{-1}\colon A\oplus C \twoheadrightarrow_\natural C)\)

18. References

Footnotes:

1

Quadratic form - Wikipedia

Author: Jeemin Kim

Created: 2026-07-12 Sun 14:29