Abstract Algebra

Table of Contents

1. Algebraic Structure

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. Module

1.3.1.1. Definition
  • A module is a vector space over a ring.

1.3.2. Vector Space

See vector space for the definition.

1.3.2.1. Symplectic Vector Space
  • \((V, \omega)\)
1.3.2.1.1. Definition
1.3.2.1.2. Standard Symplectic Space
  • \(\mathbb{R}^{2n}\) with the symplectic form given by a nonsingular, skew-symmetric matrix, typically chosen to be: \[ \omega = \begin{bmatrix} 0 & I_n \\ - I_n & 0 \end{bmatrix}. \]
1.3.2.1.3. Symplectic Map
  • Linear map \(f\colon V\to W\) between symplectic vector spaces \((V, \omega), (W, \rho)\), that the pullback preserves the symplectic form: \[ f^*\rho = \omega. \]
1.3.2.1.4. 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 ((66806d1f-9ca5-4f9d-b965-15197d286334)) called the symplectic group, denoted \(\mathrm{Sp}(V)\) or \(\mathrm{Sp}(V, \omega)\).
  • Matrix form of symplectic transformations are symplectic matrices.
  • The
1.3.2.1.5. Symplectic Complement
  • For a linear subspace \(W \subset V\), the symplectic complement of \(W\) is: \[ W^\bot := \{v\in V\mid \forall w\in W, \omega(v,w) = 0\}. \]
1.3.2.1.6. 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.

1.3.3. Algebra

2. Topological Structure

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 ((65ce2b90-4dde-4a95-964c-268f8ca744d1))s 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 Structure

3.0.1. Measure Space

See measure space

4. Order Structure

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

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 Structure

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

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. Quotient

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

7.1. Quotient Group

7.2. Quotient Ring

7.3. 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:
      • \(\alpha [x] = [\alpha x],\)
      • \([x]+[y] = [x+y].\)

7.4. Quotient Space

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

8. Form

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

8.1. Bilinear Form

8.1.1. 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.

8.1.1.1. Symmetric

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

8.1.1.2. Skew-Symmetric

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

8.1.1.3. Altenating

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

8.2. Sesquilinear Form

  • sesqui-, one and a half.

8.2.1. Definition

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

8.2.2. 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.

8.3. Symplectic Form

8.3.1. 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.

8.3.2. Example

  • 2-Form

8.4. Quadratic Form

8.4.1. 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 \(q(a\mathbf{v}) = a^2q(\mathbf{v})\).

8.4.2. 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: \[ \mathbf{x}^{\rm T}\mathbf{A}\mathbf{x} = \mathbf{x}^{\rm T} Q\Lambda Q^{\rm T}\mathbf{x} = \tilde{\mathbf{x}}^{\rm T}\Lambda \tilde{\mathbf{x}} \]

9. Isomorphism

9.1. Definition

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

9.2. Canonical Isomorphism

  • The most trivial isomorphism between two structures.

9.3. Special Isomorphisms

9.4. Isomorphism Theorems for Groups

9.4.1. First

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

9.4.2. Second

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

9.4.3. Third

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

9.5. Isomorphism Theorems for Rings

9.5.1. 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 \(\psi(r+\operatorname{ker}\varphi) = \varphi(r)\).

10. Tensor

10.1. 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

10.2. 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.

10.3. 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 \(T\): \[ T \in \bigotimes_{i=1}^nV_i. \]

10.3.1. On a Vector Space

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

10.4. Type

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

10.5. Order

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

10.6. Rank

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

10.7. 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 \(W\).

10.8. (1, 1)-Tensor

10.8.1. Geometric Interpretation

10.8.2. Examples

10.9. Tensor Contraction

  • It is the generalization of trace
    • 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: \[ T(V_1, \dots, V_n) = C^n(T\otimes V_1\otimes \cdots \otimes V_n). \]

10.10. Tensor Field

10.10.1. 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 \(M\).

11. Tensor Algebra

11.1. Definition

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

11.1.1. 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.

11.2. Properties

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

12. Exterior Algebra

12.1. Definiton

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

12.2. Exterior Power

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

12.3. Exterior Product

12.3.1. Of Tensors

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

12.3.2. Of Differential Forms

12.4. Grassmann Algebra

  • \[ \bigwedge V^* \]
  • \(\mathrm{Gr}(M)\) can be thought of as the space of all forms 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 wedge product: \(\mathrm{Gr}(M) = (\Omega(M), +, \cdot, \wedge)\).

13. Exact Sequence

13.1. 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, \(\operatorname{im}(f_i) = \ker(f_{i+1})\).

13.2. 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.

13.3. Short Exact Sequence

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

13.3.1. Properties

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

13.3.2. 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, \(B \cong A\oplus C\).

13.4. 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
      • \(\exists h\colon B \stackrel{\sim}{\to} A\oplus C, (h\circ q: A \hookrightarrow_\natural A\oplus C) \land (r\circ h^{-1}\colon A\oplus C \twoheadrightarrow_\natural C)\)

14. References

Footnotes:

1

Quadratic form - Wikipedia

Created: 2025-05-13 Tue 02:28