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Abstract Algebra

1. Algebraic Structure

1.1. Group-Like

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,ω)
1.3.2.1.1. Definition
1.3.2.1.2. Standard Symplectic Space
  • R2n with the symplectic form given by a nonsingular, skew-symmetric matrix, typically chosen to be: ω=[0InIn0].
1.3.2.1.3. Symplectic Map
  • Linear map f:VW between symplectic vector spaces (V,ω),(W,ρ), that the pullback preserves the symplectic form: fρ=ω.
1.3.2.1.4. Symplectic Group
  • Symplectic map f:VV is called a linear symplectic transformation of V. It preserves the symplectic form ω(f(u),f(v))=ω(u,v).
  • The set of symplectic transformations forms a ((66806d1f-9ca5-4f9d-b965-15197d286334)) called the symplectic group, denoted Sp(V) or Sp(V,ω).
  • Matrix form of symplectic transformations are symplectic matrices.
  • The
1.3.2.1.5. Symplectic Complement
  • For a linear subspace WV, the symplectic complement of W is: W:={vVwW,ω(v,w)=0}.
1.3.2.1.6. Properties
  • (W)=W
  • dimW+dimW=dimV
  • Unlike orthogonal complements, WW need not be {0}, distinguished four cases are:
    • W symplectic: WW={0}
      • If and only if ω restricts to a nondegenerate form on W
      • Symplectic subspace with the restricted form is a symplectic vector space.
    • W isotropic: WW
      • If and only if ω restricts to 0 on W
      • Any one-dimensional subspace is isotropic.
    • W coisotropic: WW
      • If and only if ω descends to a nondegenerate form on the quotient space W/W
      • Equivalently, W is coisotropic if and only if W is isotropic.
      • Any codimension-one subspace is coisotropic
    • W Lagrangian: W=W
      • 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

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×MR is a metric on M with:
    • d(x,x)=0
    • Positivity: xyd(x,y)>0
    • Symmetry: d(x,y)=d(y,x)
    • Triangle Inequality: d(x,z)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 τ.

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):=infaAd(p,a)=0 where inf is the infimum.
  • Similarly, a set B is close to a set A, if: d(B,A):=infbBd(b,A)=0.

2.3. Normed Vector Space

  • Normed Space
  • (V,)

2.3.1. Definition

  • Vector space V over K on which a norm is defined, such that:
    • Non-Negativity: xV,x0
    • Positive Definiteness: xV,(x=0x=0)
    • Absolute Homogeneity: λK,xV,λx=|λ|x
    • Triangle Inequality: x,yV,x+yx+y

2.3.2. Properties

  • It is also a metric space with the metric d induced by the norm: d(x,y)=yx.
  • 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: x+y2=x2+y2+2Rx,y for the induced norm.
2.3.3.1. Theorem
  • Norm satisfies the parallelogram law, if and only if, there exists an inner product , such that x2=x,x for all x.
  • The map between the norm and the inner product is bijective.
2.3.3.2. For Real Vector Space
x,y=14(x+y2xy2)=12(x+y2x2y2)=12(x2+y2xy2)
  • These forms are related by the Parallelogram law.
2.3.3.3. For Complex Vector Space

By stipulating the properties of the inner product:

x|y=14(x+y2xy2ix+iy2+ixiy2)=R(x,y)iR(x,iy)=R(x,y)+iR(ix,y)

if antilinear in the first argument,

x,y=14(x+y2xy2+ix+iy2ixiy2)=R(x,y)+iR(x,iy)=R(x,y)iR(ix,y)

if antilinear in the second argument.

2.3.4. Examples

2.3.4.1. Lp Space
  • Lp Spaces, Lebesque Spaces
2.3.4.1.1. Definition
  • Function spaces defined using a generalization of the p-norm---Lp-norm.
  • It is the space of measurable functions for which the Lp-norm is defined---|f|p is Lebesgue integrable, modulo the equivalence relation fg:fgp=0.
2.3.4.1.1.1. p-Norm
  • For a real number p1: xp=(|x1|p+|x2|p++|xn|p)1/p.
2.3.4.1.1.2. Lp-Norm
  • fp=(S|f|pdμ)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:=sup{|f(x)|:xS}.
2.3.4.2. Sobolev Space
2.3.4.2.1. Definition
  • Wk,p(F)
  • It is the subset of Lp(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
  • fk,p=(i=0kf(i)pp)1p.

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,,)

2.5.1. Definition

  • Vector space V over the field F—which can be either R or C—with an inner product ,:V×VF satisfying:
    • Conjugate Symmetry: x,y=y,x
    • Linearity in the First Argument: ax+by,z=ax,z+by,z
    • Positive Definiteness: x0x,x>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: x=x,x.

2.5.3. Vectors

2.5.3.1. Null Vector
  • x,x=0
2.5.3.2. Degenerate Vector
  • yV,x,y=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. n=0xn=Ln=0xn=L
  • This is making sure that the limit exists.

2.6.2. Examples

2.6.2.1. L² Space

2.7. Affine Space

  • (A,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 A—isplacement vectors— and a transitive free action of the additive group of A on the set A, +.
  • The action +:A×AA satisfies:
    • Right Identity: aA,a+0=a.
    • Associativity: v,wA,aA,(a+v)+w=a+(v+w).
    • Free and Transitive action: aA,the mapping AA:va+v 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, Rn with all the nice structures.

2.8.1. Definition

2.8.1.1. Euclidean Vector Space
  • (Rn,)
  • 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 Rn such that the associated vector space is equipped with the standard dot product .

3. Measure Structure

3.0.1. Measure Space

4. Order Structure

4.1. Preorder

  • Quasiorder
  • ,

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 with symmetry axiom: ababab.
  • Similarly it induces an strict partial order < with anti-symmetry axiom: a<bab¬(ab).

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: abbaa=b.
      • Therefore, the order relation on 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 on X, that satisfies:
    • Reflexivity: aa.
    • Transitivity: abbcac.
    • Antisymmetry: abbaa=b.
    • Strongly Connected(Formerly, Total): abba.
  • The set (X,) 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: ¬(a<a).
  • Asymmetry: a<b¬(b<a).
  • Transitivity: a<bb<ca<c.
  • Connected: aba<bb<a.

4.4. Directed Set

4.4.1. Definition

  • Set with a preorder , 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

  • The tensor product vw is then defined as vw+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.
    • (wV,v,w=0)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, VW=spanR{abaV,bW} forms a vector space of dimension |V||W|.

6.2.2. Tensor Product

  • Vn: V tensor producted with itself n times.
  • Tensor product of two vector spaces VW is the quotient vector space VW/I with respect to the subspace I: I=spanR{(cv)wc(vw),v(cw)c(vw),(v1+v2)w(v1w+v2w),v(w1+w2)(vw1+vw2) cR, vV, wW}
6.2.2.1. Properties
  • dim(VW)=dimVdimW
  • abRR then (ca)b=c(ab)

6.2.3. Direct Product

  • ×
  • This is what is meant by Rn.
6.2.3.1. Properties
  • dim(V×W)=dimV+dimW
  • (a,b)R×R then (ca,cb)=c(a,b)

6.2.4. Direct Sum

  • The two sets become just one set.
  • Direct Product with inclusions defined.
6.2.4.1. Properties
  • dim(VW)=dimV+dimW
  • 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 : xyxyN where N is a linear subspace of V.
    • The equivalence class (or the coset) of x is denoted: [x]=x+N={x+n:nN}.
    • The scalar multiplication and addition are defined as:
      • α[x]=[αx],
      • [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×VK 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

  • φ(ax)=aφ(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 ω:V×VF that has following properties:
    • Bilinear
    • Alternating: vV,ω(v,v)=0
    • Non-degenerate: (vV,ω(v,u)=0)u=0
  • If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry: ω(v,u)=ω(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:VK such that q(av)=a2q(v).

8.4.2. Properties

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 φ:GH,
  • G/kerφH

9.4.2. Second

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

9.4.3. Third

  • For a subgroup HG, and a normal subgroup NG,
  • H/(HN)(HN)/N

9.5. Isomorphism Theorems for Rings

9.5.1. First

  • For a ring homomorphism φ:RS, there exists a unique isomorphism ψ:R/kerφimφ such that ψ(r+kerφ)=φ(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 V1,,Vn,W (or modules over a commutative ring) f:V1××VnW that is linear in each argument when other variables are held constant.

  • Multilinear map from Vq to Vp, corresponds to the (p,q)-tensor.

10.3. Definition

  • Given a finite set of vector spaces {Vi}i=1n over a common field F, the element of their tensor product is a tensor T: Ti=1nVi.

10.3.1. On a Vector Space

  • Tensor T of type (p,q) on a vector space V is defined as: TTqp(V):=Vp(V)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: Tqp(V)L(VVpVVq;F)Lp+q(V,,Vp,V,,Vq;F), when V is finite dimensional.
    • Ln(V1,,Vn;W) is denoting the space of n-linear maps from V1××Vn 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 (p1,q1)-tensor defined by the canonical pairing of kth vector space and lth dual vector space: C:i=1pvij=1qαjαk(vl)i=1,ikpvij=1,jlqαj
    • The application of tensor is also a contraction: T(V1,,Vn)=Cn(TV1Vn).

10.10. Tensor Field

10.10.1. Definition

  • A tensor field T of type (p,q) on a manifold M is: TΓ(M,Vp(V)q),
  • where V is a vector bundle on M.

11. Tensor Algebra

11.1. Definition

  • Tensor algebra (T(V),) of a vector space V over af field K is: TV=k=0TkV=k=0Vk.

11.1.1. Tensor Power

  • TkV is called the kth tensor power of V with T0V:=K, and the multiplication is the linear extension of canonical isomorphism: TkVTVTk+V.
    • Note that of T(V) is not a tensor product.

11.2. Properties

  • It is the functor form the category of vector spaces Vect to the category of algebras Alg
    • T:VectKAlg.

12. Exterior Algebra

12.1. Definiton

  • The quotient algebra of the tensor algebra T(V) with the ideal I={xx:xV}: (V):=T(V)/I.

12.2. Exterior Power

  • kth exterior power of V is the vector subspace of the exterior algebra with grade k is kV.

12.3. Exterior Product

12.3.1. Of Tensors

  • For α,β(V), αβ=[αβ]
  • where [  ] denotes the equivalence classes with respect to I.

12.3.2. Of Differential Forms

12.4. Grassmann Algebra

  • V
  • Gr(M) can be thought of as the space of all forms on M,
    • Ω(M):=k=0dimMΩk(M)
  • equipped with addition +, the direct sum on elements, and the scalar multiplication , and the bilinear map that is a linear continuation of wedge product: Gr(M)=(Ω(M),+,,).

13. Exact Sequence

13.1. Definition

  • Sequence of morphisms between objects
    • A1f1A2f2fnAn
  • such that it is exact at each Ai, that is, im(fi)=ker(fi+1).

13.2. Properties

  • For the sequence of group homomorphisms,

13.3. Short Exact Sequence

  • Exact sequence of the form
  • 0AfBgC0.

13.3.1. Properties

  • Intuitively: A is a subobject of B and C is the correspoding factor object B/A
    • CB/im(f)=B/ker(g)

13.3.2. Split Exact Sequence

  • A short exact sequence is called split if there exists a homomorphism h:CB such that gh=idC.
  • If the objects are abelian groups, BAC.

13.4. Splitting Lemma

  • In any abelian category, for a short exact sequence 0AqBrC0,
  • the following statements are equivalent:
    1. Left Split: t:BA,tq=idA
    2. Right Split: u:CB,ru=idC
    3. Direct Sum: logseq.order-list-type:: number
      • h:BAC,(hq:AAC)(rh1:ACC)

14. References

Footnotes:

Created: 2025-05-13 Tue 02:28