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(2006-05-07)   Etymology
Vectors were so named because they "carry" the distance from the origin.

In medical and other contexts, "vector" is synonymous with "carrier".  The etymology is that of "vehicle":  The latin verb  vehere  means "to transport".

In elementary geometry, a  vector  is the  difference  between two points in space;  it's what has to be traveled to go from a given origin to a destination.  Etymologically, such a thing was perceived as "carrying" the distance between two points  (the  radius  from a fixed origin to a point).

The term  vector  started out its mathematical life as part of the French locution  "rayon vecteur"  (radius vector).  The whole expression is still used to identify a point in ordinary (Euclidean) space, as seen from a fixed origin.

As presented next, the term  vector  more generally denotes an element of a linear space  (vector space)  of unspecified  dimensionality  (possibly infinite dimension)  over any  scalar field  (not just the real numbers).

(2006-03-28)   Vector Space over a Field  K   (called the  ground field).
Vectors can be added, subtracted or  scaled.  The scalars form a field.

The vocabulary is standard:  An element of the  field  K  is called a  scalar.  Elements of the  vector space  are called  vectors.

By definition,  a  vector space  E  is a set with a well-defined internal addition  (the sum  U+V of two vectors is a vector)  and a well-defined external multiplication  (i.e.,  for a scalar  x  and a vector  U,  the scaled vector  x U   is a vector)  with the following properties:

• (E, + )  is an Abelian group.  This is to say that the addition of vectors is an associative and commutative operation and that subtraction is defined as well  (i.e.,  there's a zero vector, neutral for addition, and every vector has an  opposite  which yields  zero  when added to it).
• Scaling is compatible with arithmetic on the  field  K :

"xÎK, "yÎK, "UÎE, "VÎE,

(x + y) U   =   x (U + V)   =   (x y) U   =   1 U   =

x U  +  y U x U  +  x V x (y U) U

(2010-04-23)   Independent vectors.  Basis of a vector space.
The dimension is the largest possible number of independent vectors.

The modern definition of a vector space doesn't involve the concept of  dimension  which had a towering presence in the historical examples of vector spaces taken from Euclidean geometry:  A line has dimension 1, a plane has dimension 2, "space" has dimension 3, etc.

The concept of dimension is best retrieved by introducing two complementary notions pertaining to a set of vectors  B. 

• B  is said to consist of  independent vectors  when all  nontrivial linear combinations  of the vectors of  B  are nonzero.
• B  is said to  generate  E  when every vector of the space E is a linear combination of vectors of B.

A  linear combination  of vectors is a sum of  finitely many  of those vectors, each multiplied by a scaling factor  (called a  coefficient ).  A linear combination with at least one  nonzero  coefficient is said to be  nontrivial.

If  B  generates  E  and  consists of independent vectors,  then it's called a  basis  of E.   Note that the trivial space  {0}  has an empty basis  (the empty set does generate the space  {0}  because an empty sum is zero).

To prove that all nontrivial vector spaces have a basis requires the  Axiom of Choice  (in fact, the existence of a basis for any nontrivial vector space is  equivalent  to the  Axiom of choice ).

HINT:  Tuckey's lemma  (which is equivalent to the axiom of choice)  says that there's always at least one maximal set in a family "of finite character".  That notion is patterned after the family of all the linearly independent sets in a vector space  (which, indeed, contains a set if and only if it contains all the finite subset of that set).  A basis is a maximal set of linearly independent vectors.

Dimension theorem for vector spaces :

A not-so-obvious statement is that two bases of  E  can always be put in one-to-one correspondence with each other.  Thus,  all bases of  E  have the same  cardinal  (finite or not).  That cardinal is called the  dimension  of the space E  and it is denoted  dim (E)

Basis   |   Hamel basis (¹ Schauder basis)   |   Georg Hamel (1877-1954; Ph.D. 1901)
 
Linearly independent sets   |   Matroids   |   Matroid representations   |   Dimension theorem for vector spaces

(2010-06-21)   Intersection and Sum of Subspaces.
A vector space included in another is called a  subspace.

A subset  F  of a vector space  E  is a  subspace  of  E  if and only if it is stable by addition and scaling  (i.e., the sum of two vectors of  F  is in  F  and so is any vector of  F  multiplied into a scalar).

It's an easy exercise to show that the intersection  FÇG  of two subspaces  F  and  G  is a subspace of  E.  So is the  Minkowski sum  F+G  (defined as the set of all sums  x+y  of a vector  x  from  F  and a vector  y  from  G).

Two subspaces of  E  for which   FÇG = {0}   and   F+G = E  are said to be  supplementary.  Their sum is then called a  direct sum  and the following compact notation is used to state that fact:

E   =   F Å G

In the case of finitely many dimensions, the following relation holds:

dim ( F Å G )   =   dim (F)  +  dim (G)

The generalization to nontrivial intersections is  Grassmann's Formula :

dim ( F + G )   =   dim (F)  +  dim (G)  -  dim ( F Ç G )

A lesser-known version applies to spaces of finite codimensions:

codim ( F + G )   =   codim (F)  +  codim (G)  -  codim ( F Ç G )

Hermann Grassmann  (1809-1877)

(2010-12-03)  Linear maps.  Isomorphic vector spaces.
Two spaces are  isomorphic  if there's a linear bijection between them.

A function  f  which maps a vector space  E  into another space  F  over the same field  K  is said to be  linear  if it respects addition and scaling:

" x,y Î K,  " U,V Î E,     f ( x U + y V )   =   x f ( U )  +  y f ( V )

If such a linear function  f  is  bijective, its inverse is also a linear map and the vector spaces  E  and  F  are said to be  isomorphic. 

E   »   F

In particular, two vector spaces which have the same  finite  dimension over the same field are necessarily isomorphic.

(2010-12-03)   Quotient  E/H  of a vector space  E  by a subspace  H :
The equivalence classes (or residues) modulo  H  can be called  slices.

If  H  is a subspace of the vector space  E,  an equivalence relation can be defined by calling two vectors equivalent when their  difference  is in  H.

An equivalence class is called a  slice,  and it can be expressed as a Minkowski sum of the form  x+H.  The space  E  is thus partitioned into slices parallel to  H.  The  set  of all slices is denoted  E/H  and is called the  quotient  of E  by H.  It's clearly a vector space  (scaling a slice or adding up two slices yields a slice).

When  E/H  has  finite dimension  that dimension is called the  codimension  of  H.  A linear subspace of  codimension 1  is called an  hyperplane  of  E.

x+H  denotes the set of all sums  x+h  where  h  is an element of  H.  E/H  is indeed the quotient of  E  modulo the equivalence relation which defines as equivalent two vectors whose difference is in  H.

The canonical linear map which sends a vector  x  of  E  to the slice  x+H  is called the  quotient map  of  E  onto  E/H.

A vector space is always isomorphic to the direct sum of any subspace  H  and its quotient by that same subspace:

E   »   H Å E/H

Use this with  H  =  ker ( f )   to prove the following fundamental theorem:

(2010-12-03)  Fundamental theorem of linear algebra.  Rank theorem.
A vector space is isomorphic to the direct sum of the image and kernel  (French:  noyau)  of any linear function defined over it.

The  image  or  range  of a linear function  f  which maps a vector space  E  to a vector space  F  is a subspace of  F  defined as follows:

im ( f )   =   range ( f )   =   f (E)   =   { y Î F  |   $ x Î E ,  f (x) = y }

The  kernel  (also called  nullspace)  of  f  is the following subspace of  E :

ker ( f )  =  null ( f )  =  { x Î E  |   f (x) = 0 }

The  fundamental theorem of linear algebra  states that there's a subspace of  E  which is isomorphic to  f (E)  and  supplementary  to  ker ( f )  in E.  This results holds for a finite or an infinite number of dimensions and it's commonly expressed by the following isomorphism:

f (E)  Å  ker ( f )   »   E

Proof :   This is a corollary of the above, since  f (E)   and   E / ker ( f )   are isomorphic because a bijective map between them is obtained by associating  uniquely  f (x)  with the residue class  x + ker ( f ).  Clearly, that association doesn't depend on the choice of  x. 

The above argument is itself an incarnation of the so-called first isomorphism theorem, as published by Emmy Noether in 1927.

Restricted to vector spaces of finitely many dimensions, the theorem amounts to the following  famous result  (of great practical importance).

Rank theorem  (or rank-nullity theorem) :

For any linear function  f  over a finite-dimensional space  E, we have:

dim ( f (E) )  +  dim ( ker ( f ) )   =   dim ( E )

dim ( f (E) )  is called the  rank  of f.  The  nullity  of  f  is  dim ( ker ( f ) ).

In the language of the matrices normally associated with linear functions:  The  rank  and  nullity  of a matrix add up to its number of  columns.  The rank of a matrix  A  is defined as the largest number of linearly independent columns  (or rows)  in it.  Its nullity is the dimension of its nullspace  (consisting, by definition, of the column vectors  x  for which  A x = 0).

Wikipedia :   Fundamental theorem of linear algebra   |   Rank theorem   |   Rank of a matrix

(2007-11-06)   Normed Vector Spaces
A  normed vector space  is a linear space endowed with a  norm.

Vector spaces can be endowed with a function  (called  norm)  which associates to any vector  V  a real number  ||V||  (called the  norm  or the  length  of  V)  such that the following properties hold:

• ||V||  is positive for any nonzero vector  V.
• ||lV||  =  |l| ||V||
• || U + V ||   ≤   || U ||  +  || V ||

In this,  l  is a  scalar  and  |l|  denotes what's called a  valuation  on the field of scalars  (a valuation is a special type of one-dimensional norm; the valuation of a product is the product of the valuations of its factors).  Some examples of valuations are the  absolute value  of real numbers,  the  modulus  of complex numbers and the  p-adic metric  of p-adic numbers.

Let's insist:  The norm of a nonzero vector is always a  positive real number,  even for vector spaces whose scalars aren't real numbers.

(2020-10-09)   Non-singular bilinear form   f  on  E×E
v |® [ x |® f (x,v) ]  is an  isomorphism  from  E  to  E*.

In this  E*  is understood to be the  continuous  dual of  E*.

In finitely many dimensions,  a pseudo inner product is non-degenerate if and only if its associated  determinant  is nonzero.

Wikipedia :   Pseudo-Euclidean space   |   Nondegenerate form   |   Degenerate bilinear form

(2020-09-30)   Complex Inner-Product Spaces
Linear space endowed with a positive-definite sesquilinear form.

Parallelogram identity :

In an inner-product space, the following identity holds which reduces to Pythagoras theorem when ||u+v|| = ||u-v||

||u + v||²  +  ||u - v||²   =   2 ||u||²  +  2 ||v||²

Polarization Identity :

Conversely,  a norm which verifies the above  parallelogram identity  on a complex linear space is necessarily derived from a sesquilinear inner product,  obtained from the norm alone through the following  polarization formula:

<u,v>   =   ¼ ( ||u + v||  -  ||u - v||  +  i ||u - i v||  -  i ||u + i v|| )

A  Hilbert space  is an inner-product space which is  complete  with respect to the norm associated to the defining inner-product  (i.e.,  it's a  Banach space  with respect to that norm).  The name  Hilbert space  itself was coined by  John von Neumann (1903-1957)  in honor of the pioneering work published by David Hilbert (1862-1943).  on the  Lebesgue sequence space  ₂  (which is indeed a Hilbert space).

Parallelogram identity
 
: Separable Hilbert spaces (1:48:27)  by  Frederic Schuller  (L03, 2016-03-12).
 
: Hilbert spaces (50:17, 55:54)  by  Ole Christensen  (2013-02-20).
 
Numericana :   Hahn-Bahnach theorem on a complex or quaternionic space

(2009-09-03)   Two Flavors of Duality
Algebraic duality  & topological duality.

In a  vector space  E,  a  linear form  is a linear function which maps every  vector  of  E  to a  scalar  of the underlying field  K.  The set of all linear forms is called the  algebraic dual  of  E.  The set of all  continuous  linear forms is called the  [ topological ] dual  of  E.

With finitely many dimensions, the two concepts are identical  (i.e., every linear form is continuous).  Not so with infinitely many dimensions.  An element of the dual  (a continuous linear form)  is often called a  covector.

Unless otherwise specified, we shall use the unqualified term  dual  to denote the  topological dual.  We shall denote it  E*  (some authors use  E*  to denote the algebraic dual and  E'  for the topological dual).

The  bidual  E**  of  E  is the dual of  E*.  It's also called  second dual  or  double dual.

A  canonical  injective  homomorphism  F  exists which immerses  E  into  E**  by defining  F(v),  for any element  v  of  E,  as the linear form on  E*  which maps every element  f  of  E*  to the scalar  f (v).  That's to say:

F(v) ( f )   =   f (v)

If the canonical homomorphism is a bijection, then  E  is said to be  reflexive  and it is routinely identified with its  bidual  E**.

E   =   E**

If  E  has infinitely many dimensions, its  algebraic  bidual is  never  isomorphic to it.  That's the main reason why the notion of  topological  duality is preferred.  (Note that a Hilbert space is always reflexive in the above sense, even if it has infinitely many dimensions.)

Example of an algebraic dual :

If   E = ⁽⁾   denotes the space consisting of all complex sequences  with only finitely many nonzero values,  then the  algebraic  dual of  E  consists of  all  complex sequences without restrictions.  In other words:

E'   =  

Indeed, an element  f  of  E'  is a linear form over  E  which is uniquely determined by the unrestricted sequence of scalars formed by the images of the elements in the  countable  basis  (e₀ , e₁ , e₂  ... )  of  E.

E'  is a Banach space, but  E  is not  (it isn't complete).  As a result, an  absolutely convergent series  need not converge in  E.

For example, the series of general term   e_n / (n+1)²   doesn't converge in  E,  although it's absolutely convergent  (because the series formed by the norms of its terms is a well-known convergent real series).

Representation theorems for  [continuous]  duals :

A  representation theorem  is a statement that identifies in concrete terms some abstractly specified entity.  For example, the celebrated  Riesz representation theorem  states that the [continuous] dual of the  Lebesgue space  L^p  (an abstract specification)  is just isomorphic to the space  L^q  where  q  is a simple function of  p   (namely, 1/p+1/q = 1 ).

Lebesgue spaces are usually linear spaces with  uncountably many  dimensions  (their elements are functions over a continuum like  or ).  However, the  Lebesgue sequence spaces  described in the next section are simpler  (they have only  countably many  dimensions)  and can serve as a more accessible example.

Dual space

(2012-09-19)   Lebesgue sequence spaces
_p  and  _q  are duals of each other when   1/p + 1/q  =  1

For  p > 1,  the linear space  _p  is defined as the subspace of    consisting of all sequences for which the following series converges:

( || x || _p ) ^p   =   ( || (x₀ , x₁ , x₂ , x₃ , ... ) ||_p ) ^p   =   S_n  | x_n |^p

As the notation implies,  ||.||_p  is a  norm  on  _p  because of the following famous nontrivial inequality  (Minkowski's inequality)  which serves as the relevant  triangular inequality :

|| x+y ||_p     ≤     || x ||_p  +  || y ||_p

For the topology induced by that so-called  "p-norm",  the [topological] dual of  _p  is isomorphic to  _q ,  where:

1/p  +  1/q   =   1

Thus,  _p  is  reflexive  (i.e., isomorphic to its own bidual)  for any  p > 1.

Riesz representation theorem

(2009-09-03)   Tensorial Product and Tensors
E Ä F   is  generated  by tensor products.

Consider two vector spaces  E  and  F  over the  same  field of scalars  K.  For two  covectors  f  and  g  (respectively belonging to  E*  and  F*)  we may consider a particular  [continuous]  linear form denoted  f Ä g  and defined over the  cartesian product  E´F  via the relation:

f Ä g (u,v)   =   f (u)  g (v)

The binary operator  Ä  thus defined from  (E*)´(F*)  to  (E´F)*  is called  tensor product.  (Even when  E = F,  the operator  Ä  is  not  commutative.)

Example of Use:  The Dehn invariant  (1901)

In the Euclidean plane,  two simple polygonal loops which enlose the same  area  are always  decomposable  into each other.  That's to say,  with a finite number of  straigth cuts,  we can obtain pieces of one shape which are pairwise congruent to the pieces of the other  (William Wallace, 1807.  (Paul Gerwien, 1833.  (Farkas Bolyai, 1835.)

Hilbert's  third problem  (the equidecomposability problem, 1898)  asked whether the same is true for any pair of  polyhedra  having the same volume.

Surprisingly enough,  that's not so because  volume  is not the only invariant preserved by straight cuts in three dimensions.  The other invariant  (there are only two)  is now known as  Dehn's invariant,  in honor of Max Dehn (1878-1952)  the doctoral student of Hilbert who based his  habilitation thesis  on its discovery  (September 1901).  Here's a description:

Numericana :   Tensors
 
Max Dehn (1878-1952)   |   The Dehn Invariant (15:32)  by  Daniel Litt  (Numberphile, 2019-07-14).
 
Ars combinatoria  ou le triomphe du symbolisme  by  Pierre Cartier   |   Gian-Carlo Rota (1932-1999)
 
Gian Carlo Rota:  Matroïdes & géométrie combinatoire (1:05:05)  by  Pierre Cartier  (2014-05-06 15:00)

(2015-02-21)   Graded vector spaces and supervectors.
Direct sum of vector spaces indexed by a monoid.

When the indexing monoid is the set of natural integers  {0,1,2,3,4...}  or part thereof, the  degree  n  of a vector is the  smallest  integer such that the direct sum of the subfamily indexed by  {0,1 ... n}  contains that vector.

Grading   |   Graded vector space   |   Graded ring   |   Superalgebra   |   Graded Lie algebra

(2007-04-30)   [ Linear ]  Algebras over a Field  K
An internal product among vectors turns a  vector space  into an algebra.

They're also called  distributive algebras  because of a mandatory property they share with  rings.  Any ring is trivially a  1-dimensional  associative  algebra over the boolean field  F₂.  Any associative algebra is a ring.

Rings are always postulated to be associative.  Algebras need not be.

An  algebra  is the structure obtained when an internal binary multiplication is well-defined on the vector space  E  (the product of two vectors is a vector)  which is  bilinear  and  distributive over addition.  That's to say:

"xÎK, "UÎE, "VÎE, "WÎE,

x (U V)   =   U (V + W)   =   (V + W) U   =

(x U) V   =   U (x V) U V + U W V U + W U

The  commutator  is the following bilinear function.  If it's identically zero,  the algebra is said to be  commutative.

[ U , V ]   =   (U V)  -  (V U)

The lesser-used  anticommutator  is the following bilinear function.  If it's identically zero,  the algebra is said to be  anticommutative.

{ U , V }   =   (U V)  +  (V U)

The commutator is anticommutative.  The anticommutator is commutative.

The  associator  is defined as the following trilinear function.  It measures how the internal muliplication fails to be associative.

[ U , V , W ]   =   U (V W)  -  (U V) W

If its internal product has a neutral element, the algebra is called  unital :

$ 1 ,   " U ,       1  U   =   U  1   =   U

By definition,  a  derivation  in an algebra is a vectorial endomorphism  D  (i.e,  D  is a linear operator)  which obeys the following relation:

D ( U V )   =   D(U) V  +  U D(V)

One nontrivial example of a derivation of some historical importance is the  Dirac operator.  The derivations over an algebra form the  Lie algebra of derivations,  where the product of two derivations is defined as their  commutator.  One important example of that is the  Witt algebra,  introduced in 1909 by  Elie Cartan (1869-1951)  and studied at length in the 1930s by  Ernst Witt (1911-1991).  The Witt algebra is the Lie algebra of the derivation on the Laurent  polynomials with complex coefficients  (which may be viewed as the polynomials of two complex variables X any Y when XY = 1,  namely  [z,1/z] ).

The Witt Algebra? (36:36)  by  Michael Penn  (2022-02-02).

Associative Algebras :

When the associator defined above is identically zero, the algebra is said to be  associative  (many authors often use the word  "algebra"  to denote only  associative  algebras,  including Clifford algebras).  In other words,  associative algebras  fulfill the additional requirement:

"UÎE, "VÎE, "WÎE,       U (V W)   =   (U V) W

Distributive Algebras  (a.k.a.  nonassociative  algebras) :

In a context where distributivity is a mandatory property of algebras, that locution merely denote algebras which may or may not be associative:  A good example of this early usage was given in 1942, by a top expert:

"Alternative Algebras over an Arbitrary Field"  by  Richard D. Schafer  (1918-2014).
Bulletin of the American Mathematical Society,  49,  8,  pp. 549-555  (August 1943).

That convention is made necessary by the fact that the unqualified word  algebra  is very often used to denote only  associative algebras. 

This obeys the usual  inclusive  terminology in mathematical discourse,  whereby the properties we specify do not preclude stronger ones.  Since all algebras are distributive, this specification is indeed a good way to stress that nothing special is assumed about multiplication;  neither associativity nor any of the weaker properties presented below. 

Unfortunately,  Dick Schafer  himself later recanted and introduced a distinction between  not associative  and  nonassociative  (no hyphen)  with the latter not precuding associativity.  He explains that in the opening lines of his reference book on the subject,  thus endowed with a catchy title:

"Introduction to Nonassociative Algebras"  (1961, AP 1966, 1994, Dover 1995, 2017).

I beg to differ.  Hyphenation is too fragile a distinction and a group of experts simply can't redefine the hyphenated term  non-associative.

Ignore  Wikipedia's advice:  Refrain from calling  non-associative  (hyphenated)  something which may well be associative!   

Therefore,  unless the full associativity of multiplication is taken for granted,  I'm using  only  the following set of  inclusive  locutions:

• Distributive algebra:  Advertising nothing beyond the bare minimum.
• Flexible algebra:  A weak property related to  subassociativity.
• Cube-associative algebra:  Cubes are well-defined;  x (x x)  =  (x x) x.
• Power-associative algebra:  A  low rung in the subassociativity ladder.
• Alternative algebra:  The highest form of proper subassociativity.
• Associative algebra:  Meeting the common claim of full associativity.

Let's examine all those lesser types of algebras.  strongest first:

Alternative Algebras :

In general, a multilinear function is said to be  alternating  if its sign changes when the arguments undergo an  odd  permutation.  An algebra is said to be  alternative  when the aforementioned  associator  is alternating.

The  alternativity  condition is satisfied if and only if two of the following statements hold  (the third one then follows) :

• "U "V       U (U V)   =   (U U) V       (Left alternativity.)
• "U "V       U (V V)   =   (U V) V       (Right alternativity.)
• "U "V       U (V U)   =   (U V) U       (Flexibility.)

Octonions are a non-associative example of such an  alternative algebra.

Power-Associative Algebras :

Power-associativity  states that the n-th power of any element is well-defined regardless of the order in which multiplications are performed:

U¹   =   U
U²   =   U U
U³   =   U U²   =   U² U
U⁴   =   U U³   =   U² U²   =   U³ U
...   ...
" i > 0   " j > 0       U^i+j   =   Uⁱ U^j

The number of ways to work out a product of  n  identical factors is equal to the  Catalan number  C(2n,n)/(n+1):  1, 1, 2, 5, 14, 42, 132... (A000108)
Power-associativity  means that, for any given n, all those ways yield the same result.  The following special case  (n=3)  is  not sufficient:

"U ,       U (U U)   =   (U U) U

That just says that  cubes  are well-defined.  The accepted term for that is  third-power associativity  (I call it  cubic associativity  or  cube-associativity  for short).  It's been put to good use at least once, in print:

"Power-associative Lie-admissible algebras"  by  Georgia M. Benkart (b. 1949)
Journal of Algebra,  90,  1,  pp. 37-58  (September 1984).

The Three Standard Types of Subassociativity :

By definition,  a  subalgebra  is an algebra contained in another  (the operations of the subalgebra being restrictions of the operations in the whole algebra).  Any intersection of subalgebras is a subalgebra.  The subalgebra  generated  by a subset is the intersection of all subalgebras containing it.

The above three types of subassociativity can be fully characterized in terms of the associativity of the subalgebras generated by  1,  2  or  3  elements:

• If all subalgebras generated by one element are associative,
  then the whole algebra is  power-associative.
• If all subalgebras generated by two elements are associative,
  then the whole algebra is  alternative  (theorem of Artin).
• If all subalgebras generated by three elements are associative,
  then the whole algebra is  associative  too.

Flexibility :

A product is said to be  flexible  when:

"U "V       U (V U)   =   (U V) U

In particular, commutative  or  anticommutative  products are flexible.  Flexibility is usually  not  considered a form of subassociativity because it doesn't fit into the neat classification of the previous section.

Flexibility is preserved by the  Cayley-Dickson construction.  Therefore, all hypercomplex multiplications are flexible  (Richard D. Schafer, 1954).  In particular,  the multiplication of  sedenions  is flexible  (but not alternative).

"On the Algebras Formed by the Cayley-Dickson Process"  by  Richard D. Schafer
American Journal of Mathematics,  76,  2,  pp. 435-446  (April 1954).

Likewise,  Okubo algebras  are flexible.  So are  Lie algebras.

In a flexible algebra, the right-power of an element is equal to the matching left-power, but that doesn't make powers well-defined.  A flexible algebra is cube-associative but not necessarily power-asspciative.  In particular, fourth powers need not be well-defined:

A (A A)   =   (A A) A   =   A³
A A³   =   A³ A     may differ from     A² A²

Wikipedia :   Flexible magmas  &  flexible algebras

Antiflexible Algebras:

An algebra is called  antiflexible  when its associator is rotation-invariant:

"U "V "W       [U, V, W]   =   [V, W, U]

An antiflexible ring need not be power-associative.

Alternative Algebras over an Arbitrary Field  (Ph.D., 1942)  by  Richard Donald Schafer  (1918-2014).
Subassociative Groupoids  by  Milton S. Braitt  &  Donald Silberger   (2006).
Examples of Algebras that aren't Power-Associative  (MathOverflow, 2011-11-04).
Jordan algebras (1933)   |   Albert algebra (1934)
Power-Associative Rings (1947)  by  Abraham Adrian Albert (1905-1972).
New Results on Power-Associative Algebras (Ph.D. 1952)  by  Louis A. Kokoris (1924-1980).
Antiflexible Algebras which are not Power-Associative  by  D.J. Rodabaugh   (1964).
Power-Associativity of Antiflexible Rings  by  Hasan A. Çelik  &  David L. Outcalt   (1975).

(2015-02-14)   Lie algebras over a Field  K
Anticommutative algebras obeying  Jacobi's identity.

Hermann Weyl (1885-1955) named those structures after the Norwegian mathematician Sophus Lie (1842-1899).

The basic  internal multiplication  in a Lie algebra is a bilinear operator denoted by a square bracket  (called a  Lie bracket )  which must be  anticommutative  and obey the so-called  Jacobi identity,  namely:

• [B,A]   =   - [A,B]
• [A,[B,C]] + [B,[C,A]] + [C,[A,B]]   =   0

Anticommutativity implies  [A,A]  =  0   only in the absence of  2-torsion.

Being anticommutative,  the  Lie bracket  is  flexible.  However,  it need not be  alternative  because its  associator  need not be  alternating:

[A,B,C]  =  [A,[B,C]] - [[A,B],C]  =  [A,[B,C]] + [C,[A,B]]  =  [[C,A],B]

The operator  d  defined by   d(x)  =  [A,x]   is a  derivation, since :

d ([B,C])  =  [A,[B,C]]  =  [[A,B],C] + [B,[A,C]]  =  [d (B),C] + [B,d (C)]

d  is called the  Lie-derivative along  A.

The  cross-product  gives ordinary 3D vectors a Lie algebra structure.

Representations of a Lie Algebra :

The  bracket notation  is compatible with the key example appearing in quantum mechanics, where the Lie bracket is obtained as the  commutator  over an ordinary linear algebra of linear operators with respect to the functional  composition  of operators,  namely:

[ U , V ]   =   U o V  -  V o U

In quantum mechanics, the relevant linear operators are  Hermitian  (they're called  observables).  To make the Lie bracket an  internal  product among those Hermitian operators, we have to multiply the right-hand-side of the above defining relation into the  imaginary unit  i  (or any real multiple thereof).  This slight complication is irrelevant to the theoretical discussion of the general case and we'll ignore it here.

If the Lie bracket is so defined, the  Jacobi identity  is a simple theorem, whose proof is straightforward;  just sum up the following three equations:

• [A,[B,C]]   =   A o (BoC - CoB)  -  (BoC - CoB) o A
• [B,[C,A]]   =   B o (CoA - AoC)  -  (CoA - AoC) o B
• [C,[A,B]]   =   C o (AoB - BoA)  -  (AoB - BoA) o C

Linearity makes "o" distribute over "-",  so the sum of the right-hand-sides sonsists of 12 terms,  where the 6 possible permutations of the operators each appear twice with opposite signs.  The whole sum is thus zero.  

Conversely, an anticommutative algebra obeying the Lie identity is said to have a  representation  in terms of linear operators if it's isomorphic to the Lie algebra formed by those operators when the bracket is defined as the commutator of operators.

The  universal enveloping algebra  of a given Lie algebra  L  is the most general  unital  associative  algebra containing all representations of  L.

Lie algebra  Lie(G)  associated to a Lie group  G :

Loosely speaking,  the Lie algebra associated to a Lie group is its  tangent space  at the origin  (about the identity).

Lie(G)  is formed by all the left-invariant vector-fields on G.

A  vector field  X on a Lie Group G is said to be invariant under left translations when

"gÎG, "hÎG,   (dl_g)_h (X_h)   =   X_gh

where l_g is the left-translation within G  (l_g(x) = g x)  and  dl_g  is its differential between tangent spaces.

Lie's third theorem  states that every real finite-dimensional Lie algebra is associated to some Lie group.  However,  there exist infinite-dimensional Lie algebras not associated to any Lie group.  Adrien Douady (1935-2007)  pointed out the  first example  one late evening after a  Bourbaki  meeting...

Douady's counter-example is known as Heisenberg's Lie algebra  (non-exponential Lie algebra)  and arises naturally in  quantum mechanics  to describe the motion of particles on a straight line by means of three operators  (X,  Y  and  Z)  acting on any square-integrable function  f  of a real variable  (which form an infinite-dimensional space):

(X f ) (x)  =  x f (x)         (Y f ) (x)  =  -i ¶_x f (x)         (Z f ) (x)  =  -i f (x)

They obey the following commutation relations:

[ X, Y ]   =   Z         [ Y, Z ]   =   0         [ X, Z ]   =   0

Thus,  Jacobi's identity holds :

[X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]]  =  [X,0] + [Y,0] + [Z,Z]  =  0

This cannot be realized as the tangent space of a connected three-dimensional Lie group, because the Lie algebra associated to any such group is either abelian or solvable, and the non-exponential Lie algebra is neither.

Semisimple Lie Algebras  &  Semisimple Lie Groups :

A simple Lie algebra is a non-abelian Lie algebra without any nonzero  proper  ideal.

A simple Lie group is a connected Lie group whose Lie algebra is simple.

A semisimple Lie algebra is a direct sum of simple Lie algebras.

A semisimple Lie group is a connected Lie group whose Lie algebra is semisimple.

Any Lie algebra  L  is the semidirect sum of its  radical  R (a maximal solvable ideal)  and a semisimple algebra  S.

Lie algebra   |   Free Lie algebra   |   Lie algebra of a Lie group   |   Lie algebra representation
 
Exponential map   |   Integral curve   |   Simple Lie algebra   |   Semi-Simplicity   |   Semisimple Lie algebra
 
Levi decomposition (1888/1905)   |   Wilhelm Killing (1847-1923)   |   Elie Cartan (1869-1951)   |   Eugenio Elia Levi (1883-1917)
 
Dynkin diagrams   |   Eugene Dynkin (1924-2014)
 
Derivation algebras   |   Joseph Ritt (1893-1951)
 
Different definitions for semisimple Lie group  StackExchange  (2015-03-07).
 
Lie Algebras and Homotopy Theory (1:00:58)  by  Jacob Lurie  (IAS, 2019-11-11).
 
Lie groups and their Lie algebras (1:43:11)  by  Frederic P. Schuller  (#13, 2015-09-21).
Classification of complex Lie algebras (1:46:42)  by  Frederic P. Schuller  (#14, 2015-09-21).
Recovering (some of) a Lie group from its algebra (44:09)  Frederic P. Schuller  (#18, 2015-09-21).
Lie algebras and their representations (1:00:29)  André Henriques  (2012-09-05).

(2023-04-08)   Leibniz algebras  (Blokh 1965,  Loday, 1993)
They're  Lie algebras  if the bracket is alternative  (i.e.,  [x,x] = 0).

Much of the literature about Leibniz algebras consist in pointing out that results known for Lie algebras are also true in this more general context.

Originally,  Leibniz algebras were called D-algebras.

That early name was coined in 1965 by Alexander Mikhailovich Blok (1946-1986)  in a paper named  "Cohomology in the algebra of exterior forms"  (1965).  He switched to the name "Leibniz algebras" in a paper published in 1977 under the title  "The variety of Leibniz algebras".  Blok  (often misspelled "Bloh")  received his doctorate in 1970 from  Leningrad State University  under  Dmitry B. Fuchs (1939-)  for a thesis entitled  "The cohomology of Lie superalgebras".

The "D" stood for derivation,  a name which identifies any operator  D  obeying the following  Leibniz law  of ordinary derivatives:

D (x y)   =   x D(y)  +  D(x) y

If  D  is the operator corresponding to multiplication to the right by some element  z,  that becomes the following substitute for associativity:

(x y) z   =   x (y z)  +  (x z) y

If brackets are used for multiplication,  as is common here, that reads:

(Right)  Leibniz Identity

[[x,y],z]   =   [[x,z],y]  +  [x,[y,z]]

Indeed,  Blokh (1965) and Loday (1993) defined a  Leibniz algebra  as a vector space  (or a  module)  with a bilinear bracket verifying that identity.

In the anticommutative case,  the Leibniz identity and Jacobi's identity are equivalent.  Thus a  Lie algebra  is just an  anticommutative Leibniz algebra.

The  Koszul duals  of Leibniz algebras are called  Zinbiel  ("Leibniz" backwards).  That was a pseudonym used by Loday who created a fictitious character by that name as a joke.

Leibniz algebra (A. M. Blok 1965 "D-algebras", J-L. Loday 1993)   |   Jean-Louis Loday  (1946-2012)

(2015-02-19)   Jordan algebras are commutative.
Turning any  linear algebra  A  into a commutative one  A⁺.

Those structures were introduced in 1933 by  Pascual Jordan (1902-1980).  They were named after him by A. Adrian Albert (1902-1980)  in 1946.  (Richard Schafer,  a former student of Albert, wouldn't use that name,  possibly because Jordan was a notorious  Nazi).

Just like commutators turn linear operators into a  Lie algebra,  a  Jordan algebra  is formed by using the anti-commutator  or  Jordan product :

U V   =   ½ ( U o V  +  V o U )

Axiomatically, a  Jordan algebra  is a commutative algebra  ( U V  =  V U )  obeying  Jordan's identity :   (U V) (U U)  =  U (V (U U)).

A Jordan algebra is always  power-associative.

Jordan algebra (1933)   |   Pascual Jordan (1902-1980)

(2019-02-27)   Albert Algebra :
A  27-dimensional  exceptional  Jordan algebra.

Albert algebra (1934)   |   "AAA" = A. Adrian Albert (1905-1972)

(2007-04-30)   Clifford algebras over a Field K
Unital associative algebras with a quadratic form.

Those structures are named after the British geometer and philosopher William Clifford (1845-1879) who originated the concept in 1876.

The first description of  Clifford algebras  centered on  quadratic forms  was given in 1945 by  a founder of the  Bourbaki  collaboration:  Claude Chevalley (1909-1984).

Clifford Algebras  by John Baez  (2001).
Introduction to Clifford Algebra  (Geometric Algebra)  by John S. Denker  (2006).
A brief Introduction to Clifford Algebra  by  Silvia Franchini, Giorgio Vassallo & Filippo Sorbello  (2010)
Clifford Algebra and Spinors  by  Ivan Todorov  (2011)
Clifford Algebra: A visual introduction  by  Steven Lehar  (2014-03-18)
 
The Vector Algebra War (13:37)  by  Derek Abbott & James Chappell,  University of Adelaide  (2016-01-26).
 
Wikipedia :   Geometric algebra   |   Quadratic forms   |   Clifford algebras
 
Classification of Clifford algebras   |   Symplectic Clifford algebra (Weyl algebra)
 
Bott periodicity theorem (1957-1959)   |   Raoul Harry Bott (1923-2005, Ph.D., 1949)

(2022-02-18)   Dual Numbers  (Clifford, 1873)
Commutative two-dimensional algebra of elements of the form  x + e y

The second component  (the "Fermionic" part)  has zero square:  e² = 0.

(a + b e) (c + d e)   =   ac  +  (ad + bc) e

Dual numbers
 
Strange cousins of the complex numbers (19:13)  by  Michael Penn  (2022-02-16).

(2023-03-16)   (16-dimensional)  Spacetime Algebra :
The algebra generated by Dirac's four gamma matrices  g^m  (1927).

That's  Cl (1,3)   =   Cl_1,3 (⁴ )  namely,  the  Clifford algebra  built over real four-dimensional space using Minkowski's relativistic metric  Q :

Q ( x₀ , x₁ , x₂ , x₃ )     =     (x₀ ) ²  -  (x₁ ) ²  -  (x₂ ) ²  -  (x₃ ) ²
 
Q ( ct, x, y, z)     =     c ² t ²  -  x ²  -  y ²  -  z ²      

The grade of a product is the sum of the grades of its  factors,  modulo  4.

In the above table,  the commuting products are highlighted in blue.  The other off-diagonal products are anticommuting.

There are two opposite ways to associate the letter "a" to the time coordinate (ct).  Likewise, we can assign 3 of the 4 remaining letters (b,c,d,e) to 3 given orthogonal axes in 120 nonoriented ways or 1920 oriented ones.  Thus, a spacetime reference frame can be labeled in 1920 equivalent ways.

Covariant Differential Operators

The simplest one was found by  Dirac  in 1927.  Its eigenvectors describes particles of spin  ½  and mass  m,  like the  electron.

... and then, you could liearize the sum of four squres.
Actually. you could even linearize the sums of five squares...
Paul Dirac (Accademia Dei Lincei, Rome 1975-04-15).

The 1920 automorphisms of the Gamma group   |   Spacetime algebra (STA)

(2017-12-01)   Exterior Algebra  (Grassmann Algebra, 1844)
The exterior product  (wedge product)  is anticommutative.

Hermann Grassmann (1809-1877)

(2015-02-23)   Involutive Algebras
A special linear involution is singled out  (adjunction or conjugation).

As the  adjoint  or  conjugate  of an element  U  is usually denoted  U*  such structures are also called  *-algebras  (star-algebras).  The following properties are postulated:

Wilipedia :   Involutive algebra

(2015-02-23)   Von Neumann Algebras
Compact operators resemble ancient infinitesimals...

John von Neumann (1903-1957)  introduced those structures in 1929  (he called them simply  rings of operators).  Von Neumann  presented their basic theory in 1936 with the help of Francis Murray (1911-1996).

A  Von Neumann algebra  is an  involutive algebra  such that...

By definition,  a  factor  is a Von Neumann algebra with a trivial center  (which is to say that only the scalar multiples of identity commute with all the elements of the algebra).

Wilipedia :   Von Neumann algebras

(2009-09-25)   On multi-dimensional objects that are "not vectors"...

To a mathematician, the  juxtaposition  (or  cartesian product )  of several vector spaces over the same field  K  is  always  a vector space over that field  (as component-wise definitions of addition and scaling satisfy the above axioms).

When physicists state that some particular juxtaposition of quantities  (possibly a single numerical quantity by itself)  is "not a scalar", "not a vector" or "not a tensor"  they mean that the thing lacks an unambiguous and intrinsic definition.

Typically, a flawed vectorial definition would actually depend on the choice of a frame of reference for the physical universe.  For example, the derivative of a scalar with respect to the first spatial coordinate is "not a scalar"  (that quantity depends on what spatial frame of reference is chosen).

Less trivially, the gradient of a scalar  is  a physical covector (of which the above happens to be  one  covariant coordinate).  Indeed, the definition of a gradient specifies the same object  (in dual space)  for any choice of a physical coordinate basis.

Some physicists routinely  introduce  (especially in the context of General Relativity)  vectors  as "things that transform like elementary displacements" and  covectors  as "things that transform like gradients".  Their students are thus expected to grasp a complicated notion  (coordinate transformations)  before the stage is set.  Newbies will need several passes through that intertwined logic before they "get it".

I'd rather introduce the mathematical notion of a vector first.  Having easily absorbed that straight notion, the student may then be asked to consider whether a particular definition depends on a choice of coordinates.

For example, the  linear coefficient of thermal expansion  CTE)  cannot be properly defined as a scalar  (except for isotropic substances)  it's a  tensor.  On the other hand, the related  cubic  CTE is always a scalar  (which is equal to the trace of the aforementioned CTE tensor).

(2007-08-21)   Geometric Calculus  of  David Hestenes
Unifying some notations of mathematical physics...

Observing similitudes in distinct areas of mathematical physics and building on groundwork from  Grassmann  (1844)  and  Clifford  (1876)  David Hestenes (1933-) has been advocating a denotational unification, which has garnered quite a few enthusiastic followers.

The approach is called  Geometric Algebra  by its proponents.  The central objects are called  multivectors.  Their coordinate-free manipulation goes by the name of  multivector calculus  or  geometric calculus,  a term which first appeared in the title of Hestenes' own doctoral dissertation (1963).

That's unrelated to the abstract field of  Algebraic Geometry  (which has been at the forefront of mainstream mathematical research for decades).

Geometric Algebra   |   multivector   |   bivector (pseudovector in 3D)   |   trivector (pseudoscalar in 3D)
 
Geometric Calculus  by  David Hestenes  (Oersted Medal Lecture, 2002)   |   Video (2014)
 
Geometric Algebra Research Group  (at the  Cavendish Laboratory ).
 
Excursions en algèbre géométrique  by  Georges Ringeisen.
 
Videos :   Geometric Algebra by  Alan Macdonald  0 | 1 | 2 | 3 | 4 | 5   (2015)
 
Tutorial on Geometric Calculus  by  David Hestenes  (La Rochelle, July 2012).
 
A new language for physics (2:05:15)  by  Anthony Lasenby  (Bivector, 2021-08-23).
 
Applications of Geometric Algebra to Engineering (50:48)  by  Joan Lasenby  (Y Combinator, 2018-09-17).