In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Clifford algebra is an
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
generated by a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
with a
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
, and is a
unital associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
. As
-algebras, they generalize the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s,
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s,
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s and several other
hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
s and
orthogonal transformations. Clifford algebras have important applications in a variety of fields including
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
,
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
and
digital image processing. They are named after the English mathematician
William Kingdon Clifford
William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his ...
.
The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (''pseudo-'')''Riemannian Clifford algebras'', as distinct from ''symplectic Clifford algebras''.
[see for ex. ]
Introduction and basic properties
A Clifford algebra is a
unital associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
that contains and is generated by a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, where is equipped with a
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
. The Clifford algebra is the "freest" unital associative algebra generated by subject to the condition
where the product on the left is that of the algebra, and the is its
multiplicative identity
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
. The idea of being the "freest" or "most general" algebra subject to this identity can be formally expressed through the notion of a
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
, as done
below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
Bottom may refer to:
Anatomy and sex
* Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
.
Where is a finite-dimensional real vector space and is
nondegenerate
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case.
T ...
, may be identified by the label , indicating that has an orthogonal basis with elements with , with , and where indicates that this is a Clifford algebra over the reals; i.e. coefficients of elements of the algebra are real numbers. This basis may be found by
orthogonal diagonalization In linear algebra, an orthogonal diagonalization of a symmetric matrix is a diagonalization by means of an orthogonal change of coordinates.
The following is an orthogonal diagonalization algorithm that diagonalizes a quadratic form ''q''(''x'') o ...
.
The
free algebra generated by may be written as the
tensor algebra , that is, the sum of the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
of copies of over all , and so a Clifford algebra would be the
quotient of this tensor algebra by the two-sided
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
generated by elements of the form for all elements . The product induced by the tensor product in the quotient algebra is written using juxtaposition (e.g. ). Its associativity follows from the associativity of the tensor product.
The Clifford algebra has a distinguished
subspace , being the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of the
embedding map. Such a subspace cannot in general be uniquely determined given only a -algebra
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the Clifford algebra.
If the
characteristic of the ground field is not , then one can rewrite the fundamental identity above in the form
where
is the
symmetric bilinear form associated with , via the
polarization identity
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product then t ...
.
Quadratic forms and Clifford algebras in characteristic form an exceptional case. In particular, if it is not true that a quadratic form uniquely determines a symmetric bilinear form satisfying , nor that every quadratic form admits an
orthogonal basis In mathematics, particularly linear algebra, an orthogonal basis for an inner product space V is a basis for V whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis ...
. Many of the statements in this article include the condition that the characteristic is not , and are false if this condition is removed.
As a quantization of the exterior algebra
Clifford algebras are closely related to
exterior algebras. Indeed, if then the Clifford algebra is just the exterior algebra . For nonzero there exists a canonical ''linear'' isomorphism between and whenever the ground field does not have characteristic two. That is, they are
naturally isomorphic
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
as vector spaces, but with different multiplications (in the case of characteristic two, they are still isomorphic as vector spaces, just not naturally). Clifford multiplication together with the distinguished subspace is strictly richer than the
exterior product since it makes use of the extra information provided by .
The Clifford algebra is a
filtered algebra, the
associated graded algebra In mathematics, the associated graded ring of a ring ''R'' with respect to a proper ideal ''I'' is the graded ring:
:\operatorname_I R = \oplus_^\infty I^n/I^.
Similarly, if ''M'' is a left ''R''-module, then the associated graded module is the grad ...
is the exterior algebra.
More precisely, Clifford algebras may be thought of as ''quantizations'' (cf.
quantum group
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras) ...
) of the exterior algebra, in the same way that the
Weyl algebra is a quantization of the
symmetric algebra.
Weyl algebras and Clifford algebras admit a further structure of a
*-algebra, and can be unified as even and odd terms of a
superalgebra, as discussed in
CCR and CAR algebras
In mathematics and physics CCR algebras (after canonical commutation relations) and CAR algebras (after canonical anticommutation relations) arise from the quantum mechanical study of bosons and fermions respectively. They play a prominent role i ...
.
Universal property and construction
Let be a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, and let be a
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
on . In most cases of interest the field is either the field of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s , or the field of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s , or a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
.
A Clifford algebra is a pair , where is a
unital associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
over and is a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
satisfying for all in , defined by the following
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
: given any unital associative algebra over and any linear map such that
(where denotes the multiplicative identity of ), there is a unique
algebra homomorphism
In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in ,
* F(kx) = kF(x) ...
such that the following diagram
commutes (i.e. such that ):
The quadratic form may be replaced by a (not necessarily symmetric)
bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
that has the property , in which case an equivalent requirement on is
When the characteristic of the field is not , this may be replaced by what is then an equivalent requirement,
where the bilinear form may additionally be restricted to being symmetric without loss of generality.
A Clifford algebra as described above always exists and can be constructed as follows: start with the most general algebra that contains , namely the
tensor algebra , and then enforce the fundamental identity by taking a suitable
quotient. In our case we want to take the two-sided
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
in generated by all elements of the form
for all
and define as the quotient algebra
The
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
product inherited by this quotient is sometimes referred to as the Clifford product to distinguish it from the exterior product and the scalar product.
It is then straightforward to show that contains and satisfies the above universal property, so that is unique up to a unique isomorphism; thus one speaks of "the" Clifford algebra . It also follows from this construction that is
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
. One usually drops the and considers as a
linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...
of .
The universal characterization of the Clifford algebra shows that the construction of is in nature. Namely, can be considered as a
functor from the
category of vector spaces with quadratic forms (whose
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s are linear maps preserving the quadratic form) to the category of associative algebras. The universal property guarantees that linear maps between vector spaces (preserving the quadratic form) extend uniquely to algebra homomorphisms between the associated Clifford algebras.
Basis and dimension
Since comes equipped with a quadratic form , in characteristic not equal to there exist
bases for that are
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
. An
orthogonal basis In mathematics, particularly linear algebra, an orthogonal basis for an inner product space V is a basis for V whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis ...
is one such that for a symmetric bilinear form
for
, and
The fundamental Clifford identity implies that for an orthogonal basis
for
, and
This makes manipulation of orthogonal basis vectors quite simple. Given a product
of ''distinct'' orthogonal basis vectors of , one can put them into a standard order while including an overall sign determined by the number of
pairwise swaps needed to do so (i.e. the
signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
of the ordering
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
).
If the
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of over is and is an orthogonal basis of , then is free over with a basis
The empty product () is defined as the multiplicative
identity element. For each value of there are
basis elements, so the total dimension of the Clifford algebra is
Examples: real and complex Clifford algebras
The most important Clifford algebras are those over
real and
complex vector spaces equipped with
nondegenerate quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
s.
Each of the algebras and is isomorphic to or , where is a
full matrix ring with entries from , , or . For a complete classification of these algebras see ''
Classification of Clifford algebras
In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified. In ea ...
''.
Real numbers
Clifford algebras are also sometimes referred to as
geometric algebra
In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ge ...
s, most often over the real numbers.
Every nondegenerate quadratic form on a finite-dimensional real vector space is equivalent to the standard diagonal form:
where is the dimension of the vector space. The pair of integers is called the
signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
of the quadratic form. The real vector space with this quadratic form is often denoted The Clifford algebra on is denoted The symbol means either or depending on whether the author prefers positive-definite or negative-definite spaces.
A standard
basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
*Basis trading, a trading strategy consisting of ...
for consists of mutually orthogonal vectors, of which square to +1 and of which square to −1. Of such a basis, the algebra will therefore have vectors that square to +1 and vectors that square to −1.
A few low-dimensional cases are:
* is naturally isomorphic to since there are no nonzero vectors.
* is a two-dimensional algebra generated by that squares to −1, and is algebra-isomorphic to , the field of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s.
* is a four-dimensional algebra spanned by The latter three elements all square to −1 and anticommute, and so the algebra is isomorphic to the
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s
* is an 8-dimensional algebra isomorphic to the
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
, the
split-biquaternion
In mathematics, a split-biquaternion is a hypercomplex number of the form
:q = w + xi + yj + zk
where ''w'', ''x'', ''y'', and ''z'' are split-complex numbers and i, j, and k multiply as in the quaternion group. Since each coefficient ''w'', ''x' ...
s.
Complex numbers
One can also study Clifford algebras on complex vector spaces. Every nondegenerate quadratic form on a complex vector space of dimension ''n'' is equivalent to the standard diagonal form
Thus, for each dimension , up to isomorphism there is only one Clifford algebra of a complex vector space with a nondegenerate quadratic form. We will denote the Clifford algebra on with the standard quadratic form by .
For the first few cases one finds that
*, the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s
*, the
bicomplex number
In abstract algebra, a bicomplex number is a pair of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate (w,z)^* = (w, -z), and the product of two bicomplex numbers as
:(u,v)(w,z) = (u w - v z, u z ...
s
*, the
biquaternions
where denotes the algebra of matrices over .
Examples: constructing quaternions and dual quaternions
Quaternions
In this section, Hamilton's
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s are constructed as the even sub algebra of the Clifford algebra
Let the vector space be real three-dimensional space and the quadratic form be the negative of the usual Euclidean metric. Then, for in we have the bilinear form (or scalar product)
Now introduce the Clifford product of vectors and given by
This formulation uses the negative sign so the correspondence with
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s is easily shown.
Denote a set of orthogonal unit vectors of as then the Clifford product yields the relations
and
The general element of the Clifford algebra is given by
The linear combination of the even degree elements of defines the even subalgebra with the general element
The basis elements can be identified with the quaternion basis elements as
which shows that the even subalgebra is Hamilton's real
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
algebra.
To see this, compute
and
Finally,
Dual quaternions
In this section,
dual quaternions are constructed as the even Clifford algebra of real four-dimensional space with a degenerate quadratic form.
Let the vector space be real four-dimensional space and let the quadratic form be a degenerate form derived from the Euclidean metric on For in introduce the degenerate bilinear form
This degenerate scalar product projects distance measurements in onto the hyperplane.
The Clifford product of vectors and is given by
Note the negative sign is introduced to simplify the correspondence with quaternions.
Denote a set of mutually orthogonal unit vectors of as then the Clifford product yields the relations
and
The general element of the Clifford algebra has 16 components. The linear combination of the even degree elements defines the even subalgebra with the general element
The basis elements can be identified with the quaternion basis elements and the dual unit as
This provides the correspondence of with
dual quaternion algebra.
To see this, compute
and
The exchanges of and alternate signs an even number of times, and show the dual unit commutes with the quaternion basis elements
Examples: in small dimension
Let be any field of characteristic not .
Dimension 1
For , if has diagonalization , that is there is a non-zero vector such that , then is algebra-isomorphic to a -algebra generated by an element satisfying , the quadratic algebra .
In particular, if (that is, is the zero quadratic form) then is algebra-isomorphic to the
dual numbers algebra over .
If is a non-zero square in , then .
Otherwise, is isomorphic to the quadratic field extension of .
Dimension 2
For , if has diagonalization with non-zero and (which always exists if is non-degenerate), then is isomorphic to a -algebra generated by elements and satisfying , and .
Thus is isomorphic to the (generalized)
quaternion algebra . We retrieve Hamilton's quaternions when , since .
As a special case, if some in satisfies , then .
Properties
Relation to the exterior algebra
Given a vector space , one can construct the
exterior algebra , whose definition is independent of any quadratic form on . It turns out that if does not have characteristic then there is a
natural isomorphism between and considered as vector spaces (and there exists an isomorphism in characteristic two, which may not be natural). This is an algebra isomorphism if and only if . One can thus consider the Clifford algebra as an enrichment (or more precisely, a quantization, cf. the Introduction) of the exterior algebra on with a multiplication that depends on (one can still define the exterior product independently of ).
The easiest way to establish the isomorphism is to choose an ''orthogonal'' basis for and extend it to a basis for as described
above. The map is determined by
Note that this only works if the basis is orthogonal. One can show that this map is independent of the choice of orthogonal basis and so gives a natural isomorphism.
If the
characteristic of is , one can also establish the isomorphism by antisymmetrizing. Define functions by
where the sum is taken over the
symmetric group on elements, . Since is
alternating it induces a unique linear map . The
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of these maps gives a linear map between and . This map can be shown to be a linear isomorphism, and it is natural.
A more sophisticated way to view the relationship is to construct a
filtration
Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...
on . Recall that the
tensor algebra has a natural filtration: , where contains sums of tensors with
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
. Projecting this down to the Clifford algebra gives a filtration on . The
associated graded algebra In mathematics, the associated graded ring of a ring ''R'' with respect to a proper ideal ''I'' is the graded ring:
:\operatorname_I R = \oplus_^\infty I^n/I^.
Similarly, if ''M'' is a left ''R''-module, then the associated graded module is the grad ...
is naturally isomorphic to the exterior algebra . Since the associated graded algebra of a filtered algebra is always isomorphic to the filtered algebra as filtered vector spaces (by choosing complements of in for all ), this provides an isomorphism (although not a natural one) in any characteristic, even two.
Grading
In the following, assume that the characteristic is not 2.
Clifford algebras are Z
2-
graded algebra
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the se ...
s (also known as
superalgebras). Indeed, the linear map on ''V'' defined by (
reflection through the origin
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
) preserves the quadratic form ''Q'' and so by the universal property of Clifford algebras extends to an algebra
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
Since is an
involution (i.e. it squares to the
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
) one can decompose into positive and negative eigenspaces of
where
Since is an automorphism it follows that:
where the bracketed superscripts are read modulo 2. This gives the structure of a Z
2-
graded algebra
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the se ...
. The subspace forms a
subalgebra of , called the ''even subalgebra''. The subspace is called the ''odd part'' of (it is not a subalgebra). This Z
2-grading plays an important role in the analysis and application of Clifford algebras. The automorphism is called the ''main
involution'' or ''grade involution''. Elements that are pure in this Z
2-grading are simply said to be even or odd.
''Remark''. In characteristic not 2 the underlying vector space of inherits an N-grading and a Z-grading from the canonical isomorphism with the underlying vector space of the exterior algebra . It is important to note, however, that this is a ''vector space grading only''. That is, Clifford multiplication does not respect the N-grading or Z-grading, only the Z
2-grading: for instance if , then , but , not in . Happily, the gradings are related in the natural way: . Further, the Clifford algebra is Z-
filtered:
The ''degree'' of a Clifford number usually refers to the degree in the N-grading.
The even subalgebra of a Clifford algebra is itself isomorphic to a Clifford algebra. If is the
orthogonal direct sum of a vector of nonzero norm and a subspace , then is isomorphic to , where is the form restricted to and multiplied by . In particular over the reals this implies that:
In the negative-definite case this gives an inclusion , which extends the sequence
Likewise, in the complex case, one can show that the even subalgebra of is isomorphic to .
Antiautomorphisms
In addition to the automorphism , there are two
antiautomorphisms that play an important role in the analysis of Clifford algebras. Recall that the
tensor algebra comes with an antiautomorphism that reverses the order in all products of vectors:
Since the ideal is invariant under this reversal, this operation descends to an antiautomorphism of called the ''transpose'' or ''reversal'' operation, denoted by . The transpose is an antiautomorphism: . The transpose operation makes no use of the Z
2-grading so we define a second antiautomorphism by composing and the transpose. We call this operation ''Clifford conjugation'' denoted
Of the two antiautomorphisms, the transpose is the more fundamental.
Note that all of these operations are
involutions. One can show that they act as ±1 on elements which are pure in the Z-grading. In fact, all three operations depend only on the degree modulo 4. That is, if ''x'' is pure with degree ''k'' then
where the signs are given by the following table:
Clifford scalar product
When the characteristic is not 2, the quadratic form ''Q'' on ''V'' can be extended to a quadratic form on all of (which we also denoted by ''Q''). A basis-independent definition of one such extension is
where ⟨''a''⟩ denotes the scalar part of ''a'' (the degree-0 part in the Z-grading). One can show that
where the ''v
i'' are elements of ''V'' – this identity is ''not'' true for arbitrary elements of .
The associated symmetric bilinear form on is given by
One can check that this reduces to the original bilinear form when restricted to ''V''. The bilinear form on all of is
nondegenerate
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case.
T ...
if and only if it is nondegenerate on ''V''.
The operator of left (respectively right) Clifford multiplication by the transpose ''a'' of an element ''a'' is the
adjoint of left (respectively right) Clifford multiplication by ''a'' with respect to this inner product. That is,
and
Structure of Clifford algebras
''In this section we assume that characteristic is not 2, the vector space ''V'' is finite-dimensional and that the associated symmetric bilinear form of ''Q'' is nondegenerate.''
A
central simple algebra
In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple a ...
over is a matrix algebra over a (finite-dimensional) division algebra with center . For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions.
*If ''V'' has even dimension then is a central simple algebra over ''K''.
*If ''V'' has even dimension then the even subalgebra is a central simple algebra over a quadratic extension of ''K'' or a sum of two isomorphic central simple algebras over ''K''.
*If ''V'' has odd dimension then is a central simple algebra over a quadratic extension of ''K'' or a sum of two isomorphic central simple algebras over ''K''.
*If ''V'' has odd dimension then the even subalgebra is a central simple algebra over ''K''.
The structure of Clifford algebras can be worked out explicitly using the following result. Suppose that has even dimension and a non-singular bilinear form with
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
, and suppose that is another vector space with a quadratic form. The Clifford algebra of is isomorphic to the tensor product of the Clifford algebras of and , which is the space with its quadratic form multiplied by . Over the reals, this implies in particular that
These formulas can be used to find the structure of all real Clifford algebras and all complex Clifford algebras; see the
classification of Clifford algebras
In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified. In ea ...
.
Notably, the
Morita equivalence
In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely two rings like ''R'', ''S'' are Morita equivalent (denoted by R\approx S) if their categories of modules ...
class of a Clifford algebra (its representation theory: the equivalence class of the category of modules over it) depends only on the signature . This is an algebraic form of
Bott periodicity.
Lipschitz group
The class of Lipschitz groups ( Clifford groups or Clifford–Lipschitz groups) was discovered by
Rudolf Lipschitz.
In this section we assume that is finite-dimensional and the quadratic form is
nondegenerate
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case.
T ...
.
An action on the elements of a Clifford algebra by its
group of units
In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that
vu = uv = 1,
where is the multiplicative identity; the element is unique for this ...
may be defined in terms of a twisted conjugation: twisted conjugation by maps , where is the ''main involution'' defined
above.
The Lipschitz group is defined to be the set of invertible elements that ''stabilize the set of vectors'' under this action, meaning that for all in we have:
This formula also defines an action of the Lipschitz group on the vector space ''V'' that preserves the quadratic form ''Q'', and so gives a homomorphism from the Lipschitz group to the orthogonal group. The Lipschitz group contains all elements ''r'' of ''V'' for which ''Q''(''r'') is invertible in ''K'', and these act on ''V'' by the corresponding reflections that take ''v'' to . (In characteristic these are called orthogonal transvections rather than reflections.)
If ''V'' is a finite-dimensional real vector space with a
non-degenerate
In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'' ) given by is not an isomorphism. An equivalent definiti ...
quadratic form then the Lipschitz group maps onto the orthogonal group of ''V'' with respect to the form (by the
Cartan–Dieudonné theorem) and the kernel consists of the nonzero elements of the field ''K''. This leads to exact sequences
Over other fields or with indefinite forms, the map is not in general onto, and the failure is captured by the spinor norm.
Spinor norm
In arbitrary characteristic, the
spinor norm ''Q'' is defined on the Lipschitz group by
It is a homomorphism from the Lipschitz group to the group ''K''
× of non-zero elements of ''K''. It coincides with the quadratic form ''Q'' of ''V'' when ''V'' is identified with a subspace of the Clifford algebra. Several authors define the spinor norm slightly differently, so that it differs from the one here by a factor of −1, 2, or −2 on Γ
1. The difference is not very important in characteristic other than 2.
The nonzero elements of ''K'' have spinor norm in the group (''K''
×)
2 of squares of nonzero elements of the field ''K''. So when ''V'' is finite-dimensional and non-singular we get an induced map from the orthogonal group of ''V'' to the group ''K''
×/(''K''
×)
2, also called the spinor norm. The spinor norm of the reflection about ''r''
⊥, for any vector ''r'', has image ''Q''(''r'') in ''K''
×/(''K''
×)
2, and this property uniquely defines it on the orthogonal group. This gives exact sequences:
Note that in characteristic 2 the group has just one element.
From the point of view of
Galois cohomology of
algebraic groups, the spinor norm is a
connecting homomorphism
The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance ...
on cohomology. Writing ''μ''
2 for the
algebraic group of square roots of 1 (over a field of characteristic not 2 it is roughly the same as a two-element group with trivial Galois action), the short exact sequence
yields a long exact sequence on cohomology, which begins
The 0th Galois cohomology group of an algebraic group with coefficients in ''K'' is just the group of ''K''-valued points: , and , which recovers the previous sequence
where the spinor norm is the connecting homomorphism .
Spin and Pin groups
In this section we assume that is finite-dimensional and its bilinear form is non-singular.
The
pin group
The PIN Group was a German courier and postal services company. It belonged to PIN Group S.A., a Luxembourg-based corporate affiliation made up of several German postal companies.
History and shareholding
The PIN Group originally traded under ...
is the subgroup of the Lipschitz group of elements of spinor norm , and similarly the
spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a L ...
is the subgroup of elements of
Dickson invariant in . When the characteristic is not , these are the elements of determinant . The spin group usually has index in the pin group.
Recall from the previous section that there is a homomorphism from the Lipschitz group onto the orthogonal group. We define the
special orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. T ...
to be the image of . If does not have characteristic this is just the group of elements of the orthogonal group of determinant . If does have characteristic , then all elements of the orthogonal group have determinant , and the special orthogonal group is the set of elements of Dickson invariant .
There is a homomorphism from the pin group to the orthogonal group. The image consists of the elements of spinor norm . The kernel consists of the elements and , and has order unless has characteristic . Similarly there is a homomorphism from the Spin group to the special orthogonal group of .
In the common case when is a positive or negative definite space over the reals, the spin group maps onto the special orthogonal group, and is simply connected when has dimension at least . Further the kernel of this homomorphism consists of and . So in this case the spin group, , is a double cover of . Please note, however, that the simple connectedness of the spin group is not true in general: if is for and both at least then the spin group is not simply connected. In this case the algebraic group is simply connected as an algebraic group, even though its group of real valued points is not simply connected. This is a rather subtle point, which completely confused the authors of at least one standard book about spin groups.
Spinors
Clifford algebras , with even, are matrix algebras which have a complex representation of dimension . By restricting to the group we get a complex representation of the Pin group of the same dimension, called the
spin representation. If we restrict this to the spin group then it splits as the sum of two ''half spin representations'' (or ''Weyl representations'') of dimension .
If is odd then the Clifford algebra is a sum of two matrix algebras, each of which has a representation of dimension , and these are also both representations of the Pin group . On restriction to the spin group these become isomorphic, so the spin group has a complex spinor representation of dimension .
More generally, spinor groups and pin groups over any field have similar representations whose exact structure depends on the
structure of the corresponding Clifford algebras: whenever a Clifford algebra has a factor that is a matrix algebra over some division algebra, we get a corresponding representation of the pin and spin groups over that division algebra.
For examples over the reals see the article on
spinors.
Real spinors
To describe the real spin representations, one must know how the spin group sits inside its Clifford algebra. The
pin group
The PIN Group was a German courier and postal services company. It belonged to PIN Group S.A., a Luxembourg-based corporate affiliation made up of several German postal companies.
History and shareholding
The PIN Group originally traded under ...
, is the set of invertible elements in that can be written as a product of unit vectors:
Comparing with the above concrete realizations of the Clifford algebras, the pin group corresponds to the products of arbitrarily many reflections: it is a cover of the full orthogonal group . The
spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a L ...
consists of those elements of that are products of an even number of unit vectors. Thus by the
Cartan–Dieudonné theorem Spin is a cover of the group of proper rotations .
Let be the automorphism which is given by the mapping acting on pure vectors. Then in particular, is the subgroup of whose elements are fixed by . Let
(These are precisely the elements of even degree in .) Then the spin group lies within .
The irreducible representations of restrict to give representations of the pin group. Conversely, since the pin group is generated by unit vectors, all of its irreducible representation are induced in this manner. Thus the two representations coincide. For the same reasons, the irreducible representations of the spin coincide with the irreducible representations of .
To classify the pin representations, one need only appeal to the
classification of Clifford algebras
In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified. In ea ...
. To find the spin representations (which are representations of the even subalgebra), one can first make use of either of the isomorphisms (see above)
and realize a spin representation in signature as a pin representation in either signature or .
Applications
Differential geometry
One of the principal applications of the exterior algebra is in
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
where it is used to define the
bundle
Bundle or Bundling may refer to:
* Bundling (packaging), the process of using straps to bundle up items
Biology
* Bundle of His, a collection of heart muscle cells specialized for electrical conduction
* Bundle of Kent, an extra conduction pat ...
of
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s on a
smooth manifold. In the case of a (
pseudo-)
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
, the
tangent spaces come equipped with a natural quadratic form induced by the
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathema ...
. Thus, one can define a
Clifford bundle In mathematics, a Clifford bundle is an algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the algebra structure. There is a natural Clifford bundle associated to any (pseudo) Riemannian mani ...
in analogy with the
exterior bundle In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor produ ...
. This has a number of important applications in
Riemannian geometry. Perhaps more important is the link to a
spin manifold In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry.
Spin structures have wide applications to mathematical ...
, its associated
spinor bundle and manifolds.
Physics
Clifford algebras have numerous important applications in physics. Physicists usually consider a Clifford algebra to be an algebra with a basis generated by the matrices called
Dirac matrices
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
which have the property that
where is the matrix of a quadratic form of signature (or corresponding to the two equivalent choices of metric signature). These are exactly the defining relations for the Clifford algebra , whose
complexification is which, by the
classification of Clifford algebras
In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified. In ea ...
, is isomorphic to the algebra of complex matrices . However, it is best to retain the notation , since any transformation that takes the bilinear form to the canonical form is ''not'' a Lorentz transformation of the underlying spacetime.
The Clifford algebra of spacetime used in physics thus has more structure than . It has in addition a set of preferred transformations – Lorentz transformations. Whether complexification is necessary to begin with depends in part on conventions used and in part on how much one wants to incorporate straightforwardly, but complexification is most often necessary in quantum mechanics where the spin representation of the Lie algebra sitting inside the Clifford algebra conventionally requires a complex Clifford algebra. For reference, the spin Lie algebra is given by
This is in the convention, hence fits in .
The Dirac matrices were first written down by
Paul Dirac
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
when he was trying to write a relativistic first-order wave equation for the
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
, and give an explicit isomorphism from the Clifford algebra to the algebra of complex matrices. The result was used to define the
Dirac equation and introduce the
Dirac operator
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formall ...
. The entire Clifford algebra shows up in
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
in the form of
Dirac field bilinear
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
s.
The use of Clifford algebras to describe quantum theory has been advanced among others by
Mario Schönberg
is a character created by Japanese video game designer Shigeru Miyamoto. He is the title character of the ''Mario'' franchise and the mascot of Japanese video game company Nintendo. Mario has appeared in over 200 video games since his cre ...
, by
David Hestenes in terms of
geometric calculus, by
David Bohm and
Basil Hiley
Basil J. Hiley (born 1935), is a British people, British Quantum mechanics, quantum physicist and professor emeritus of the University of London.
Long-time colleague of David Bohm, Hiley is known for his work with Bohm on implicate orders and for ...
and co-workers in form of a
hierarchy of Clifford algebras, and by Elio Conte et al.
Computer vision
Clifford algebras have been applied in the problem of action recognition and classification in
computer vision
Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the hum ...
. Rodriguez et al.
propose a Clifford embedding to generalize traditional MACH filters to video (3D spatiotemporal volume), and vector-valued data such as
optical flow. Vector-valued data is analyzed using the
Clifford Fourier Transform. Based on these vectors action filters are synthesized in the Clifford Fourier domain and recognition of actions is performed using Clifford correlation. The authors demonstrate the effectiveness of the Clifford embedding by recognizing actions typically performed in classic feature films and sports broadcast television.
Generalizations
* While this article focuses on a Clifford algebra of a vector space over a field, the definition extends without change to a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
over any unital, associative, commutative ring.
* Clifford algebras may be generalized to a form of degree higher than quadratic over a vector space.
[
]
See also
*
Algebra of physical space, APS
*
Cayley–Dickson construction
*
Classification of Clifford algebras
In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified. In ea ...
*
Clifford analysis
*
Clifford module In mathematics, a Clifford module is a representation of a Clifford algebra. In general a Clifford algebra ''C'' is a central simple algebra over some field extension ''L'' of the field ''K'' over which the quadratic form ''Q'' defining ''C'' is d ...
*
Complex spin structure In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry.
Spin structures have wide applications to mathematica ...
*
Dirac operator
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formall ...
*
Exterior algebra
*
Fierz identity
In theoretical physics, a Fierz identity is an identity that allows one to rewrite '' bilinears of the product'' of two spinors as a linear combination of ''products of the bilinears'' of the individual spinors. It is named after Swiss physicist M ...
*
Gamma matrices
*
Generalized Clifford algebra
In mathematics, a Generalized Clifford algebra (GCA) is a unital associative algebra that generalizes the Clifford algebra, and goes back to the work of Hermann Weyl, who utilized and formalized these clock-and-shift operators introduced b ...
*
Geometric algebra
In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ge ...
*
Higher-dimensional gamma matrices
*
Hypercomplex number
*
Octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...
*
Paravector
*
Quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
*
Spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a L ...
*
Spin structure
*
Spinor
*
Spinor bundle
Notes
References
Sources
* , section IX.9.
* Carnahan, S. ''Borcherds Seminar Notes, Uncut.'' Week 5, "Spinors and Clifford Algebras".
*
*
*
* . An advanced textbook on Clifford algebras and their applications to differential geometry.
*
*
* ; ibid II (1883) 46; ibid III (1884) 7–9. Summarized in ''The Collected Mathematics Papers of James Joseph Sylvester'' (Cambridge University Press, 1909) v III
onlinean
further
*
*
Further reading
*
External links
*
(unverified)
John Baez on Clifford algebrasClifford Algebra: A Visual Introduction
{{authority control
Ring theory
Quadratic forms