Symplectic representation
Encyclopedia
In mathematical
field of representation theory
, a quaternionic representation is a representation
on a complex
vector space V with an invariant quaternionic structure, i.e., an antilinear equivariant map
which satisfies
Together with the imaginary unit i and the antilinear map k := ij, j equips V with the structure of a quaternionic vector space
(i.e., V becomes a module
over the division algebra
of quaternion
s). From this point of view, quaternionic representation of a group
G is a group homomorphism
φ: G → GL(V, H), the group of invertible quaternion-linear transformations of V. In particular, a quaternionic matrix representation of g assigns a square matrix of quaternions ρ(g) to each element g of G such that ρ(e) is the identity matrix and
Quaternionic representations of associative
and Lie algebra
s can be defined in a similar way.
and the quaternionic structure j is a unitary operator, then V admits an invariant complex symplectic form ω, and hence is a symplectic representation
. This always holds if V is a representation of a compact group
(e.g. a finite group
) and in this case quaternionic representations are also known as symplectic representations. Such representations, amongst irreducible representations, can be picked out by the Frobenius-Schur indicator
.
Quaternionic representations are similar to real representation
s in that they are isomorphic to their complex conjugate representation
. Here a real representation is taken to be a complex representation with an invariant real structure
, i.e., an antilinear equivariant map
which satisfies
A representation which is isomorphic to its complex conjugate, but which is not a real representation, is sometimes called a pseudoreal representation.
Real and pseudoreal representations of a group G can be understood by viewing them as representations of the real group algebra
R[G]. Such a representation will be a direct sum of central simple R-algebras, which, by the Artin-Wedderburn theorem, must be matrix algebras over the real numbers or the quaternions. Thus a real or pseudoreal representation is a direct sum of irreducible real representations and irreducible quaternionic representations. It is real if no quaternionic representations occur in the decomposition.
s in three dimensions. Each (proper) rotation is represented by a quaternion with unit norm. There is an obvious one-dimensional quaternionic vector space, namely the space H of quaternions themselves under left multiplication. By restricting this to the unit quaternions, we obtain a quaternionic representation of the spinor group Spin(3).
This representation ρ: Spin(3) → GL(1,H) also happens to be a unitary quaternionic representation because
for all g in Spin(3).
Another unitary example is the spin representation of Spin(5). An example of a nonunitary quaternionic representation would be the two dimensional irreducible representation of Spin(5,1).
More generally, the spin representations of Spin(d) are quaternionic when d equals 3 + 8k, 4 + 8k, and 5 + 8k dimensions, where k is an integer. In physics, one often encounters the spinor
s of Spin(d, 1). These representations have the same type of real or quaternionic structure as the spinors of Spin(d − 1).
Among the compact real forms of the simple Lie groups, irreducible quaternionic representations only exist for the Lie groups of type A4k+1, B4k+1, B4k+2, Ck, D4k+2, and E7.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
field of representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
, a quaternionic representation is a representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
on a complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
vector space V with an invariant quaternionic structure, i.e., an antilinear equivariant map
which satisfies
Together with the imaginary unit i and the antilinear map k := ij, j equips V with the structure of a quaternionic vector space
Quaternionic vector space
In mathematics, a left quaternionic vector space is a left H-module where H denotes the noncommutative ring of the quaternions....
(i.e., V becomes a module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over the division algebra
Division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field, in which division is possible.- Definitions :...
of quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
s). From this point of view, quaternionic representation of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G is a group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
φ: G → GL(V, H), the group of invertible quaternion-linear transformations of V. In particular, a quaternionic matrix representation of g assigns a square matrix of quaternions ρ(g) to each element g of G such that ρ(e) is the identity matrix and
Quaternionic representations of associative
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
and Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s can be defined in a similar way.
Properties and related concepts
If V is a unitary representationUnitary representation
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...
and the quaternionic structure j is a unitary operator, then V admits an invariant complex symplectic form ω, and hence is a symplectic representation
Symplectic representation
In mathematical field of representation theory, a quaternionic representation is a representation on a complex vector space V with an invariant quaternionic structure, i.e., an antilinear equivariant mapj\colon V\to V\,which satisfiesj^2=-1.\,...
. This always holds if V is a representation of a compact group
Compact group
In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...
(e.g. a finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
) and in this case quaternionic representations are also known as symplectic representations. Such representations, amongst irreducible representations, can be picked out by the Frobenius-Schur indicator
Frobenius-Schur indicator
In mathematics the Schur indicator, named after Issai Schur, or Frobenius–Schur indicator describes what invariant bilinear forms a given irreducible representation of a compact group on a complex vector space has...
.
Quaternionic representations are similar to real representation
Real representation
In the mathematical field of representation theory a real representation is usually a representation on a real vector space U, but it can also mean a representation on a complex vector space V with an invariant real structure, i.e., an antilinear equivariant mapj\colon V\to V\,which...
s in that they are isomorphic to their complex conjugate representation
Complex conjugate representation
In mathematics, if G is a group and ρ is a representation of it over the complex vector space V, then the complex conjugate representation ρ* is defined over the conjugate vector space V* as follows:...
. Here a real representation is taken to be a complex representation with an invariant real structure
Real structure
In mathematics, a real structure on a complex vector space is a way to decompose the complex vector space in the direct sum of two real vector spaces...
, i.e., an antilinear equivariant map
which satisfies
A representation which is isomorphic to its complex conjugate, but which is not a real representation, is sometimes called a pseudoreal representation.
Real and pseudoreal representations of a group G can be understood by viewing them as representations of the real group algebra
Group algebra
In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra , such that representations of the algebra are related to representations of the group...
R[G]. Such a representation will be a direct sum of central simple R-algebras, which, by the Artin-Wedderburn theorem, must be matrix algebras over the real numbers or the quaternions. Thus a real or pseudoreal representation is a direct sum of irreducible real representations and irreducible quaternionic representations. It is real if no quaternionic representations occur in the decomposition.
Examples
A common example involves the quaternionic representation of rotationRotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
s in three dimensions. Each (proper) rotation is represented by a quaternion with unit norm. There is an obvious one-dimensional quaternionic vector space, namely the space H of quaternions themselves under left multiplication. By restricting this to the unit quaternions, we obtain a quaternionic representation of the spinor group Spin(3).
This representation ρ: Spin(3) → GL(1,H) also happens to be a unitary quaternionic representation because
for all g in Spin(3).
Another unitary example is the spin representation of Spin(5). An example of a nonunitary quaternionic representation would be the two dimensional irreducible representation of Spin(5,1).
More generally, the spin representations of Spin(d) are quaternionic when d equals 3 + 8k, 4 + 8k, and 5 + 8k dimensions, where k is an integer. In physics, one often encounters the spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
s of Spin(d, 1). These representations have the same type of real or quaternionic structure as the spinors of Spin(d − 1).
Among the compact real forms of the simple Lie groups, irreducible quaternionic representations only exist for the Lie groups of type A4k+1, B4k+1, B4k+2, Ck, D4k+2, and E7.