nLab
pseudoscalar
Contents
Context
Representation theory
representation theory

geometric representation theory

Ingredients
Definitions
representation , 2-representation , ∞-representation

group , ∞-group

group algebra , algebraic group , Lie algebra

vector space , n-vector space

affine space , symplectic vector space

action , ∞-action

module , equivariant object

bimodule , Morita equivalence

induced representation , Frobenius reciprocity

Hilbert space , Banach space , Fourier transform , functional analysis

orbit , coadjoint orbit , Killing form

unitary representation

geometric quantization , coherent state

socle , quiver

module algebra , comodule algebra , Hopf action , measuring

Geometric representation theory
D-module , perverse sheaf ,

Grothendieck group , lambda-ring , symmetric function , formal group

principal bundle , torsor , vector bundle , Atiyah Lie algebroid

geometric function theory , groupoidification

Eilenberg-Moore category , algebra over an operad , actegory , crossed module

reconstruction theorems

Theorems
Contents
Idea
Given a symmetry group $G$ equipped with a homomorphism $G \overset{\phi}{\longrightarrow} O(s,t)$ to an orthogonal group (for instance a Pin group ), a pseudoscalar is an element of the 1-dimensional linear representation (over a given ground field $k$ )

$\mathbf{1}_{{}_{sgn}}
\;\in\;
Rep_k(G)$

that is given by forming the determinant (sign representation ):

(1) $\mathbf{1}_{{}_{sgn}}
\;\;\coloneqq\;\;
\left(
\array{
G \times k
&\longrightarrow&
k
\\
(g, c ) &\mapsto& det\big( \phi(g) \big)
}
\right)
\,.$

More generally, given a function with values in $\mathbf{1}_{{}_{sgn}}$ , or yet more generally a section of a fiber bundle with typical fiber $\mathbf{1}_{{}_{sgn}}$ , this whole function/section is often called a pseudoscalar ; more precisely: a pseudoscalar field . This as opposed to scalar fields , which take values in the 1-dimensional trivial representation $\mathbf{1}$ .

If the fiber bundle in question is a “canonical bundle ”/determinant line bundle of a Riemannian manifold of dimension $n$ , hence the top exterior power of a tangent bundle (i.e. top degree (Kähler ) differential form -bundle) then such “pseudoscalar fields” of physics are what are called densities in mathematics .

References
See also

Last revised on April 28, 2020 at 15:12:21.
See the history of this page for a list of all contributions to it.