Special orthogonal group. The algebra of \( SO(N) \) is \( B_{(N-1)/2} \) if \( N \) is odd, \( D_{N/2} \) else. See mty::algebra::B and mty::algebra::D. More...

#include <group.h>

Inheritance diagram for mty::group::SO:

[legend]

Public Member Functions
	SO (std::string const &t_name, int t_dim=3)
	Constructor with two parameters. More...

	SO (int t_dim=3)
	Constructor with one parameter. More...

mty::group::Type	getType () const override
	Returns the type of the group. See mty::group::Type. More...

mty::Irrep	getAdjointRep () const override
	Returns the adjoint representation of \( SO(N) \), i.e. the representation of the gauge boson. More...

Public Member Functions inherited from mty::SemiSimpleGroup
	SemiSimpleGroup (const std::string &name)
	Constructor with one parameter. More...

	SemiSimpleGroup (const SemiSimpleAlgebra &t_algebra, const std::string &t_name)
	Constructor with two parameters. More...

	~SemiSimpleGroup () override
	Destructor.

SemiSimpleAlgebra *	getAlgebra () const override
	Returns a pointer to the algebra of the group. More...

void	setAlgebra (const SemiSimpleAlgebra &t_algebra)
	Replaces the algebra owned by the group. More...

Irrep	highestWeightRep (const std::vector< int > &highestWeight) const override
	Returns an Irrep of highest weight highestWeight. More...

csl::Expr	getQuadraticCasimir (const AlgebraState &highest) const
	Returns the expression (symbolic, rational number) of the quadratic Casimir operator for a representation. More...

Irrep	getTrivialRep () const override
	Returns the trivial (dimension 1) representation of the group. More...

SumIrrep	tensorProduct (const Irrep &rep1, const Irrep &rep2) const
	Computes the product of two representations and returns its decomposition in a sum of irreducible representations (SumIrrep). See SemiSimpleAlgebra::tensorProduct(). More...

SumIrrep	tensorProduct (const std::initializer_list< int > &rep1, const std::initializer_list< int > &rep2) const
	Computes and the product of two representations and returns its decomposition in a sum of irreducible representations. See SemiSimpleAlgebra::tensorProduct(). More...

const csl::Space *	getVectorSpace (const Irrep &irrep) const override
	Returns the vector space (csl::Space) associated with a certain irreducible representation of the group. More...

AlgebraState	getHighestWeight (const csl::Space *t_space) const override
	Searches and returns the highest weight of the representation living in t_space (csl::Space). More...

std::vector< const csl::Space * >	getAllVectorSpace () const override
	Returns all vector spaces (csl::Space) created by the group. More...

csl::Index	getCorrespondingIndex (const std::string &index_name, const QuantumFieldParent &parent) override
	Returns a csl::Index of the vector space corresponding to a given mty::QuantumFieldParent's representation of the group. More...

Public Member Functions inherited from mty::AbstractGroup
	AbstractGroup (const std::string &t_name="")
	Constructor with one optional parameter. More...

	AbstractGroup (int t_dim, const std::string &t_name="")
	Constructor with two parameters. More...

virtual	~AbstractGroup ()
	Destructor.

mty::GaugedGroup *	getGaugedGroup () const

void	setGaugedGroup (mty::GaugedGroup *t_gauged)

std::string	getName () const
	Returns the name of the group. More...

int	getDim () const
	Returns the dimension (integer) of the group. More...

Additional Inherited Members
Protected Attributes inherited from mty::SemiSimpleGroup
std::unique_ptr< SemiSimpleAlgebra >	algebra
	Owner std::unique_ptr to the SemiSimpleAlgebra associated with the group.

std::vector< std::pair< AlgebraState, const csl::Space * > >	vectorSpaces
	Association of AlgebraState (highest weights of irreps) and vector spaces. To each irrep (i.e. each highest weight) is associated a different and unique vector space (csl::Space). See getVectorSpace().

std::map< AlgebraState, csl::Expr >	quadraticCasimir
	Association of AlgebraState (highest weights of irreps) and quadratic Casimirs. To each irrep (i.e. each highest weight) is associated a quadratic casimir operator. It is computed the first time asked, and stored.

Protected Attributes inherited from mty::AbstractGroup
int	dim
	Dimension of the group.

std::string	name
	Name of the group.

mty::GaugedGroup *	gauged

Detailed Description

Special orthogonal group. The algebra of \( SO(N) \) is \( B_{(N-1)/2} \) if \( N \) is odd, \( D_{N/2} \) else. See mty::algebra::B and mty::algebra::D.

Constructor & Destructor Documentation

◆ SO() [1/2]

mty::group::SO::SO	(	std::string const &	t_name,
		int	t_dim = `3`
	)

explicit

Constructor with two parameters.

Parameters

t_name	Name of the group.
t_dim	Dimension (optional, default = 3) of the group.

◆ SO() [2/2]

mty::group::SO::SO ( int t_dim = 3 )

explicit

Constructor with one parameter.

Parameters

t_dim Dimension (optional, default = 3) of the group.

Member Function Documentation

◆ getAdjointRep()

mty::Irrep mty::group::SO::getAdjointRep ( ) const

overridevirtual

Returns the adjoint representation of \( SO(N) \), i.e. the representation of the gauge boson.

The adjoint representation is needed automatically in general in order to define gauge bosons without the help of the user. Adjoint representations expressed in terms of dynkin labels (highest weight state) can be found in the literature, see the manual for references.