Algebra (R, +) of the U(1) group. More...

#include <semiSimpleAlgebra.h>

Inheritance diagram for mty::algebra::R:

Public Member Functions
	R ()
	Default constructor.

	~R ()
	Destructor.

algebra::Type	getType () const override
	Pure virtual function overriden in derived classes. More...

void	setCartanMatrix () override
	Overrides the function of mty::SemiSimpleAlgebra and does nothing (no Cartan matrix).

void	setSquaredNorm () override
	Overrides the function of mty::SemiSimpleAlgebra and does nothing (no root).

mty::Irrep	highestWeightRep (const mty::AlgebraState &highestWeight, bool mult=true) const override
	Creates a representation from an AlgebraState containing only the numerator and the denominator of the U(1) charge. More...

mty::Irrep	highestWeightRep (const std::vector< int > &highestWeight, bool mult=true) const override
	Creates a representation from a std::vector of int containing only the numerator and the denominator of the U(1) charge. More...

mty::Irrep	getTrivialRep () const override

mty::SumIrrep	tensorProduct (const mty::Irrep &A, const mty::Irrep &B, bool mult=true) const override

void	sortRep (std::vector< mty::AlgebraState > &rep, std::vector< int > &mult) const override
	Sorts the different states of the rep. More...

csl::Expr	getEta () const override

Public Member Functions inherited from mty::SemiSimpleAlgebra
	SemiSimpleAlgebra (int t_l)
	Constructor with one parameter. More...

virtual	~SemiSimpleAlgebra ()
	Destructor.

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

virtual csl::Expr	getIndex (const Irrep &irrep, size_t n) const

virtual Irrep	getDefiningRep () const

virtual SumIrrep	tensorProduct (const SumIrrep &A, const Irrep &B, bool mult=true) const

virtual SumIrrep	tensorProduct (const Irrep &A, const SumIrrep &B, bool mult=true) const

SumIrrep	tensorProduct (const SumIrrep &A, const SumIrrep &B, bool mult=true) const

int	getOrderL () const
	Returns the order of the algebra, i.e. the number of simple roots l. More...

Additional Inherited Members
Protected Member Functions inherited from mty::SemiSimpleAlgebra
void	init ()
	Initialization function called in derived classes to initialize the cartan matrix, the Weyl group and the norms of simple roots.

Protected Attributes inherited from mty::SemiSimpleAlgebra
const int	l
	Order of the semi-simple algebra, i.e. number of simple roots.

std::vector< int >	rho
	std::vector of size l. Particular root (half-sum of simple roots) useful in calculations. More...

csl::vector_expr	absoluteNorms
	Absolute norms of simple roots used in the computation of the quadratic Casimir operator. See SemiSimpleAlgebra::getQuadraticCasimir().

std::vector< int >	squaredNorm
	std::vector of integers of size l. Contains squared norms of all simple roots.

std::vector< std::vector< int > >	positiveRoots
	Contains all positive roots in a std::vector. Each positive root is a std::vector of integers of size l: dinkin labels in co-root space.

std::vector< std::vector< int > >	cartanMatrix
	Cartan matrix of the algebra.

std::vector< std::vector< int > >	rootMetric
	Metric for roots in co-root space, i.e. metrix to apply in order to compute scalar dot between roots expressed with their dinkin labels.

csl::Expr	symbolicRho
	Symbolic version of the half-sum of simple roots (see positiveRoots). Contains the factor 1/2 that positiveRoots does not.

csl::Expr	symbolicCartan
	Symbolic version of the cartan matrix.

csl::Expr	inverseCartan
	Inverse cartan matrix.

csl::Expr	symbolicMetric
	Symbolic metric in root space. i.e. different from rootMetric that is the metric in co-root space.

csl::vector_expr	symbolicRoots
	Symbolic version of positiveRoots, each root is a csl::Vector of csl::Integer in co-root space (dinkin labels).

csl::vector_expr	symbolicSquaredNorms
	Symbolic version of squaredNorm.

std::vector< std::pair< AlgebraState, Irrep > >	irreps

std::vector< IndexData >	indices

Detailed Description

Algebra (R, +) of the U(1) group.

This particular algebra has a different treatment as the other semi-simple algebras and reimplements some of the main functions. R needs only one charge (possibly fractionnal) to fully describe a representation. No need then for the cartan formalism. A representation is then represented by a two-labels AlgebraState. The first label corresponds to the charge's numerator, and the second corresponds to the denominator.

Member Function Documentation

◆ getTrivialRep()

mty::Irrep R::getTrivialRep ( ) const

overridevirtual

Returns: The trivial rep: charge 0.

Reimplemented from mty::SemiSimpleAlgebra.

◆ getType()

mty::algebra::Type R::getType ( ) const

overridevirtual

Pure virtual function overriden in derived classes.

Returns: The type of the algebra, see the enum algebra::Type.

Implements mty::SemiSimpleAlgebra.

◆ highestWeightRep() [1/2]

mty::Irrep R::highestWeightRep	(	const mty::AlgebraState &	highestWeight,
		bool	mult = `true`
	)		const

overridevirtual

Creates a representation from an AlgebraState containing only the numerator and the denominator of the U(1) charge.

Returns: The U(1) Irrep with the right charge.

Reimplemented from mty::SemiSimpleAlgebra.

◆ highestWeightRep() [2/2]

mty::Irrep R::highestWeightRep	(	const std::vector< int > &	highestWeight,
		bool	mult = `true`
	)		const

overridevirtual

Creates a representation from a std::vector of int containing only the numerator and the denominator of the U(1) charge.

Returns: The U(1) Irrep with the right charge.

Reimplemented from mty::SemiSimpleAlgebra.

◆ sortRep()

void R::sortRep	(	std::vector< mty::AlgebraState > &	rep,
		std::vector< int > &	mult
	)		const

overridevirtual

Sorts the different states of the rep.

Sorts the different states by increasing value of charge.

Parameters

rep	std::vector of AlgebraState to sort (modified).
mult	multiplicities of the states sorted the same way (modified).

Reimplemented from mty::SemiSimpleAlgebra.

◆ tensorProduct()

mty::SumIrrep R::tensorProduct	(	const mty::Irrep &	A,
		const mty::Irrep &	B,
		bool	mult = `true`
	)		const

overridevirtual

Parameters

A	First Irrep in the product.
B	Second Irrep in the procut.

Returns: The tensor procuct of the two Irrep, i.e. another U(1) rep whose charge is the sum of those of A and B, embedded in a SumIrrep of one term.

Reimplemented from mty::SemiSimpleAlgebra.

The documentation for this class was generated from the following files:

include/semiSimpleAlgebra.h
src/semiSimpleAlgebra.cpp

Public Member Functions

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ getTrivialRep()

◆ getType()

◆ highestWeightRep() [1/2]

◆ highestWeightRep() [2/2]

◆ sortRep()

◆ tensorProduct()