NCAlgebra

Description

This package is used to define and manipulate noncommutative algebras. Many of the commands contain calls to the existing noncommutative algebra package Bergman.

Detailed instructions for installing Bergman, as well as the NCAlgebra system, can be found in the file installNCAlgebra.txt file contained in the NCAlgebra package directory. It may also be found at https://github.com/Macaulay2/M2/blob/master/M2/Macaulay2/packages/NCAlgebra/installNCAlgebra.txt.

Authors

Version

This documentation describes version 1.0 of NCAlgebra.

Source code

The source code from which this documentation is derived is in the file NCAlgebra.m2. The auxiliary files accompanying it are in the directory NCAlgebra/.

Exports

Types
- NCGroebnerBasis -- Type of a Groebner basis for an NCIdeal in an NCRing.
- NCIdeal -- Type of a two-sided ideal in a noncommutative ring
- NCLeftIdeal -- Type of a left ideal in a noncommutative ring
- NCMatrix -- Type of a matrix over a noncommutative ring
- NCPolynomialRing -- Type of a noncommutative polynomial ring
- NCQuotientRing -- Type of a noncommutative ring
- NCRightIdeal -- Type of a right ideal in a noncommutative ring
- NCRing -- Type of a noncommutative ring
- NCRingElement -- Type of an element in a noncommutative ring
- NCRingMap -- Type of a map to or from a noncommutative ring.
- NCChainComplex -- see resolution(NCMatrix) -- Compute the resolution of coker M as a map of free right modules
Functions and commands
- assignDegrees -- Weights entries of a matrix to make associated map of free modules graded
- centralElements -- Finds central elements in a given degree
- coordinates -- Computes coordinates relative to a given basis
- endomorphismRing -- Methods for creating endomorphism rings of modules over a commutative ring
- envelopingAlgebra -- Create the enveloping algebra
- fourDimSklyanin -- Defines a four-dimensional Sklyanin with given parameters
- freeProduct -- Define the free product of two algebras
- gbFromOutputFile -- Read in a NCGroebnerBasis from a Bergman output file.
- gddKernel -- Computes a homogeneous generating set of the kernel of a ring map.
- hilbertBergman -- Calls Bergman to compute the Hilbert series of an NCQuotientRing
- homogDual -- Computes the dual of a pure homogeneous ideal
- isCentral -- Determines if an element is central
- isExterior -- see isCommutative(NCRing) -- Returns whether an NCRing is commutative
- isLeftRegular -- Determines if a given (homogeneous) element is regular in a given degree
- isRightRegular -- see isLeftRegular -- Determines if a given (homogeneous) element is regular in a given degree
- kernelComponent -- Computes a basis of the kernel of a ring map in a specified degree.
- leftMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
- rightMultiplicationMap -- see leftMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
- minimizeRelations -- Minimizes a list of NCRingElements
- ncGroebnerBasis -- Compute a noncommutative Groebner basis.
- ncIdeal -- Define a two-sided ideal in a noncommutative ring
- ncLeftIdeal -- Define a left ideal in a noncommutative ring
- ncMap -- Make a map to or from an NCRing
- ncMatrix -- Create an NCMatrix
- ncRightIdeal -- Define a right ideal in a noncommutative ring
- normalAutomorphism -- Computes the automorphism determined by a normal homogeneous element
- normalElements -- Finds normal elements
- normalFormBergman -- Calls Bergman for a normal form calculation
- oppositeRing -- Creates the opposite ring of a noncommutative ring
- oreExtension -- Creates an Ore extension of a noncommutative ring
- oreIdeal -- Creates the defining ideal of an Ore extension of a noncommutative ring
- qTensorProduct -- Define the (q-)commuting tensor product
- quadraticClosure -- Creates the subideal generated by quadratic elements of a given ideal
- rightKernel -- Method for computing kernels of matrices over noncommutative rings in a given degree without using Bergman
- rightKernelBergman -- Methods for computing kernels of matrices over noncommutative rings using Bergman
- setWeights -- Set a nonstandard grading for a NCRing.
- skewPolynomialRing -- Defines a skew polynomial ring via a skewing matrix
- sparseCoeffs -- Converts ring elements into vectors over the coefficient ring
- threeDimSklyanin -- Defines a three-dimensional Sklyanin with given parameters
- toM2Ring -- Compute the abelianization of an NCRing and returns a Ring.
- toNCRing -- Converts a Ring to an NCRing
- twoSidedNCGroebnerBasisBergman -- Calls Bergman to compute a two sided noncommutative Groebner Basis.
Methods
- - NCMatrix -- Negates NCMatrices
- ambient(NCQuotientRing) -- Ambient ring of an NCQuotientRing
- ambient(NCRingMap) -- Extends an NCRingMap to the ambient ring of the source.
- assignDegrees(NCMatrix) -- see assignDegrees -- Weights entries of a matrix to make associated map of free modules graded
- assignDegrees(NCMatrix,List,List) -- see assignDegrees -- Weights entries of a matrix to make associated map of free modules graded
- baseName(NCRingElement) -- Returns the base name of a generator of an NCRing
- basis(ZZ,NCIdeal) -- Returns a basis of an NCIdeal in a particular degree.
- basis(ZZ,NCLeftIdeal) -- Returns a basis of an NCLeftIdeal in a particular degree.
- basis(ZZ,NCRightIdeal) -- Returns a basis of an NCRightIdeal in a particular degree.
- basis(ZZ,NCRing) -- Returns a basis of an NCRing in a particular degree.
- centralElements(NCRing,ZZ) -- see centralElements -- Finds central elements in a given degree
- coefficientRing(NCRing) -- Returns the base ring of an NCRing
- coordinates(List) -- see coordinates -- Computes coordinates relative to a given basis
- coordinates(NCRingElement) -- see coordinates -- Computes coordinates relative to a given basis
- degree(NCRingElement) -- Returns the degree of an NCRingElement
- endomorphismRing(Module,Symbol) -- see endomorphismRing -- Methods for creating endomorphism rings of modules over a commutative ring
- entries(NCMatrix) -- Returns the entries of the NCMatrix
- envelopingAlgebra(NCRing,Symbol) -- see envelopingAlgebra -- Create the enveloping algebra
- fourDimSklyanin(Ring,List) -- see fourDimSklyanin -- Defines a four-dimensional Sklyanin with given parameters
- fourDimSklyanin(Ring,List,List) -- see fourDimSklyanin -- Defines a four-dimensional Sklyanin with given parameters
- freeProduct(NCRing,NCRing) -- see freeProduct -- Define the free product of two algebras
- gbFromOutputFile(NCPolynomialRing,String) -- see gbFromOutputFile -- Read in a NCGroebnerBasis from a Bergman output file.
- gddKernel(ZZ,NCRingMap) -- see gddKernel -- Computes a homogeneous generating set of the kernel of a ring map.
- generators(NCGroebnerBasis) -- The list of algebra generators of an NCGroebnerBasis
- generators(NCIdeal) -- Returns the generators of an NCIdeal
- generators(NCLeftIdeal) -- Returns the generators of an NCLeftIdeal
- generators(NCRightIdeal) -- Returns the generators of an NCRightIdeal
- generators(NCRing) -- The list of algebra generators of an NCRing
- hilbertBergman(NCQuotientRing) -- see hilbertBergman -- Calls Bergman to compute the Hilbert series of an NCQuotientRing
- hilbertSeries(NCRing) -- Computes the Hilbert series of an NCRing
- Hom(ZZ,NCMatrix,NCMatrix) -- Compute a graded component of Hom(M,N)
- homogDual(NCIdeal) -- see homogDual -- Computes the dual of a pure homogeneous ideal
- homogDual(NCQuotientRing) -- see homogDual -- Computes the dual of a pure homogeneous ideal
- homogDual(Ring) -- see homogDual -- Computes the dual of a pure homogeneous ideal
- ideal(NCPolynomialRing) -- The defining ideal of an NCPolynomialRing
- ideal(NCQuotientRing) -- Defining ideal of an NCQuotientRing in its ambient ring
- isCentral(NCRingElement) -- see isCentral -- Determines if an element is central
- isCentral(NCRingElement,NCGroebnerBasis) -- see isCentral -- Determines if an element is central
- isCommutative(NCRing) -- Returns whether an NCRing is commutative
- isExterior(NCRing) -- see isCommutative(NCRing) -- Returns whether an NCRing is commutative
- isExterior(Ring) -- see isCommutative(NCRing) -- Returns whether an NCRing is commutative
- isConstant(NCRingElement) -- Returns whether the NCRingElement is constant
- isHomogeneous(NCIdeal) -- Determines whether the input defines a homogeneous object
- isHomogeneous(NCLeftIdeal) -- see isHomogeneous(NCIdeal) -- Determines whether the input defines a homogeneous object
- isHomogeneous(NCMatrix) -- see isHomogeneous(NCIdeal) -- Determines whether the input defines a homogeneous object
- isHomogeneous(NCRightIdeal) -- see isHomogeneous(NCIdeal) -- Determines whether the input defines a homogeneous object
- isHomogeneous(NCRing) -- see isHomogeneous(NCIdeal) -- Determines whether the input defines a homogeneous object
- isHomogeneous(NCRingElement) -- see isHomogeneous(NCIdeal) -- Determines whether the input defines a homogeneous object
- isHomogeneous(NCRingMap) -- Determines if an NCRingMap preserves the natural grading
- isLeftRegular(NCRingElement,ZZ) -- see isLeftRegular -- Determines if a given (homogeneous) element is regular in a given degree
- isRightRegular(NCRingElement,ZZ) -- see isLeftRegular -- Determines if a given (homogeneous) element is regular in a given degree
- isNormal(NCRingElement) -- Determines if a given NCRingElement is normal
- isWellDefined(NCRingMap) -- Determines if an NCRingMap is well-defined.
- kernelComponent(ZZ,NCRingMap) -- see kernelComponent -- Computes a basis of the kernel of a ring map in a specified degree.
- leadCoefficient(NCRingElement) -- Returns the lead monomial of an NCRingElement
- leadMonomial(NCRingElement) -- Returns the lead monomial of an NCRingElement
- leadTerm(NCRingElement) -- Returns the lead term of an NCRingElement
- leftMultiplicationMap(NCRingElement,List,List) -- see leftMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
- leftMultiplicationMap(NCRingElement,ZZ) -- see leftMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
- leftMultiplicationMap(NCRingElement,ZZ,ZZ) -- see leftMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
- rightMultiplicationMap(NCRingElement,List,List) -- see leftMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
- rightMultiplicationMap(NCRingElement,ZZ) -- see leftMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
- rightMultiplicationMap(NCRingElement,ZZ,ZZ) -- see leftMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
- lift(NCMatrix) -- Lifts an NCMatrix
- List * NCRingElement -- Scales a list by an NCRingElement on the right
- List / NCRingMap -- Applies an NCRingMap to each element of a list
- Matrix * NCMatrix -- Product of NCMatrices
- matrix(NCRingMap) -- An NCMatrix associated to an NCRingMap.
- minimizeRelations(List) -- see minimizeRelations -- Minimizes a list of NCRingElements
- monomials(NCRingElement) -- Returns the monomials appearing in the NCRingElement
- ncGroebnerBasis(List) -- see ncGroebnerBasis -- Compute a noncommutative Groebner basis.
- ncGroebnerBasis(NCIdeal) -- see ncGroebnerBasis -- Compute a noncommutative Groebner basis.
- ncIdeal(List) -- see ncIdeal -- Define a two-sided ideal in a noncommutative ring
- ncIdeal(NCRingElement) -- see ncIdeal -- Define a two-sided ideal in a noncommutative ring
- NCIdeal + NCIdeal -- Sum of NCIdeals
- ncLeftIdeal(List) -- see ncLeftIdeal -- Define a left ideal in a noncommutative ring
- ncLeftIdeal(NCRingElement) -- see ncLeftIdeal -- Define a left ideal in a noncommutative ring
- NCLeftIdeal + NCLeftIdeal -- Sum of NCLeftIdeals
- ncMap(NCRing,NCRing,List) -- see ncMap -- Make a map to or from an NCRing
- ncMap(NCRing,Ring,List) -- see ncMap -- Make a map to or from an NCRing
- ncMap(Ring,NCRing,List) -- see ncMap -- Make a map to or from an NCRing
- ncMatrix(List) -- see ncMatrix -- Create an NCMatrix
- ncMatrix(NCRing,List,List) -- see ncMatrix -- Create an NCMatrix
- NCMatrix % NCGroebnerBasis -- Reduces the entries of an NCMatrix with respect to an NCGroebnerBasis
- NCMatrix * Matrix -- Product of NCMatrices
- NCMatrix * NCMatrix -- Product of NCMatrices
- NCMatrix * NCRingElement -- Product of NCMatrices
- NCMatrix * QQ -- Product of NCMatrices
- NCMatrix * RingElement -- Product of NCMatrices
- NCMatrix * ZZ -- Product of NCMatrices
- NCMatrix + NCMatrix -- Add NCMatrices
- NCMatrix - NCMatrix -- Subtract NCMatrices
- NCMatrix // NCMatrix -- Factor one map through another
- NCMatrix == NCMatrix -- Test equality of matrices
- NCMatrix == ZZ -- see NCMatrix == NCMatrix -- Test equality of matrices
- ZZ == NCMatrix -- see NCMatrix == NCMatrix -- Test equality of matrices
- NCMatrix ^ List -- Select some rows of an NCMatrix
- NCMatrix ^ ZZ -- Exponentiate an NCMatrix
- NCMatrix _ List -- Select some columns of an NCMatrix
- NCMatrix _ ZZ -- Induced map in homogeneous degree n
- NCMatrix | NCMatrix -- Join NCMatrices horizontally
- NCMatrix || NCMatrix -- Join NCMatrices vertically
- NCMatrix Array -- Graded shift of an NCMatrix.
- NCPolynomialRing / NCIdeal -- Construct a NCQuotientRing
- ncRightIdeal(List) -- see ncRightIdeal -- Define a right ideal in a noncommutative ring
- ncRightIdeal(NCRingElement) -- see ncRightIdeal -- Define a right ideal in a noncommutative ring
- NCRightIdeal + NCRightIdeal -- Sum of NCRightIdeals
- NCRingElement % NCGroebnerBasis -- Reduces a NCRingElement by a NCGroebnerBasis
- QQ % NCGroebnerBasis -- see NCRingElement % NCGroebnerBasis -- Reduces a NCRingElement by a NCGroebnerBasis
- ZZ % NCGroebnerBasis -- see NCRingElement % NCGroebnerBasis -- Reduces a NCRingElement by a NCGroebnerBasis
- NCRingElement * List -- Scales a list by an NCRingElement on the left
- NCRingElement * NCMatrix -- Product of NCMatrices
- NCRingMap + NCRingMap -- Basic operations with NCRingMaps
- NCRingMap ^ ZZ -- see NCRingMap + NCRingMap -- Basic operations with NCRingMaps
- QQ * NCRingMap -- see NCRingMap + NCRingMap -- Basic operations with NCRingMaps
- RingElement * NCRingMap -- see NCRingMap + NCRingMap -- Basic operations with NCRingMaps
- ZZ * NCRingMap -- see NCRingMap + NCRingMap -- Basic operations with NCRingMaps
- NCRingMap @@ NCRingMap -- Compose two NCRingMaps
- NCRingMap _ ZZ -- Matrix of one homogeneous component of an NCRingMap
- NCRingMap NCGroebnerBasis -- see NCRingMap NCIdeal -- Apply a ring map to the generators of an ideal
- NCRingMap NCIdeal -- Apply a ring map to the generators of an ideal
- NCRingMap NCMatrix -- see NCRingMap NCRingElement -- Apply an NCRingMap to an element or matrix
- NCRingMap NCRingElement -- Apply an NCRingMap to an element or matrix
- NCRingMap RingElement -- see NCRingMap NCRingElement -- Apply an NCRingMap to an element or matrix
- normalAutomorphism(NCRingElement) -- see normalAutomorphism -- Computes the automorphism determined by a normal homogeneous element
- normalElements(NCQuotientRing,ZZ,Symbol,Symbol) -- see normalElements -- Finds normal elements
- normalElements(NCRingMap,ZZ) -- Finds elements normalized by a ring map
- normalFormBergman(List,NCGroebnerBasis) -- see normalFormBergman -- Calls Bergman for a normal form calculation
- normalFormBergman(NCRingElement,NCGroebnerBasis) -- see normalFormBergman -- Calls Bergman for a normal form calculation
- numgens(NCRing) -- The number of algebra generators of an NCRing
- oppositeRing(NCRing) -- see oppositeRing -- Creates the opposite ring of a noncommutative ring
- oreExtension(NCRing,NCRingMap,NCRingElement) -- see oreExtension -- Creates an Ore extension of a noncommutative ring
- oreExtension(NCRing,NCRingMap,NCRingMap,NCRingElement) -- see oreExtension -- Creates an Ore extension of a noncommutative ring
- oreExtension(NCRing,NCRingMap,NCRingMap,Symbol) -- see oreExtension -- Creates an Ore extension of a noncommutative ring
- oreExtension(NCRing,NCRingMap,Symbol) -- see oreExtension -- Creates an Ore extension of a noncommutative ring
- oreIdeal(NCRing,NCRingMap,NCRingElement) -- see oreIdeal -- Creates the defining ideal of an Ore extension of a noncommutative ring
- oreIdeal(NCRing,NCRingMap,NCRingMap,NCRingElement) -- see oreIdeal -- Creates the defining ideal of an Ore extension of a noncommutative ring
- oreIdeal(NCRing,NCRingMap,NCRingMap,Symbol) -- see oreIdeal -- Creates the defining ideal of an Ore extension of a noncommutative ring
- oreIdeal(NCRing,NCRingMap,Symbol) -- see oreIdeal -- Creates the defining ideal of an Ore extension of a noncommutative ring
- QQ * NCMatrix -- Product of NCMatrices
- NCRing ** NCRing -- see qTensorProduct -- Define the (q-)commuting tensor product
- qTensorProduct(NCRing,NCRing,QQ) -- see qTensorProduct -- Define the (q-)commuting tensor product
- qTensorProduct(NCRing,NCRing,RingElement) -- see qTensorProduct -- Define the (q-)commuting tensor product
- qTensorProduct(NCRing,NCRing,ZZ) -- see qTensorProduct -- Define the (q-)commuting tensor product
- quadraticClosure(NCIdeal) -- see quadraticClosure -- Creates the subideal generated by quadratic elements of a given ideal
- quadraticClosure(NCQuotientRing) -- see quadraticClosure -- Creates the subideal generated by quadratic elements of a given ideal
- betti(NCChainComplex) -- see resolution(NCMatrix) -- Compute the resolution of coker M as a map of free right modules
- resolution(NCMatrix) -- Compute the resolution of coker M as a map of free right modules
- rightKernel(NCMatrix,ZZ) -- see rightKernel -- Method for computing kernels of matrices over noncommutative rings in a given degree without using Bergman
- rightKernelBergman(NCMatrix) -- see rightKernelBergman -- Methods for computing kernels of matrices over noncommutative rings using Bergman
- ring(NCGroebnerBasis) -- see ring(NCIdeal) -- Returns the ring of an NCIdeal or NCGroebnerBasis
- ring(NCIdeal) -- Returns the ring of an NCIdeal or NCGroebnerBasis
- ring(NCLeftIdeal) -- Returns the ring of an NCLeftIdeal
- ring(NCMatrix) -- Gives the ring of the NCMatrix
- ring(NCRightIdeal) -- Returns the ring of an NCRightIdeal
- ring(NCRingElement) -- Returns the NCRing of an NCRingElement
- RingElement * NCMatrix -- Product of NCMatrices
- setWeights(NCRing,List) -- see setWeights -- Set a nonstandard grading for a NCRing.
- size(NCRingElement) -- Returns the number of terms of an NCRingElement
- skewPolynomialRing(Ring,Matrix,List) -- see skewPolynomialRing -- Defines a skew polynomial ring via a skewing matrix
- skewPolynomialRing(Ring,QQ,List) -- see skewPolynomialRing(Ring,RingElement,List) -- Defines a skew polynomial ring via a scaling factor
- skewPolynomialRing(Ring,RingElement,List) -- Defines a skew polynomial ring via a scaling factor
- skewPolynomialRing(Ring,ZZ,List) -- see skewPolynomialRing(Ring,RingElement,List) -- Defines a skew polynomial ring via a scaling factor
- source(NCRingMap) -- Source of a map
- sparseCoeffs(List) -- see sparseCoeffs -- Converts ring elements into vectors over the coefficient ring
- sparseCoeffs(NCRingElement) -- see sparseCoeffs -- Converts ring elements into vectors over the coefficient ring
- support(NCRingElement) -- Returns the variables appearing in the NCRingElement
- target(NCRingMap) -- Target of a map
- terms(NCRingElement) -- Returns the terms of an NCRingElement
- threeDimSklyanin(Ring,List) -- see threeDimSklyanin -- Defines a three-dimensional Sklyanin with given parameters
- threeDimSklyanin(Ring,List,List) -- see threeDimSklyanin -- Defines a three-dimensional Sklyanin with given parameters
- toM2Ring(NCRing) -- see toM2Ring -- Compute the abelianization of an NCRing and returns a Ring.
- toNCRing(Ring) -- see toNCRing -- Converts a Ring to an NCRing
- toString(NCRingElement) -- Converts an NCRingElement to a string
- transpose(NCMatrix) -- Transposes an NCMatrix
- twoSidedNCGroebnerBasisBergman(List) -- see twoSidedNCGroebnerBasisBergman -- Calls Bergman to compute a two sided noncommutative Groebner Basis.
- twoSidedNCGroebnerBasisBergman(NCIdeal) -- see twoSidedNCGroebnerBasisBergman -- Calls Bergman to compute a two sided noncommutative Groebner Basis.
- use(NCRing) -- Brings the variables of a particular NCRing in scope
- ZZ * NCMatrix -- Product of NCMatrices
Symbols
- endomorphismRingGens -- see endomorphismRing -- Methods for creating endomorphism rings of modules over a commutative ring
- rightKernelDegreeLimit -- see rightKernelBergman -- Methods for computing kernels of matrices over noncommutative rings using Bergman

For the programmer

The object NCAlgebra is a package.