NCAlgebra : Index
- - NCMatrix -- Negates NCMatrices
- ambient(NCQuotientRing) -- Ambient ring of an NCQuotientRing
- ambient(NCRingMap) -- Extends an NCRingMap to the ambient ring of the source.
- assignDegrees -- Weights entries of a matrix to make associated map of free modules graded
- assignDegrees(NCMatrix) -- Weights entries of a matrix to make associated map of free modules graded
- assignDegrees(NCMatrix,List,List) -- 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
- Basic operations on noncommutative algebras
- 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.
- betti(NCChainComplex) -- Compute the resolution of coker M as a map of free right modules
- centralElements -- Finds central elements in a given degree
- centralElements(NCRing,ZZ) -- Finds central elements in a given degree
- coefficientRing(NCRing) -- Returns the base ring of an NCRing
- coordinates -- Computes coordinates relative to a given basis
- coordinates(..., Basis => ...) -- Computes coordinates relative to a given basis
- coordinates(List) -- Computes coordinates relative to a given basis
- coordinates(NCRingElement) -- Computes coordinates relative to a given basis
- degree(NCRingElement) -- Returns the degree of an NCRingElement
- endomorphismRing -- Methods for creating endomorphism rings of modules over a commutative ring
- endomorphismRing(Module,Symbol) -- Methods for creating endomorphism rings of modules over a commutative ring
- endomorphismRingGens -- Methods for creating endomorphism rings of modules over a commutative ring
- entries(NCMatrix) -- Returns the entries of the NCMatrix
- envelopingAlgebra -- Create the enveloping algebra
- envelopingAlgebra(NCRing,Symbol) -- Create the enveloping algebra
- fourDimSklyanin -- Defines a four-dimensional Sklyanin with given parameters
- fourDimSklyanin(Ring,List) -- Defines a four-dimensional Sklyanin with given parameters
- fourDimSklyanin(Ring,List,List) -- Defines a four-dimensional Sklyanin with given parameters
- freeProduct -- Define the free product of two algebras
- freeProduct(NCRing,NCRing) -- Define the free product of two algebras
- gbFromOutputFile -- Read in a NCGroebnerBasis from a Bergman output file.
- gbFromOutputFile(..., CacheBergmanGB => ...) -- Read in a NCGroebnerBasis from a Bergman output file.
- gbFromOutputFile(..., MakeMonic => ...) -- Read in a NCGroebnerBasis from a Bergman output file.
- gbFromOutputFile(..., ReturnIdeal => ...) -- Read in a NCGroebnerBasis from a Bergman output file.
- gbFromOutputFile(NCPolynomialRing,String) -- Read in a NCGroebnerBasis from a Bergman output file.
- gddKernel -- Computes a homogeneous generating set of the kernel of a ring map.
- gddKernel(ZZ,NCRingMap) -- Computes a homogeneous generating set of the kernel of a ring map.
- General setup information
- 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 -- Calls Bergman to compute the Hilbert series of an NCQuotientRing
- hilbertBergman(..., DegreeLimit => ...) -- Calls Bergman to compute the Hilbert series of an NCQuotientRing
- hilbertBergman(NCQuotientRing) -- 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 -- Computes the dual of a pure homogeneous ideal
- homogDual(NCIdeal) -- Computes the dual of a pure homogeneous ideal
- homogDual(NCQuotientRing) -- Computes the dual of a pure homogeneous ideal
- homogDual(Ring) -- 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 -- Determines if an element is central
- isCentral(NCRingElement) -- Determines if an element is central
- isCentral(NCRingElement,NCGroebnerBasis) -- Determines if an element is central
- isCommutative(NCRing) -- Returns whether an NCRing is commutative
- isConstant(NCRingElement) -- Returns whether the NCRingElement is constant
- isExterior -- Returns whether an NCRing is commutative
- isExterior(NCRing) -- Returns whether an NCRing is commutative
- isExterior(Ring) -- Returns whether an NCRing is commutative
- isHomogeneous(NCIdeal) -- Determines whether the input defines a homogeneous object
- isHomogeneous(NCLeftIdeal) -- Determines whether the input defines a homogeneous object
- isHomogeneous(NCMatrix) -- Determines whether the input defines a homogeneous object
- isHomogeneous(NCRightIdeal) -- Determines whether the input defines a homogeneous object
- isHomogeneous(NCRing) -- Determines whether the input defines a homogeneous object
- isHomogeneous(NCRingElement) -- Determines whether the input defines a homogeneous object
- isHomogeneous(NCRingMap) -- Determines if an NCRingMap preserves the natural grading
- isLeftRegular -- Determines if a given (homogeneous) element is regular in a given degree
- isLeftRegular(NCRingElement,ZZ) -- Determines if a given (homogeneous) element is regular in a given degree
- isNormal(NCRingElement) -- Determines if a given NCRingElement is normal
- isRightRegular -- Determines if a given (homogeneous) element is regular in a given degree
- isRightRegular(NCRingElement,ZZ) -- Determines if a given (homogeneous) element is regular in a given degree
- isWellDefined(NCRingMap) -- Determines if an NCRingMap is well-defined.
- kernelComponent -- Computes a basis of the kernel of a ring map in a specified degree.
- kernelComponent(ZZ,NCRingMap) -- 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 -- Computes a matrix for left or right multiplication by a homogeneous element
- leftMultiplicationMap(NCRingElement,List,List) -- Computes a matrix for left or right multiplication by a homogeneous element
- leftMultiplicationMap(NCRingElement,ZZ) -- Computes a matrix for left or right multiplication by a homogeneous element
- leftMultiplicationMap(NCRingElement,ZZ,ZZ) -- 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 -- Minimizes a list of NCRingElements
- minimizeRelations(..., Verbosity => ...) -- Minimizes a list of NCRingElements
- minimizeRelations(List) -- Minimizes a list of NCRingElements
- monomials(NCRingElement) -- Returns the monomials appearing in the NCRingElement
- NCAlgebra
- NCChainComplex -- Compute the resolution of coker M as a map of free right modules
- NCGroebnerBasis -- Type of a Groebner basis for an NCIdeal in an NCRing.
- ncGroebnerBasis -- Compute a noncommutative Groebner basis.
- ncGroebnerBasis(..., InstallGB => ...) -- Compute a noncommutative Groebner basis.
- ncGroebnerBasis(List) -- Compute a noncommutative Groebner basis.
- ncGroebnerBasis(NCIdeal) -- Compute a noncommutative Groebner basis.
- NCIdeal -- Type of a two-sided ideal in a noncommutative ring
- ncIdeal -- Define a two-sided ideal in a noncommutative ring
- NCIdeal + NCIdeal -- Sum of NCIdeals
- ncIdeal(List) -- Define a two-sided ideal in a noncommutative ring
- ncIdeal(NCRingElement) -- Define a two-sided ideal in a noncommutative ring
- NCLeftIdeal -- Type of a left ideal in a noncommutative ring
- ncLeftIdeal -- Define a left ideal in a noncommutative ring
- NCLeftIdeal + NCLeftIdeal -- Sum of NCLeftIdeals
- ncLeftIdeal(List) -- Define a left ideal in a noncommutative ring
- ncLeftIdeal(NCRingElement) -- Define a left ideal in a noncommutative ring
- ncMap -- Make a map to or from an NCRing
- ncMap(..., Derivation => ...) -- Make a map to or from an NCRing
- ncMap(NCRing,NCRing,List) -- Make a map to or from an NCRing
- ncMap(NCRing,Ring,List) -- Make a map to or from an NCRing
- ncMap(Ring,NCRing,List) -- Make a map to or from an NCRing
- NCMatrix -- Type of a matrix over a noncommutative ring
- 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 -- 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.
- ncMatrix(List) -- Create an NCMatrix
- ncMatrix(NCRing,List,List) -- Create an NCMatrix
- NCPolynomialRing -- Type of a noncommutative polynomial ring
- NCPolynomialRing / NCIdeal -- Construct a NCQuotientRing
- NCQuotientRing -- Type of a noncommutative ring
- NCRightIdeal -- Type of a right ideal in a noncommutative ring
- ncRightIdeal -- Define a right ideal in a noncommutative ring
- NCRightIdeal + NCRightIdeal -- Sum of NCRightIdeals
- ncRightIdeal(List) -- Define a right ideal in a noncommutative ring
- ncRightIdeal(NCRingElement) -- Define a right ideal in a noncommutative ring
- NCRing -- Type of a noncommutative ring
- NCRing ** NCRing -- Define the (q-)commuting tensor product
- NCRingElement -- Type of an element in a noncommutative ring
- 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 -- Type of a map to or from a noncommutative ring.
- NCRingMap + NCRingMap -- Basic operations with NCRingMaps
- NCRingMap @@ NCRingMap -- Compose two NCRingMaps
- NCRingMap ^ ZZ -- Basic operations with NCRingMaps
- NCRingMap _ ZZ -- Matrix of one homogeneous component of an NCRingMap
- NCRingMap NCGroebnerBasis -- Apply a ring map to the generators of an ideal
- NCRingMap NCIdeal -- Apply a ring map to the generators of an ideal
- NCRingMap NCMatrix -- Apply an NCRingMap to an element or matrix
- NCRingMap NCRingElement -- Apply an NCRingMap to an element or matrix
- NCRingMap RingElement -- Apply an NCRingMap to an element or matrix
- normalAutomorphism -- Computes the automorphism determined by a normal homogeneous element
- normalAutomorphism(NCRingElement) -- Computes the automorphism determined by a normal homogeneous element
- normalElements -- Finds normal elements
- normalElements(NCQuotientRing,ZZ,Symbol,Symbol) -- Finds normal elements
- normalElements(NCRingMap,ZZ) -- Finds elements normalized by a ring map
- normalFormBergman -- Calls Bergman for a normal form calculation
- normalFormBergman(List,NCGroebnerBasis) -- Calls Bergman for a normal form calculation
- normalFormBergman(NCRingElement,NCGroebnerBasis) -- Calls Bergman for a normal form calculation
- numgens(NCRing) -- The number of algebra generators of an NCRing
- oppositeRing -- Creates the opposite ring of a noncommutative ring
- oppositeRing(NCRing) -- Creates the opposite ring of a noncommutative ring
- oreExtension -- Creates an Ore extension of a noncommutative ring
- oreExtension(NCRing,NCRingMap,NCRingElement) -- Creates an Ore extension of a noncommutative ring
- oreExtension(NCRing,NCRingMap,NCRingMap,NCRingElement) -- Creates an Ore extension of a noncommutative ring
- oreExtension(NCRing,NCRingMap,NCRingMap,Symbol) -- Creates an Ore extension of a noncommutative ring
- oreExtension(NCRing,NCRingMap,Symbol) -- Creates an Ore extension of a noncommutative ring
- oreIdeal -- Creates the defining ideal of an Ore extension of a noncommutative ring
- oreIdeal(NCRing,NCRingMap,NCRingElement) -- Creates the defining ideal of an Ore extension of a noncommutative ring
- oreIdeal(NCRing,NCRingMap,NCRingMap,NCRingElement) -- Creates the defining ideal of an Ore extension of a noncommutative ring
- oreIdeal(NCRing,NCRingMap,NCRingMap,Symbol) -- Creates the defining ideal of an Ore extension of a noncommutative ring
- oreIdeal(NCRing,NCRingMap,Symbol) -- Creates the defining ideal of an Ore extension of a noncommutative ring
- QQ % NCGroebnerBasis -- Reduces a NCRingElement by a NCGroebnerBasis
- QQ * NCMatrix -- Product of NCMatrices
- QQ * NCRingMap -- Basic operations with NCRingMaps
- qTensorProduct -- Define the (q-)commuting tensor product
- qTensorProduct(NCRing,NCRing,QQ) -- Define the (q-)commuting tensor product
- qTensorProduct(NCRing,NCRing,RingElement) -- Define the (q-)commuting tensor product
- qTensorProduct(NCRing,NCRing,ZZ) -- Define the (q-)commuting tensor product
- quadraticClosure -- Creates the subideal generated by quadratic elements of a given ideal
- quadraticClosure(NCIdeal) -- Creates the subideal generated by quadratic elements of a given ideal
- quadraticClosure(NCQuotientRing) -- Creates the subideal generated by quadratic elements of a given ideal
- resolution(NCMatrix) -- Compute the resolution of coker M as a map of free right modules
- rightKernel -- Method for computing kernels of matrices over noncommutative rings in a given degree without using Bergman
- rightKernel(..., NumberOfBins => ...) -- Method for computing kernels of matrices over noncommutative rings in a given degree without using Bergman
- rightKernel(..., Verbosity => ...) -- Method for computing kernels of matrices over noncommutative rings in a given degree without using Bergman
- rightKernel(NCMatrix,ZZ) -- 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
- rightKernelBergman(..., DegreeLimit => ...) -- Methods for computing kernels of matrices over noncommutative rings using Bergman
- rightKernelBergman(NCMatrix) -- Methods for computing kernels of matrices over noncommutative rings using Bergman
- rightKernelDegreeLimit -- Methods for computing kernels of matrices over noncommutative rings using Bergman
- rightMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
- rightMultiplicationMap(NCRingElement,List,List) -- Computes a matrix for left or right multiplication by a homogeneous element
- rightMultiplicationMap(NCRingElement,ZZ) -- Computes a matrix for left or right multiplication by a homogeneous element
- rightMultiplicationMap(NCRingElement,ZZ,ZZ) -- Computes a matrix for left or right multiplication by a homogeneous element
- ring(NCGroebnerBasis) -- 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
- RingElement * NCRingMap -- Basic operations with NCRingMaps
- setWeights -- Set a nonstandard grading for a NCRing.
- setWeights(NCRing,List) -- Set a nonstandard grading for a NCRing.
- size(NCRingElement) -- Returns the number of terms of an NCRingElement
- skewPolynomialRing -- Defines a skew polynomial ring via a skewing matrix
- skewPolynomialRing(Ring,Matrix,List) -- Defines a skew polynomial ring via a skewing matrix
- skewPolynomialRing(Ring,QQ,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) -- Defines a skew polynomial ring via a scaling factor
- source(NCRingMap) -- Source of a map
- sparseCoeffs -- Converts ring elements into vectors over the coefficient ring
- sparseCoeffs(..., Monomials => ...) -- Converts ring elements into vectors over the coefficient ring
- sparseCoeffs(List) -- Converts ring elements into vectors over the coefficient ring
- sparseCoeffs(NCRingElement) -- 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 -- Defines a three-dimensional Sklyanin with given parameters
- threeDimSklyanin(Ring,List) -- Defines a three-dimensional Sklyanin with given parameters
- threeDimSklyanin(Ring,List,List) -- Defines a three-dimensional Sklyanin with given parameters
- toM2Ring -- Compute the abelianization of an NCRing and returns a Ring.
- toM2Ring(..., SkewCommutative => ...) -- Compute the abelianization of an NCRing and returns a Ring.
- toM2Ring(NCRing) -- Compute the abelianization of an NCRing and returns a Ring.
- toNCRing -- Converts a Ring to an NCRing
- toNCRing(Ring) -- Converts a Ring to an NCRing
- toString(NCRingElement) -- Converts an NCRingElement to a string
- transpose(NCMatrix) -- Transposes an NCMatrix
- twoSidedNCGroebnerBasisBergman -- Calls Bergman to compute a two sided noncommutative Groebner Basis.
- twoSidedNCGroebnerBasisBergman(..., CacheBergmanGB => ...) -- Calls Bergman to compute a two sided noncommutative Groebner Basis.
- twoSidedNCGroebnerBasisBergman(..., DegreeLimit => ...) -- Calls Bergman to compute a two sided noncommutative Groebner Basis.
- twoSidedNCGroebnerBasisBergman(..., MakeMonic => ...) -- Calls Bergman to compute a two sided noncommutative Groebner Basis.
- twoSidedNCGroebnerBasisBergman(..., NumModuleVars => ...) -- Calls Bergman to compute a two sided noncommutative Groebner Basis.
- twoSidedNCGroebnerBasisBergman(List) -- Calls Bergman to compute a two sided noncommutative Groebner Basis.
- twoSidedNCGroebnerBasisBergman(NCIdeal) -- Calls Bergman to compute a two sided noncommutative Groebner Basis.
- use(NCRing) -- Brings the variables of a particular NCRing in scope
- Using the Bergman interface
- ZZ % NCGroebnerBasis -- Reduces a NCRingElement by a NCGroebnerBasis
- ZZ * NCMatrix -- Product of NCMatrices
- ZZ * NCRingMap -- Basic operations with NCRingMaps
- ZZ == NCMatrix -- Test equality of matrices