NCAlgebra : Table of Contents

• NCAlgebra

  • Basic operations on noncommutative algebras
  • General setup information
  • Using the Bergman interface
• - 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
• 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 -- Finds central elements in a given degree
• coefficientRing(NCRing) -- Returns the base ring of an NCRing
• coordinates -- 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
• entries(NCMatrix) -- Returns the entries of the NCMatrix
• 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.
• 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
• 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
• 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
• 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(NCRingMap) -- Determines if an NCRingMap preserves the natural grading
• 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 -- 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
• 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
• monomials(NCRingElement) -- Returns the monomials appearing in the NCRingElement
• NCGroebnerBasis -- Type of a Groebner basis for an NCIdeal in an NCRing.
• ncGroebnerBasis -- 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
• NCLeftIdeal -- Type of a left ideal in a noncommutative ring
• ncLeftIdeal -- Define a left ideal in a noncommutative ring
• NCLeftIdeal + NCLeftIdeal -- Sum of NCLeftIdeals
• ncMap -- 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 ^ 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 -- 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
• NCRing -- Type of a noncommutative ring
• 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 -- Matrix of one homogeneous component of an NCRingMap
• NCRingMap NCIdeal -- Apply a ring map to the generators of an ideal
• NCRingMap NCRingElement -- Apply an NCRingMap to an element or matrix
• normalAutomorphism -- Computes the automorphism determined by a normal homogeneous element
• normalElements -- Finds normal elements
• normalElements(NCRingMap,ZZ) -- Finds elements normalized by a ring map
• normalFormBergman -- 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
• oreExtension -- Creates an Ore extension of a noncommutative ring
• oreIdeal -- Creates the defining ideal of an Ore extension of a noncommutative ring
• QQ * NCMatrix -- Product of NCMatrices
• qTensorProduct -- Define the (q-)commuting tensor product
• quadraticClosure -- 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
• rightKernelBergman -- Methods for computing kernels of matrices over noncommutative rings using Bergman
• 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 -- 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,RingElement,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
• 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
• toM2Ring -- Compute the abelianization of an NCRing and returns a Ring.
• toNCRing -- 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.
• use(NCRing) -- Brings the variables of a particular NCRing in scope
• ZZ * NCMatrix -- Product of NCMatrices