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NCAlgebra :: NCAlgebra

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
  • 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.

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