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<!DOCTYPE html> <html lang="en"> <head> <title>LieTypes : Table of Contents</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="Common types for Lie groups and Lie algebras" href="index.html">LieTypes</a> :: <a href="toc.html">Table of Contents</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> next | previous | forward | backward | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <h1>LieTypes : Table of Contents</h1> <ul> <li><span><a title="Common types for Lie groups and Lie algebras" href="index.html">LieTypes</a> -- Common types for Lie groups and Lie algebras</span></li> <li><span><a title="Computes the action of the nth Adams operator on a Lie algebra module" href="_adams.html">adams</a> -- Computes the action of the nth Adams operator on a Lie algebra module</span></li> <li><span><a title="The adjoint module of a Lie algebra" href="_adjoint__Module.html">adjointModule</a> -- The adjoint module of a Lie algebra</span></li> <li><span><a title="A Lie algebra module viewed as a module over a Lie subalgebra" href="_branching__Rule.html">branchingRule</a> -- A Lie algebra module viewed as a module over a Lie subalgebra</span></li> <li><span><a title="Provide the Cartan matrix of a simple Lie algebra" href="_cartan__Matrix.html">cartanMatrix</a> -- Provide the Cartan matrix of a simple Lie algebra</span></li> <li><span><a title="computes the scalar by which the Casimir operator acts on an irreducible Lie algebra module" href="_casimir__Scalar.html">casimirScalar</a> -- computes the scalar by which the Casimir operator acts on an irreducible Lie algebra module</span></li> <li><span><a title="Computes the character of a Lie algebra module" href="_character.html">character</a> -- Computes the character of a Lie algebra module</span></li> <li><span><a title="computes the dimension of a Lie algebra module as a vector space over the ground field" href="_dim_lp__Lie__Algebra__Module_rp.html">dim(LieAlgebraModule)</a> -- computes the dimension of a Lie algebra module as a vector space over the ground field</span></li> <li><span><a title="returns the dual Coxeter number of a simple Lie algebra" href="_dual__Coxeter__Number.html">dualCoxeterNumber</a> -- returns the dual Coxeter number of a simple Lie algebra</span></li> <li><span><a title="Provide the Dynkin diagram of a simple Lie algebra" href="_dynkin__Diagram.html">dynkinDiagram</a> -- Provide the Dynkin diagram of a simple Lie algebra</span></li> <li><span><a title="gives the embedding of Cartan subalgebras of one Lie algebra into another" href="_embedding.html">embedding</a> -- gives the embedding of Cartan subalgebras of one Lie algebra into another</span></li> <li><span><a title="computes the multiplicity of W in the fusion product of U and V" href="_fusion__Coefficient.html">fusionCoefficient</a> -- computes the multiplicity of W in the fusion product of U and V</span></li> <li><span><a title="computes the multiplicities of irreducibles in the decomposition of the fusion product of U and V" href="_fusion__Product.html">fusionProduct</a> -- computes the multiplicities of irreducibles in the decomposition of the fusion product of U and V</span></li> <li><span><a title="returns the highest root of a simple Lie algebra" href="_highest__Root.html">highestRoot</a> -- returns the highest root of a simple Lie algebra</span></li> <li><span><a title="construct the irreducible Lie algebra module with given highest weight" href="_irreducible__Lie__Algebra__Module.html">irreducibleLieAlgebraModule</a> -- construct the irreducible Lie algebra module with given highest weight</span></li> <li><span><a title="Whether a Lie algebra module is irreducible or not" href="_is__Irreducible.html">isIrreducible</a> -- Whether a Lie algebra module is irreducible or not</span></li> <li><span><a title="tests whether two Lie algebra are isomorphic" href="_is__Isomorphic.html">isIsomorphic</a> -- tests whether two Lie algebra are isomorphic</span></li> <li><span><a title="computes the scaled Killing form applied to two weights" href="_killing__Form.html">killingForm</a> -- computes the scaled Killing form applied to two weights</span></li> <li><span><a title="class for Lie algebras" href="___Lie__Algebra.html">LieAlgebra</a> -- class for Lie algebras</span></li> <li><span><a title="Take the direct sum of Lie algebras" href="___Lie__Algebra_sp_pl_pl_sp__Lie__Algebra.html">LieAlgebra ++ LieAlgebra</a> -- Take the direct sum of Lie algebras</span></li> <li><span><a title="tests equality of LieAlgebra" href="___Lie__Algebra_sp_eq_eq_sp__Lie__Algebra.html">LieAlgebra == LieAlgebra</a> -- tests equality of LieAlgebra</span></li> <li><span><a title="selects one summand of a semi-simple Lie Algebra" href="___Lie__Algebra_sp_us_sp__Z__Z.html">LieAlgebra _ ZZ</a> -- selects one summand of a semi-simple Lie Algebra</span></li> <li><span><a title="gives the list of summands of a semi-simple Lie Algebra" href="___Lie__Algebra_sp_us_st.html">LieAlgebra _*</a> -- gives the list of summands of a semi-simple Lie Algebra</span></li> <li><span><a title="class for Lie algebra modules" href="___Lie__Algebra__Module.html">LieAlgebraModule</a> -- class for Lie algebra modules</span></li> <li><span><a title="tensor product of LieAlgebraModules" href="___Lie__Algebra__Module_sp_st_st_sp__Lie__Algebra__Module.html">LieAlgebraModule ** LieAlgebraModule</a> -- tensor product of LieAlgebraModules</span></li> <li><span><a title="direct sum of LieAlgebraModules" href="___Lie__Algebra__Module_sp_pl_pl_sp__Lie__Algebra__Module.html">LieAlgebraModule ++ LieAlgebraModule</a> -- direct sum of LieAlgebraModules</span></li> <li><span><a title="Take the tensor product of modules over different Lie algebras" href="___Lie__Algebra__Module_sp_at_sp__Lie__Algebra__Module.html">LieAlgebraModule @ LieAlgebraModule</a> -- Take the tensor product of modules over different Lie algebras</span></li> <li><span><a title="Computes the nth tensor power of a Lie algebra module" href="___Lie__Algebra__Module_sp%5E_st_st_sp__Z__Z.html">LieAlgebraModule ^** ZZ</a> -- Computes the nth tensor power of a Lie algebra module</span></li> <li><span><a title="Pick out one irreducible submodule of a Lie algebra module" href="___Lie__Algebra__Module_sp_us_sp__Z__Z.html">LieAlgebraModule _ ZZ</a> -- Pick out one irreducible submodule of a Lie algebra module</span></li> <li><span><a title="List irreducible submodules of a Lie algebra module" href="___Lie__Algebra__Module_sp_us_st.html">LieAlgebraModule _*</a> -- List irreducible submodules of a Lie algebra module</span></li> <li><span><a title="finds a Lie algebra module based on its weights" href="___Lie__Algebra__Module__From__Weights.html">LieAlgebraModuleFromWeights</a> -- finds a Lie algebra module based on its weights</span></li> <li><span><a title="compute the multiplicity of a weight in a Lie algebra module" href="_multiplicity_lp__List_cm__Lie__Algebra__Module_rp.html">multiplicity(List,LieAlgebraModule)</a> -- compute the multiplicity of a weight in a Lie algebra module</span></li> <li><span><a title="Define a Lie algebra from its Cartan matrix" href="_new_sp__Lie__Algebra_spfrom_sp__Matrix.html">new LieAlgebra from Matrix</a> -- Define a Lie algebra from its Cartan matrix</span></li> <li><span><a title="returns the positive (co)roots of a simple Lie algebra" href="_positive__Roots.html">positiveRoots</a> -- returns the positive (co)roots of a simple Lie algebra</span></li> <li><span><a title="Compute principal specialization of character or quantum dimension" href="_qdim.html">qdim</a> -- Compute principal specialization of character or quantum dimension</span></li> <li><span><a title="construct a simple Lie algebra" href="_simple__Lie__Algebra.html">simpleLieAlgebra</a> -- construct a simple Lie algebra</span></li> <li><span><a title="returns the simple roots of a simple Lie algebra" href="_simple__Roots.html">simpleRoots</a> -- returns the simple roots of a simple Lie algebra</span></li> <li><span><a title="computes w* for a weight w" href="_star__Involution.html">starInvolution</a> -- computes w* for a weight w</span></li> <li><span><a title="Define a sub-Lie algebra of an existing one" href="_sub__Lie__Algebra.html">subLieAlgebra</a> -- Define a sub-Lie algebra of an existing one</span></li> <li><span><a title="Computes the nth symmetric / exterior tensor power of a Lie algebra module" href="_symmetric__Power_lp__Z__Z_cm__Lie__Algebra__Module_rp.html">symmetricPower(ZZ,LieAlgebraModule)</a> -- Computes the nth symmetric / exterior tensor power of a Lie algebra module</span></li> <li><span><a title="computes the multiplicity of W in U tensor V" href="_tensor__Coefficient.html">tensorCoefficient</a> -- computes the multiplicity of W in U tensor V</span></li> <li><span><a title="The trivial module of a Lie algebra" href="_trivial__Module.html">trivialModule</a> -- The trivial module of a Lie algebra</span></li> <li><span><a title="computes the weights in a Lie algebra module and their multiplicities" href="_weight__Diagram.html">weightDiagram</a> -- computes the weights in a Lie algebra module and their multiplicities</span></li> <li><span><a title="the dominant integral weights of level less than or equal to l" href="_weyl__Alcove.html">weylAlcove</a> -- the dominant integral weights of level less than or equal to l</span></li> <li><span><a title="The zero module of a Lie algebra" href="_zero__Module.html">zeroModule</a> -- The zero module of a Lie algebra</span></li> </ul> </body> </html>