% \iffalse meta-comment % % Copyright (C) 2018 - 2021 by ChairX % % This file may be distributed and/or modified under the % conditions of the LaTeX Project Public License, either % version 1.3 of this license or (at your option) any later % version. The latest version of this license is in: % % http://www.latex-project.org/lppl.txt % % and version 1.3 or later is part of all distributions of % LaTeX version 2005/12/01 or later. % % This file contains the documentation of all algebra related macros. % % Macros have to be described by (delete the first %) % \DescribeMacro{\macro} % Description and usage of the macro. % % The description will appear in the usage % part of the documentation. Use \subsubsection{} etc. for structuring. % % The implementation of the macros defined here has to be written in % chairxmathAlgebra.dtx %\fi % %\subsubsection{Fonts for Rings and Things} % %\DescribeMacro{\field} % Font for rings |\field{R}|: $\field{R}$\\ % Uses |fieldfont|. % %\DescribeMacro{\ring} % Font for rings |\ring{C}|: $\ring{C}$\\ % Uses |ringfont|. % %\DescribeMacro{\group} % Font for particular (matrix) groups |\group{SO}(3)|: $\group{SO}(3)$\\ % Uses |groupfont|. % %\DescribeMacro{\algebra} % Font for algebras |\algebra{A}|: $\algebra{A}$\\ % Uses |algebrafont|. % %\DescribeMacro{\module} % Font for modules |\module{M}|: $\module{M}$\\ % Uses |modulefont|. % %\DescribeMacro{\liealg} % Font for Lie algebras |\liealg{g}|: $\liealg{g}$\\ % Uses |liealgfont|. % %\DescribeMacro{\MC} % MC for Maurer-Cartan as a tiny index |\mu_\MC \in \liealg{g}^1|: $\mu_\MC \in \liealg{g}^1$\\ % Uses |scriptfont|. % %\DescribeMacro{\gerstenhaber} % Font for Gerstenhaber algebras |\gerstenhaber{G}|: $\gerstenhaber{G}$\\ % Uses |gerstenhaberfont|. % %\subsubsection{Some Symbols needed in Algebra} % %\DescribeMacro{\Pol} % Polynomials and polynomial functions |\Pol(T^*Q)|: $\Pol(T^*Q)$\\ % Uses |operatorfont|. % %\DescribeMacro{\lmult} % Left multiplications |\lmult_a|: $\lmult_a$\\ % Uses |operatorfont|. % %\DescribeMacro{\rmult} % Right multiplications |\rmult_b|: $\rmult_b$\\ % Uses |operatorfont|. % %\DescribeMacro{\Lmult} % Left multiplications |\Lmult_a|: $\Lmult_a$\\ % Uses |operatorfont|. % %\DescribeMacro{\Rmult} % Right multiplications |\Rmult_b|: $\Rmult_b$\\ % Uses |operatorfont|. % %\DescribeMacro{\Center} % Center |\Center(\algebra{A})|: $\Center(\algebra{A})$ % %\DescribeMacro{\ad} % Adjoint action (infinitesimal) |\ad(a)|: $\ad(a)$\\ % Uses |operatorfont|. % %\DescribeMacro{\Ad} % Adjoint action |\Ad_g|: $\Ad_g$\\ % Uses |operatorfont|. % %\DescribeMacro{\Conj} % Conjugation |\Conj_g|: $\Conj_g$\\ % Uses |operatorfont|. % %\DescribeMacro{\acts} % A generic (left) action map |g \acts a|: $g \acts a$ % %\DescribeMacro{\racts} % A generic right action map |a \racts g|: $a \racts g$ % %\DescribeMacro{\Char} % Characteristics of a field |\Char(\mathbb{k})|: $\Char(\mathbb{k})$\\ % Uses |operatorfont|. % %\DescribeMacro{\modulo} % Yet another modulo |n \modulo 2|: $n \modulo 2$\\ % Uses |operatorfont|. % % \DescribeMacro{\Clifford} % Clifford algebra generated by a vector space and a bilinear form: % |\Clifford(V, h)|: $\Clifford(V, h)$ \\ % Uses |operatorfont|. % % \DescribeMacro{\cClifford} % Complex Clifford algebra |\cClifford(V, h)|: $\cClifford(V, h)$ \\ % Uses |operatorfont|. % % \DescribeMacro{\Der} % ($*$-)Derivations |\Der(\algebra{A})|: $\Der(\algebra{A})$\\ % |\Der*(\algebra{A})|: $\Der*(\algebra{A})$\\ % Uses |operatorfont|. % % \DescribeMacro{\InnDer} % Inner ($*$-)derivations |\InnDer(\algebra{A})|: $\InnDer(\algebra{A})$\\ % |\InnDer*(\algebra{A})|: $\InnDer*(\algebra{A})$\\ % Uses |operatorfont|. % % \DescribeMacro{\OutDer} % Outer ($*$-)derivations |\OutDer(\algebra{A})|: $\OutDer(\algebra{A})$\\ % |\OutDer*(\algebra{A})|: $\OutDer*(\algebra{A})$\\ % Uses |operatorfont|. % % \DescribeMacro{\InnAut} % Inner ($*$-)automorphisms |\InnAut(\algebra{A})|: $\InnAut(\algebra{A})$\\ % |\InnAut*(\algebra{A})|: $\InnAut*(\algebra{A})$\\ % Uses |operatorfont|. % % \DescribeMacro{\OutAut} % Outer ($*$-)automorphisms |\OutAut(\algebra{A})|: $\OutAut(\algebra{A})$\\ % |\OutAut*(\algebra{A})|: $\OutAut*(\algebra{A})$\\ % Uses |operatorfont|. % % \DescribeMacro{\formal} % Formal power series in some variables |V\formal{\lambda}|: % $V \formal{\lambda}$ % % \DescribeMacro{\laurent} % Formal Laurent series in some variables |V\laurent{\lambda}|: % $V \laurent{\lambda}$ % % \DescribeMacro{\sweedler} % Smaller index for Sweedler notation in Hopf algebra theory \\ % |\Delta(a) = a_\sweedler{1} \tensor a_\sweedler{2}|: % $\Delta(a) = a_\sweedler{1} \tensor a_\sweedler{2}$ % %\subsubsection{Categories from Algebra} % %\DescribeMacro{\algebras} % Category of algebras |\algebras|: $\algebras$ \\ % Category of $*$-algebras |\algebras*|: $\algebras*$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\Algebras} % Category of unital algebras |\Algebras|: $\Algebras$ \\ % Category of unital $*$-algebras |\Algebras*|: $\Algebras*$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\reps} % Category of ($*$-)representations % |\reps_\algebra{C}(\algebra{B})|: $\reps_\algebra{C}(\algebra{B})$ \\ % |\reps*_\algebra{C}(\algebra{B})|: $\reps*_\algebra{C}(\algebra{B})$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\Reps} % Category of strongly non-degenerate ($^*$)-representations % |\Reps_\algebra{A}(\algebra{B})|:$\Reps_\algebra{A}(\algebra{B})$ \\ % |\Reps*_\algebra{A}(\algebra{B})|:$\Reps*_\algebra{A}(\algebra{B})$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\PoissonAlg} % Category of ($*$-)Poisson algebras |\PoissonAlg|: $\PoissonAlg$ \\ % |\PoissonAlg*|: $\PoissonAlg*$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\modules} % Category of (inner product) modules % |\modules_\algebra{A}(\algebra{B})|: $\modules_\algebra{A}(\algebra{B})$ \\ % |\modules*_\algebra{A}(\algebra{B})|: $\modules*_\algebra{A}(\algebra{B})$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\Leftmodules} % Category of left modules % |\Leftmodules{\algebra{A}}|: $\Leftmodules{\algebra{A}}$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\Rightmodules} % Category of right modules with optional subscript % |\Rightmodules[\category{C}]{\algebra{A}}|: $\Rightmodules[\category{C}]{\algebra{A}}$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\Modules} % Category of strongly non-degenerate (inner product) modules % |\Modules_\algebra{A}(\algebra{B})|: $\Modules_\algebra{A}(\algebra{B})$ \\ % |\Modules*_\algebra{A}(\algebra{B})|: $\Modules*_\algebra{A}(\algebra{B})$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\LeftModules} % Category of strongly non-degenerate left modules % |\LeftModules{\algebra{A}}|: $\LeftModules{\algebra{A}}$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\RightModules} % Category of strongly non-degenerate right modules with optional subscript % |\RightModules{\algebra{A}}|: $\RightModules{\algebra{A}}$ or % |\RightModules[\category{C}]{\algebra{A}}|: $\RightModules[\category{C}]{\algebra{A}}$\\ % Uses |categorynamefont|. % %\DescribeMacro{\Bimodules} % Category of (inner product) bimodules % |\Bimodules(\algebra{A},\algebra{B})|: $\Bimodules(\algebra{A},\algebra{B})$ \\ % |\Bimodules*(\algebra{A},\algebra{B})|: $\Bimodules*(\algebra{A},\algebra{B})$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\Rings} % Category of unital rings (meant to be associative) |\Rings|: $\Rings$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\Groups} % Category of groups |\Groups|: $\Groups$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\Ab} % Category of abelian groups |\Ab|: $\Ab$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\Lattices} % Category of lattices |\Lattices|: $\Lattices$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\Sets} % Category of sets |\Sets|: $\Sets$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\Vect} % Category of vector spaces |\Vect|: $\Vect$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\LieAlgs} % Category of Lie algebras |\LieAlgs|: $\LieAlgs$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\Posets} % Category of partially ordered sets |\Posets|: $\Posets$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\Directed} % Category of directed sets |\Directed|: $\Directed$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\GSets} % Category of $G$-Sets |\GSets|: $\GSets$ and |\Gsets[H]|: $\GSets[H]$ \\ % Uses |categorynamefont|. % %\DescribeMacro{\Groupoids} % Category of groupoids |\Groupoids|: $\Groupoids$ \\ % Uses |categorynamefont|.