Yurie/Algebra

A Mathematica paclet for implementing and managing finitely presented associative algebras.

Dependency:

"Yurie/Cluster"->">=2.0.0"

Attributes

$algebraDefine - list of the defined algebras.
$algebraDefault - list of the default algebras.
algebraDefine[algList_|algs___] - define the algebras.
algebraDefault[algList_|algs___] - set the default algebras.
algebraReset|algebraUnset[algList_|algs___] - reset/unset the algebras.
algebraAdd|algebraMinus[algList_|algs___][assoc_] - add/drop elements to/from the algebras.

The argument assoc_ accepts _Rule|{___Rule}|_Association.
algebraShow[alg_] - show the algebra.

For an algebra $A$ over a base field $k$, the following operations are implemented.

The predefined algebras with their implemented structures are listed below.

Algebra	Operation	Structure
`"Algebra"`	multiplication	linearity, zero, identity
`"Conjugate"`	conjugate	anti-linearity, anti-morphism, identity
`"Inverse"`	inverse	inverse, anti-morphism, identity
`"Tensor"`	tensor product	linearity, composition
`"Coalgebra"`	comultiplication, counit	linearity
`"Bialgebra"`	comultiplication, counit	morphism, identity
`"Antipode"`	antipode	linearity, anti-morphism, identity

There are several remarks.

The algebra "Algebra" will be enabled by default.
The associativity of multiplication and tensor product is implemented by the attributes {Flat,OneIdentity}.
The base field $k$ and its embedding in the algebra $A$ is distinguished, and the identity operator id is the image $\eta(1)\in A$.

id - identity operator.
generator[alg_] - return the generators of the default/specified algebra.
relation[alg_] - return the defining relations of the default/specified algebra.
printing[alg_] - return the formatting rules of the default/specified algebra.
generatorQ[_] - check whether the expression is a generator by the default algebra.
scalarQ|operatorQ[_] - check whether the expression is a scalar/operator by the default algebra.
algebraSimplify[_] - simplify the expression by the default algebra.
algebraPrint[_] - format the expression by the default algebra.
tensorank[_] - return the tensor rank of the expression by the default algebra.
tensorankUnsafe[_] - return the tensor rank of the expression by the default algebra without validating the expression.
parity[_] - return the parity of the expression by the default algebra.
parityUnsafe[_] - return the parity of the expression by the default algebra without validating the expression.

Table of shortcuts:

Name	Meaning
`algS[_]`	`algebraSimplify`
`algP[_]`	`algebraPrint`
`algSP[_]`	`algebraSimplify + algebraPrint`
`algEqualQ[_,_]`	`x==y` for operators.
`algSameQ[_,_]`	`x===y` for operators.

This functionality needs the algebras "Conjugate" and/or "Inverse".

conjugate[_] - conjugate of the operator.
innerProduct[_,_] - inner product of the two operators.

For operators $x$ and $y$ this returns $x^{\dagger}\cdot y$, and for a single operator $x$ this returns $x^{\dagger}\cdot x$.
inverse[_] - inverse of the operator.

This functionality needs the algebras "Tensor", "Coalgebra", "Bialgebra" and/or "Antipode".

tensor[__] - tensor product.
comultiply[_] - comultiplication of coalgebra.
counit[_] - counit of coalgebra.
antipode[_] - antipode of Hopf algebra.
tensorCompose[_] - composite the tensors over multiplication according to tensor rank.
tensorPermute[cycle_][_] - permute the arguments of the tensor.

comm[__] - (anti-)commutator.
commSim[__] - simplify the (anti-)commutator.
commIn[_,_]:>_/;_ - define the (anti-)commutator with the given order and condition.

adjoint[op_,order_:1][expr_] - the adjoint action of Lie algebra.

For $x,y\in \glie$, $\ad^{0}_x(y)=y$ and

\[\begin{equation} \ad^n_x(y) =[x,\ad^{n-1}_x(y)] \, . \end{equation}\]
adjointExp[op_,max_,t_:1][expr_] - the adjoint action of formal Lie group truncated at the given order.

For $x,y\in \glie$ and $t\in \CC$,

\[\begin{equation} \Ad_{t x}(y)|_{n} = e^{t x}y e^{-t x}|_{n} = \sum_{i=0}^{n}\frac{t^{i}}{i!}\ad^{i}_{x}(y) \, . \end{equation}\]

operatorPower[op_,order_:1] - power of the operator.
operatorExp[op_,max_,t_:1] - exponential of the operator truncated at the given order.

For $x\in A$ and $t\in\CC$,

\[\begin{equation} e^{t x}|_n=\sum_{i=0}^{n}\frac{t^i}{i!} x^i \, . \end{equation}\]

operatorSeparate[_] - separate scalars and operators in the given linear expression.
scalarSimplify[_] - simplify the scalar part of the expression.

User-defined relations are not guaranteed to be consistent with the internal ones. The following functions are designed for consistency checks.

checkLieBracket[_,_,_] - check the Jacobi identity of Lie algebra.

For $x,y,z\in\glie$,

\[\begin{equation} [[x,y],z]=[x,[y,z]]-[y,[x,z]] \, . \end{equation}\]
checkLieModule[_,_,_] - check the action on Lie module.

For $x,y\in \glie $ and $v\in V$,

\[\begin{equation} [x,y] (v)=x(y(v))-y(x(v)) \, . \end{equation}\]
checkCoassociativity[_] - check the coassociativity of comultiplication.

For $x\in A$,

\[\begin{equation} \sum x_{(1)(1)}\ox x_{(1)(2)}\ox x_{(2)} = \sum x_{(1)}\ox x_{(2)(1)}\ox x_{(2)(2)} \, . \end{equation}\]
checkCounitality[_] - check the counitality of counit.

For $x\in A$,

\[\begin{equation} \sum \epsilon(x_{(1)})x_{(2)} = \sum x_{(1)}\epsilon(x_{(2)}) = x \, . \end{equation}\]
checkAntipode[_] - check the antipode of Hopf algebra.

For $x\in A$,

\[\begin{equation} \sum S(x_{(1)})x_{(2)} = \sum x_{(1)}S(x_{(2)}) = \epsilon(x)\id \, . \end{equation}\]

Needs["Yurie`Algebra`"] will preserve the previously defined algebras, while Get["Yurie`Algebra`"] will clear them.
From version 14.2 to 14.3, the behavior of Simplify is changed to convert e.g. x**x to GeneralizedPower[x,2].

The function scalarSimplify mimics Simplify before version 14.3.