Action (mathematics)

The action (yes action, operation) in the mathematics considered a meeting of the upper conversion representation as algebraic structure in an algebraic system. It is a concept to describe a state of the exercise of the space (if I say in a common meaning a figure) and the structure of the thing causing it geometrically.

The field is called an expression theory for one minute of the mathematics developed based on the technique that I characterize by expressing the algebraic structure to be given abstractly such as abstract groups as the structure that the whole exercise on the concrete space makes through the action.

Table of contents

Definition

Gathering is conversion θ on meeting A with an operator (Sayo lie, operator) on the algebraic system that is A a stand: It is A → A. Give set Ω of the letter, and allow for a different letter of Ω to give the same operator; and the representation from Ω to an operator group on algebraic system A (temporarily write this down with Trans(A) here)

${\displaystyle \sigma \colon \Omega \to \mathrm {Trans} (A);\ \omega \mapsto \sigma (\omega)}$

It is said that action of meeting Ω is determined by algebraic system A where meeting Ω performs action (act, operate) in algebraic system A when I gave を. In addition, I call A a Ω-algebraic system [¹], expression space of Ω then and call Ω a scope (yes I go operation domain) of A or an action group.

Of scope Ω, representation σ, a group (σ, Ω, A) of expression space A Ω call it with expression (representation which does not jest) of Ω in action (action) to A or A; of the fear of the misunderstanding when there is not it, merely express it in representation σ. In addition, letter ω is expressed in ω as well as the original letter that it is often operator σ (ω) to set of A (I omit action σ) by action σ.

Put back x oneself to meeting A for each former x; when because "did nothing", and was provided with conversion (恒等変換) naturally, was given meeting Ω which was different as a scope optionally; each operator ω of Ω as 恒等変換 (恒等作用素) to A of Ω, as for "what, do not do it", and can establish action. Self-evident action to A of Ω means this.

When subset B of A is closed about action σ for action (σ, Ω, A), is clogged up; operator σ (ω) on A: Limit σ(ω)| of the domain of A → A _B: I am represented by B → B

${\displaystyle \sigma |_{B}\colon \Omega \times B\to B;\ (\omega ,x)\mapsto \sigma (\omega) |_{B}(x)}$

When a group (σ|B,Ω, B) acts again when I think about を, in B, it is said that it is Ω-immutability (ふへん, invariant) stable Ω-(stability, stable) or peculiarity (こゆう, proper) about action σ, and it is said that B is a partial Ω-algebraic system of A or part expression again.

Distinction of right and left

The image by operator θ of former a of A is expressed according to θ(a),θa,^θa or the right scale according to the left scale in (a)θ, aθ, a^θ.

${\displaystyle \theta \colon A\to A;\ a\mapsto \theta (a).}$
${\displaystyle \theta \colon A\to A;\ a\mapsto a^{\theta}}.$

After the fashion of the distinction of right and left of the scale of the image of the operator, the right action is determined the left action to algebraic system A of scope Ω. With action σ from the left to A of Ω being given, I represent, for example, it

${\displaystyle \Omega \times A\to A;\ (\omega ,a)\mapsto \sigma (\omega) a}$

を is equivalent with giving it. When σ serves right

${\displaystyle A\times \Omega \to A;\ (a,\omega) \mapsto a^{\sigma (\omega)}}$

などに is equivalent. When a scope makes an algebraic system with the product that is non-commutation, right and left of the action receive distinction about the structure and expression particularly consciously, but, on the other hand, (decided by upsetting the order of products) right and left are subsumed by the concept of the reverse algebraic system and do not rarely sell only one (in the case of most, I serve left) when they lecture on generalization.

Structure and action

Because the algebraic system has a family of the algorithm as the structure, and a thing called the good associate same model of the affinity with the structure for the upper representation is particularly important, it is custom that treat such a thing as an operator. For example, the associate same model of the top that is the meeting that it is free if I think in particular about algebraic system E having no operation (of course the meeting may have mathematical structure except the algebraic structure) is simple representation. If, in other words, I think about E bijection from E and write down the whole with Aut(E), in Aut(E), it is as the only operation by composition of the representation substitution on E in particular as representation in a group. I give different group G then and represent it

${\displaystyle \rho \colon G\to \mathrm {Aut} (E)}$

But, group action (ぐんさよう, group action) to meeting E of group G is defined by thinking about the thing which is the group associate same model in particular. As for this, the composition of the operator moves to the product of the group and

${\displaystyle \rho (g)\rho (h)=\rho (gh)}$

It is the limit that is stronger than action of meeting G being decided in a meaning to become the のようにまた operator in merely set E. Similarly, it is often that I think about a thing called the action to A of U only for Trans(A) and algebraic system U named the associate same model when I do it, and conversion group Trans(A) on algebraic system A forms a new algebraic system again. Particularly, because whole End(M) of the self-associate same model of module M has the structure of the ring; is ring associate same model π from selfish ring R: Ring action (かんさよう, ring action) from ring R to module M is defined by R → End(M). I call ring R which is a scope then an M coefficient ring (I suck Kei feel) in particular and call the cause of coefficient ring R with a coefficient.

When different structure appears on space X with action, action to X which is a stand may cause action to structure on X. It is, for example, group action G X X → X; (g, x) → gx

${\displaystyle f\mapsto \sigma f;\quad \sigma f(x):=f(\sigma ^{-1}x)}$

I can shift to a function ring on によって X by action. It is said that structure is compatible with action at such time.

Action associate same model

Algebraic system A, B and associate same model f: of the meantime where similar algorithm group R fixes structure for I think about A → B. In addition, A, B has same scope Λ, and it is said that action of Λ is given in (πA,Λ, A), (πB,Λ, B). Associate same model f as representation

${\displaystyle f\circ \pi _{A}(\lambda) =\pi _{B}(\lambda) \circ f}$

If meet it for λ of the を option; associate same model f: It is said that A → B is the associate same model including the action of Λ. I easily call it the action associate same model or the Λ-associate same model.

When action of Λ is given A, B together from the right; the condition that A → B is the Λ-associate same model

${\displaystyle f(x^{\pi _{A}(\lambda) })=f(x)^{\pi _{B}(\lambda)}}$

If it is what I meet for any former x of any former λ and A of を Λ and is the left action together

${\displaystyle f(\pi _{A}(\lambda) x)=\pi _{B}(\lambda) f(x)}$

It is spoken with what I meet for any former x of any former λ and A of を Λ. One is from the left and, in the case of from the right, can write the other equally. In addition, if I omit action and write it

${\displaystyle f\circ \lambda =\lambda \circ f}$

But, it will be managed, and this condition is spoken with "f being action and commutation of Λ" from the situation that considers composition of the representation to be a binary operation.

In addition, it is Λ-associate same model f: For A → B, it is an image

${\displaystyle \mathrm {Im} (f):= f(a)\in B\mid a\in A}$

Not only it is a partial algebraic system of は B, but also is the partial system as the Λ-algebraic system. Referring to this, it is said that the Λ-associate same model keeps the structure of the Λ-algebraic system. In addition, this considers the upper action to be an unary operation group from an algebraic system (A, R); and S = R ∪ {π_A(λ) | By putting it with λ ∈ Λ}; as a new algebraic system (A, S) = (A, R, (π_A,Λ)) を is the same in keeping the structure of the algebraic system in a normal meaning and thinking about the associate same model if I think.

Constitution and resolution of the action

Two action (π, Ω, A), (τ, Λ, B) determine prompt shipment action (π X τ, Ω X Λ, A X B) by the action every ingredient. In addition, it is expressed the tensor product if the tensor product is defined as A by B when these are linear expression (the outside)

${\displaystyle (\pi \boxtimes \tau ,\Omega \times \Lambda ,A\otimes B)}$

I pass and can extend it and give linear expression again. For action (π, Ω, A), it is representation φ: A → B,ψ₂: B → A is the new action that exchanged a scope and expression space by composition Λ → Ω and ψ ₁

${\displaystyle (\pi \circ \varphi ,\Lambda ,A),}$
${\displaystyle (\psi _{1}\circ \pi \circ \psi _{2},\Omega ,B)}$

I lead を. If inclusion map and ψ ₂ and composition B → A → B with ψ ₁ are divided ψ ₂ a limit of the expression if φ in particular is inclusion map, part expression is defined each.

Allied item

• Expression (mathematics)
• Algebraic structure
• Coefficient
• Operator
• Module

Footnote

1. If ^ algebraic system A gathers, a Ω-meeting, a group say with Ω-groups.

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