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Group Action on Quantum Field Theory International Mathematical Forum, Vol. 14, 2019, no. 4, 169 - 179 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2019.9728 Group Action on Quantum Field Theory Ahmad M. Alghamdi and Roa M. Makki Department of Mathematical Sciences Faculty of Applied Sciences Umm Alqura University P.O. Box 14035, Makkah 21955, Saudi Arabia This article is distributed under the Creative Commons by-nc-nd Attribution License. Copyright c 2019 Hikari Ltd. Abstract The main aim of this paper is to develop some tools for understand- ing and explaining the concept of group action on a space and group action on algebra by using subgroups with finite index and both re- striction and transfer maps. We extend this concept on quantum field theory. 0 Introduction Group action is a very important tool in mathematics. Namely, if a certain group acts on a certain object, then many results and phenomena can be resolved. Let G be a group, the object can be a set and then we have the notion of a group acts on a set see [7]. Such object can be a vector space which results the so called G-space or G-module see for instance [6]. In some cases, the object can be a group and then we have the notion of group acts on a group see [3]. The topic of a group acting on an algebra can be seen in [5, 9]. This approaches open many gates in the research area called the G-algebras over a field. If we have an action from the group G to an object, then there is a fixed point under this action. This is the key idea in the theory of group action. In fact, by restriction, each subgroup of G acts on the same object too. Therefore, that subgroup has also fixed points. This theory goes to back the work of Cauchy Frobenius Burnside see [10]. That is the counting lemma which has many application in real life. However, the systematic method which had been initiated by Green in [4] is a very significant approach to unify the study of group action on certain objects. 170 Ahmad M. Alghamdi and Roa M. Makki That paper gives a fundamental study of the group action on finite dimensional algebras over a field. But such study was for finite groups. Our main object in this work is to deal with infinite groups and then make restriction for the subgroups of those infinite groups with certain condition. To further study these actions, we use the restriction and transfer maps as main tools. We assume all the time that the index of any subgroup H of the group G is finite even if G was infinite to insure that the sum of certain series is convergent. As a notion of the index, we may write that [ G : H] . We write V to denote a vector space over a field F . We organize the paper as follows. Section 1 for some definitions and prelim- inaries. Section 2 is devoted to the general construction of the related quotient space and a subsection for the action of group on algebra. Section 3 is for the new notion which we initiated, namely group action on quantum field theory. 1 Definitions and Preliminaries In this section, we recall the basic definitions which are related to our work and that we shall use frequently. We start by defining what is a G-space. Definition 1.1 Let G be a group that acts on a vector space V over a vector field F . For all g 2 G and v 2 V , there exist a unique element say vg 2 V such g that there exists a unique element say v:g or v in V where (v1+v2)g = v1g+v2g and v(g1g2) = (vg1)g2 for all v1; v2 2 V and g1; g2 2 G. If G is a group that acts on a vector space V over a field F , then we define the fixed point, which is still a subspace, as follows: Definition 1.2 Let G be a group and V a vector space over a field F . Assume that G acts on V , then we define a fixed point of such action by InvG(V ) = fv 2 V : v = vg; 8g 2 Gg: The invariant set defined in Definition 1.2 is a G-subspace. Let H be a subgroup of G, then the fixed point of G acting on each H can be denoted by InvH (V ): Lemma 1.3 Let G be a group that acts on a subgroup H of G, then Hg = fg−1hg ; for h 2 Hg is a subgroup of G and InvG(V ) ⊆ InvH (V ): Group action on quantum field theory 171 g −1 −1 Proof: H is a nonempty set since, for 1G 2 G we have g 1G g = g g = g −1 −1 1G 2 H. Now take h1; h2 2 H such that h1 = g xg and h2 = g yg for −1 −1 −1 −1 −1 x; y 2 H, then h1h2 = g xgg yg = g (xy)g 2 g Hg: Lemma 1.4 Let G be a group that acts on V , then for all g 2 G InvHg (V ) = InvH (V ):g Proof: One inclusion is clear and the other inclusion follows from the defi- nition of the action. We mentioned that our work will be an action of infinite group on certain objects in QFT. That infinite group is mainly the automorphism group of that object or a subgroup of it. Let us remind the reader that the general linear group which is the automorphism group of a certain vector space contains all groups under consideration. That group can be equipped with certain length and norm which can be used to look at it as a topological group. We refer the reader for more information to see the book by Baker [1]. We define the restriction and transfer maps as follows: G Definition 1.5 A restriction map is defined as RH : InvG(V ) ! InvH (V ) G such that for a 2 InvG(V ), we have RH (a) = a. For a group G and a subgroup H of that G, one can define a relation on G such that for all x; y 2 G, x is related to y if and only if Hx = Hy. It follows that such relation is an equivalence relation. This equivalence relation generates equivalence classes which is called co-sets. We shall assume in this work that all subgroups we are dealing with have finite index. We fix T to be an orbit of this relation which is called a co-set representative or a transversal set. Definition 1.6 Let G be a group and H a subgroup of G such that the index [G : H] = n < 1. Fix a transversal set T = fg1; g2; :::; gng. A transfer map is G defined as TH : InvH (V ) ! InvG(V ), such that for a 2 InvH (V ), we have: n G X TH (a) = ag1 + ag2 + ::: + agn = agi: i=1 We remark that the transfer map has more than one name in the literature such as relative trace map as well as Gasch¨utzoperator (see [5] page 36). In the language of representation theory, one can consider the relationship with 172 Ahmad M. Alghamdi and Roa M. Makki G the induction of an H-space to be the G-space IndH (V ) as a consequence of the transfer map. It is clear that the transfer map is a linear map and if a is fixed under the G H action, then its image is fixed under G action. For TH is a linear map, we G G G G G see that TH (a + b) = TH (a) + TH (b) and TH (ka) = kTH (a) for all k 2 F . Also G TH (a) 2 InvG(V ); 8a 2 InvH (V ): We have more properties such as: G G Lemma 1.7 TH (V ) = THg (V ), for g 2 G. Proof: Clear. We have the following tools: Definition 1.8 G G TH ◦ RH : InvG(V ) ! InvG(V ); G G RH ◦ TH : InvH (V ) ! InvH (V ): G G By this definition, we see that the composite map: TH ◦ RH can be seen as an G G endomorphism of the space InvG(V ). Similarly, the map RH ◦ TH can be seen as an endomorphism of the space InvH (V ): Definition 1.9 Let H 6 K 6 G, then InvG(V ) ⊆ InvK (V ) ⊆ InvH (V ). G G K We note that, if H 6 K 6 G, then TH = TK ◦ TH . This property is called the transitivity of the transfer map. For notstion use, we have the following remark. g Remark 1.10 If G is a group and H 6 K for some g 2 G, then we write H 6G K: G G Corollary 1.11 If H 6G K, then TH (V ) ⊆ TK (V ). The main idea in this paper is to consider the notion of G-algebra in the case of infinite group G. The key assumption in our attempt is to consider subgroups of G which have finite index. In fact, these types of subgroups play a significant part in the theory of groups. Also, the action of the group G on certain co-sets leads to interesting results such as the following old results. Lemma 1.12 Let G be an infinite group with a subgroup H of finite index, say n. Then considering the action of G on the co-sets of H, we can identify the factor group G=K with a subgroup of the symmetric group Sn.
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