![UtilitasMathematica
ISSN 0315-3681 Volume 120, 2023
84
Action of Representation for Lie Groups
Zinah Kadhim1, Taghreed Majeed2
1
Mathematics Department - College of Education, Mustansiriyah University,
zenah73@uomustansiriyah.edu.iq
2
Mathematics Department - College of Education, Mustansiriyah University,
taghreedmajeed@uomustansiriyah.edu.iq
Abstract
The primary purpose of this research is to work out a new action of Lie group through dual
representation. In our paper we mention the basic definitions, the tensor product of two
representations, We’ll discuss the study of action for Lie group on Hom-spase using equivalence
relationship between Hom and tensor product. We’ll their action to study on a structure
consisting of five vector spaces. In the end we obtain new generalizations using dual action of
representation for Lie group Ǥ.
Keywords: Lie group, representation of Lie group, dual representation of Lie group, tensor
product for representation of Lie group.
1. Introduction
A Lie group Ǥ is finite dimensional smooth manifold, together with a group structure on Ǥ, such
that the multiplication Ǥ×Ǥ→Ǥ and attaching of an inverse Ǥ→Ǥ∶ ǥ → ǥ−1
being smooth maps
[1]. Lie group homomorphism from Ǥ to ℋ is linear map such that Ǥ and ℋ represent Lie
groups , Э: Ǥ → ℋ represents the group homomorphism , and Э represent 𝐶∞
−map on ℋ [6].
The direct sum of 𝑟𝑖 of Lie group Ǥ acting on the vector spaces 𝑊𝑖 over the field M is:
𝑟1(𝑠)𝑊1, … , 𝑟𝑖(𝑠)𝑊𝑖 [4]. 𝐻𝑜𝑚(𝑊1, 𝑊2) is linear map from vector space 𝑊1 to vector space 𝑊2
such that: 𝐻𝑜𝑚(𝑊1, 𝑊2) ≅ 𝑊1
∗
⊗ 𝑊2[2].
And, in this paper, We will symbolize for representations quadrilaterals and pentagons by (𝑄𝑍𝐴)
and (𝑃𝑍𝐴) respectively. see [7] and [3].](https://image.slidesharecdn.com/23-230811122415-2311e5c4/85/publishable-paper-1-320.jpg)
![UtilitasMathematica
ISSN 0315-3681 Volume 120, 2023
85
2. Dual representation of Lie Groups:
Definition (2.1): [5]
Let Lie group Ǥ , be a finite dimensional real or complex representation of Ǥ being a Lie group
homomorphism 𝑟: Ǥ → Ǥ𝐿(𝑛, ℛ) = (𝑛 ≥ 1). In general, a Lie group homomorphism is 𝑟: Ǥ →
Ǥ𝐿(𝑊) where 𝑊 being a finite dimensional real or complex vector space with a dim 𝑊 ≥ 1.
Example (2.2):
The 1-dimensional complex vector space (₵). For any Lie group Ǥ , one can describe the trivial
representation of Ǥ, 𝑟: Ǥ → Ǥ𝐿(1, ₵), via the formular:
𝑟(𝑠) = 1, for all 𝑠 ∈ Ǥ.
Definition (2.3): [1]
Let Ǥ be a Lie group and 𝑟𝑖, 𝑖 = 1,2, … , 𝑚 be representation of Ǥ affects the vector spaces
𝑊𝑖, 𝑖 = 1,2, … , 𝑚, then the direct sum of 𝑟𝑖, bring the representation defined by: {𝑟1 ⊕ 𝑟2 ⊕ … ⊕
𝑟𝑚(𝑠)}(𝑊1, 𝑊2, … , 𝑊
𝑚) = 𝑟1(𝑠)𝑊1, 𝑟2(𝑠)𝑊2, … , 𝑟𝑚(𝑠)𝑊
𝑚 for all (𝑠 ∈ Ǥ), 𝑊1, 𝑊2, … , 𝑊
𝑚 ∈
𝑊1, 𝑊2 × … × 𝑊
𝑚.
Example (2.4):
Let 𝑟1: ℛ → Ǥ𝐿(2, ℛ), such that 𝑟1(𝑠) = (
1 𝑠
0 1
) for all 𝑠 ∈ ℛ.
And 𝑟2: ℛ → Ǥ𝐿(2, ℛ), such that 𝑟2(𝑡) = (
−1 0
𝑡 1
) for all 𝑠 ∈ ℛ.
{𝑟1 + 𝑟2(𝑠)}(𝑊1, 𝑊2) = (𝑟1(𝑠)𝑊1, 𝑟2(𝑠)𝑊2) = ((
1 𝑠
0 1
) 𝑊1 , (
−1 0
𝑡 1
) 𝑊2 ).
Definition (2.5): [9]
Let both Ǥ and ℋ be the Lie groups , let 𝑟1 be a representation of Ǥ affects the space 𝑊1 and let
𝑟2 be representation of ℋ affects the space 𝑊2, then the tensor product of 𝑟1and 𝑟2 being the
representation on (𝑟1 ⊗ 𝑟2)(𝑠, 𝑡) = 𝑟(𝑠) ⊗ 𝑟2(𝑡).
for all 𝑠 ∈ Ǥ and 𝑡 ∈ ℋ.
Example (2.6):
Let 𝑟1: ℛ → Ǥ𝐿(2, ℛ), such that 𝑟1(𝑠) = (
1 𝑠
0 1
) for all 𝑠 ∈ ℛ.](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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- 1. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 84 Action of Representation for Lie Groups Zinah Kadhim1, Taghreed Majeed2 1 Mathematics Department - College of Education, Mustansiriyah University, zenah73@uomustansiriyah.edu.iq 2 Mathematics Department - College of Education, Mustansiriyah University, taghreedmajeed@uomustansiriyah.edu.iq Abstract The primary purpose of this research is to work out a new action of Lie group through dual representation. In our paper we mention the basic definitions, the tensor product of two representations, We’ll discuss the study of action for Lie group on Hom-spase using equivalence relationship between Hom and tensor product. We’ll their action to study on a structure consisting of five vector spaces. In the end we obtain new generalizations using dual action of representation for Lie group Ǥ. Keywords: Lie group, representation of Lie group, dual representation of Lie group, tensor product for representation of Lie group. 1. Introduction A Lie group Ǥ is finite dimensional smooth manifold, together with a group structure on Ǥ, such that the multiplication Ǥ×Ǥ→Ǥ and attaching of an inverse Ǥ→Ǥ∶ ǥ → ǥ−1 being smooth maps [1]. Lie group homomorphism from Ǥ to ℋ is linear map such that Ǥ and ℋ represent Lie groups , Э: Ǥ → ℋ represents the group homomorphism , and Э represent 𝐶∞ −map on ℋ [6]. The direct sum of 𝑟𝑖 of Lie group Ǥ acting on the vector spaces 𝑊𝑖 over the field M is: 𝑟1(𝑠)𝑊1, … , 𝑟𝑖(𝑠)𝑊𝑖 [4]. 𝐻𝑜𝑚(𝑊1, 𝑊2) is linear map from vector space 𝑊1 to vector space 𝑊2 such that: 𝐻𝑜𝑚(𝑊1, 𝑊2) ≅ 𝑊1 ∗ ⊗ 𝑊2[2]. And, in this paper, We will symbolize for representations quadrilaterals and pentagons by (𝑄𝑍𝐴) and (𝑃𝑍𝐴) respectively. see [7] and [3].
- 2. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 85 2. Dual representation of Lie Groups: Definition (2.1): [5] Let Lie group Ǥ , be a finite dimensional real or complex representation of Ǥ being a Lie group homomorphism 𝑟: Ǥ → Ǥ𝐿(𝑛, ℛ) = (𝑛 ≥ 1). In general, a Lie group homomorphism is 𝑟: Ǥ → Ǥ𝐿(𝑊) where 𝑊 being a finite dimensional real or complex vector space with a dim 𝑊 ≥ 1. Example (2.2): The 1-dimensional complex vector space (₵). For any Lie group Ǥ , one can describe the trivial representation of Ǥ, 𝑟: Ǥ → Ǥ𝐿(1, ₵), via the formular: 𝑟(𝑠) = 1, for all 𝑠 ∈ Ǥ. Definition (2.3): [1] Let Ǥ be a Lie group and 𝑟𝑖, 𝑖 = 1,2, … , 𝑚 be representation of Ǥ affects the vector spaces 𝑊𝑖, 𝑖 = 1,2, … , 𝑚, then the direct sum of 𝑟𝑖, bring the representation defined by: {𝑟1 ⊕ 𝑟2 ⊕ … ⊕ 𝑟𝑚(𝑠)}(𝑊1, 𝑊2, … , 𝑊 𝑚) = 𝑟1(𝑠)𝑊1, 𝑟2(𝑠)𝑊2, … , 𝑟𝑚(𝑠)𝑊 𝑚 for all (𝑠 ∈ Ǥ), 𝑊1, 𝑊2, … , 𝑊 𝑚 ∈ 𝑊1, 𝑊2 × … × 𝑊 𝑚. Example (2.4): Let 𝑟1: ℛ → Ǥ𝐿(2, ℛ), such that 𝑟1(𝑠) = ( 1 𝑠 0 1 ) for all 𝑠 ∈ ℛ. And 𝑟2: ℛ → Ǥ𝐿(2, ℛ), such that 𝑟2(𝑡) = ( −1 0 𝑡 1 ) for all 𝑠 ∈ ℛ. {𝑟1 + 𝑟2(𝑠)}(𝑊1, 𝑊2) = (𝑟1(𝑠)𝑊1, 𝑟2(𝑠)𝑊2) = (( 1 𝑠 0 1 ) 𝑊1 , ( −1 0 𝑡 1 ) 𝑊2 ). Definition (2.5): [9] Let both Ǥ and ℋ be the Lie groups , let 𝑟1 be a representation of Ǥ affects the space 𝑊1 and let 𝑟2 be representation of ℋ affects the space 𝑊2, then the tensor product of 𝑟1and 𝑟2 being the representation on (𝑟1 ⊗ 𝑟2)(𝑠, 𝑡) = 𝑟(𝑠) ⊗ 𝑟2(𝑡). for all 𝑠 ∈ Ǥ and 𝑡 ∈ ℋ. Example (2.6): Let 𝑟1: ℛ → Ǥ𝐿(2, ℛ), such that 𝑟1(𝑠) = ( 1 𝑠 0 1 ) for all 𝑠 ∈ ℛ.
- 3. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 86 Let 𝑟2: ℛ → Ǥ𝐿(2, ℛ), such that 𝑟2(𝑡) = ( 1 0 −𝑡 −1 ) for all 𝑠 ∈ ℛ. (𝑟1 ⊗ 𝑟2)(𝑠, 𝑡) = 𝑟1(𝑠) ⊗ 𝑟2(𝑡) = ( 1 𝑠 0 1 ) ⊗ ( 1 0 −𝑡 −1 ) = ( ( 1 0 −𝑡 −1 ) ( 𝑠 0 −𝑡𝑠 −𝑠 ) 0 ( 1 0 −𝑡 −1 ) ) = ( 1 0 −𝑡 −1 𝑠 0 −𝑡𝑠 −𝑠 0 0 0 0 1 0 −𝑡 −1 ) 4×4 , the matrix in Ǥ𝐿(4, ℛ). Proposition (2.7): [7] Let 𝑟 be a representation for Lie group Ǥ affects the finite dimensional vector space 𝑊, then representation of dual of 𝑟 is representation of Ǥ on 𝑊∗ given by: 𝑟∗(𝑠) = [𝑟(𝑠−1)]𝑡𝑟 , the dual representation is also called contragredient representation. Example (2.8): Let 𝑟: 𝑠−1 → 𝑆𝑜(2, ₵), where 𝑠1{(cos ∝ , sin ∝), 0 ≤∝≤ 2𝜋}, and 𝑠1 = 𝑒𝑖∝ = cos ∝ + 𝑖 sin ∝ 𝑆𝑜(2, ⊄) = {( cos ∝ − sin ∝ sin ∝ cos ∝ ) , 0 ≤∝≤ 2𝜋} such that 𝑟(cos ∝ , sin ∝) = ( cos ∝ − sin ∝ sin ∝ cos ∝ ) 𝑟(𝑒𝑖∝ ) = ( cos ∝ − sin ∝ sin ∝ cos ∝ ) , 0 ≤∝≤ 2𝜋. 𝑟 is a representation for Lie group 𝑆1 , let 𝑆 = 𝑒𝑖∝ , 𝑟(𝑒𝑖∝ ) = ( cos ∝ sin ∝ −sin ∝ cos ∝ ) 𝑆1 = cos ∝ − 𝑖 sin ∝ , 𝑟(𝑠)−1 = ( cos ∝ − sin ∝ sin ∝ cos ∝ ) {𝑟(𝑠)−1}𝑡𝑟 = ( cos ∝ − sin ∝ sin ∝ cos ∝ ). 3. The action of representation for Lie groups Through the lemma Schur's is the action idea upon the tensor product of two Lie algebra's representation where it is : Assume that 𝑟′1 and 𝑟′2 are representation of Lie algebra (ǥ) affects the finite-dimensional spaces 𝑊1 as well as 𝑊2, correspondingly. defined as an action of (ǥ) on 𝐻𝑜𝑚𝑀(𝑊2, 𝑊1), where 𝑀 be field, Э: ǥ → ǥ𝐿(𝐻𝑜𝑚𝑀(𝑊2, 𝑊1)). By for all 𝑠 ∈ ǥ, ℎ ∈ 𝐻𝑜𝑚𝑀(𝑊2, 𝑊1).
- 4. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 87 Э(𝑠) = 𝑟′1(𝑠)ℎ = ℎ𝑟′2(𝑠). And 𝐻𝑜𝑚𝑀(𝑊2, 𝑊1) ≅ 𝑊2 ∗ ⊗ 𝑊1 as equivalence of representations, see [2] . Proposition (3.1): [4] Let 𝑟𝑖: Ǥ → Ǥ𝐿(𝑊𝑖) be representations of Lie group Ǥ on 𝑀 − finite dimensional vector space (𝑊𝑖), for 𝑖 = 1,2, correspondingly, and 𝑟𝑖 ∗ : Ǥ → Ǥ𝐿(𝑊𝑖 ∗ ) the dual representation on (𝑊𝑖 ∗ ),for 𝑖 = 1,2, which is give by 𝑟𝑖 ∗ (𝑠) = ℎ𝑖 ∘ 𝑟𝑖(𝑠) , for all 𝑠 ∈ Ǥ, where ℎ𝑖: 𝑊𝑖 → 𝑀. See the diagram: Proposition (3.2): Let 𝑟𝑖, 𝑖 = 1,2,3, 𝑎𝑛𝑑 4 be for representations of Lie group affects the vector spaces 𝑊𝑖, 𝑖 = 1,2,3, 𝑎𝑛𝑑 4 respectively, and let 𝐻𝑜𝑚𝑀(𝑊4 ∗ , Hom (𝑊3 ∗ , 𝑊2), 𝐻𝑜𝑚(𝑊2, 𝑊1)), be vector space of the whole linear mapping from 𝑊2 and from Hom (𝑊3 ∗ , 𝑊2) to Hom (𝑊3 ∗ , 𝑊1) as well as from (𝑊4, Hom (𝑊3 ∗ , 𝑊2) to Hom(𝑊3, 𝑊1)), then the (𝑄𝑍𝐴) of Lie group on 𝐻𝑜𝑚𝑀(𝑊4 ∗ , Hom (𝑊3 ∗ , 𝑊2), 𝐻𝑜𝑚(𝑊2, 𝑊1)). Proof: Define Э ́ : Ǥ → Ǥ𝐿(𝐻𝑜𝑚𝑀(𝑊4 ∗ , Hom (𝑊3 ∗ , 𝑊2), 𝐻𝑜𝑚(𝑊2, 𝑊1)). Such that: Э ́ (𝑠)ℎ ́ [(𝑟1(𝑠) ∘ ℎ ́1 ∘ 𝑟2(𝑠)) ∘ (𝑟2(𝑠) ∘ ℎ ́2𝑟3(𝑠)−1 )] ∘ ℎ ́3 ∘ 𝑟4(𝑠)−1 , for all (𝑠 ∈ Ǥ) and ℎ𝑖: 𝑊𝑖 → 𝑀. The following diagram shows the use acting of group Ǥ into Ǥ𝐿(𝐻𝑜𝑚𝑀(𝑊4 ∗ , Hom (𝑊3 ∗ , 𝑊2), 𝐻𝑜𝑚(𝑊2, 𝑊1)).
- 5. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 88 Proposition (3.3): Let 𝑟𝑖, 𝑖 = 1,2,3, 𝑎𝑛𝑑 4 be for representations of Lie group affects the vector spaces 𝑊𝑖, 𝑖 = 1,2,3, 𝑎𝑛𝑑 4 respectively, and let 𝐻𝑜𝑚𝑀(𝑊4, 𝐻𝑜𝑚(𝑊3, 𝑊2), 𝐻𝑜𝑚(𝑊2 ∗ , 𝑊1 ∗)), be M-vector space of the whole linear mapping from 𝑊4 to 𝐻𝑜𝑚(𝑊3, 𝑊2), 𝐻𝑜𝑚(𝑊2 ∗ , 𝑊1 ∗). Then the (𝑄𝑍𝐴) of Lie group on 𝐻𝑜𝑚𝑀(𝑊4, 𝐻𝑜𝑚(𝑊3, 𝑊2), 𝐻𝑜𝑚(𝑊2 ∗ , 𝑊1 ∗)). Proof: Define Э ́ : Ǥ → Ǥ𝐿𝐻𝑜𝑚𝑀(𝑊4, 𝐻𝑜𝑚(𝑊3, 𝑊2), 𝐻𝑜𝑚(𝑊2 ∗ , 𝑊1 ∗)), such that Э ́ (𝑠)ℎ ́ [(𝑟1(𝑠)−1 ∘ ℎ ́1 ∘ 𝑟2(𝑠)−1 ) ∘ (𝑟2(𝑠) ∘ ℎ ́2 ∘ 𝑟3(𝑠))] ∘ ℎ ́3 ∘ 𝑟4(𝑠) , for all 𝑠 ∈ Ǥ and ℎ𝑖: 𝑊𝑖 → 𝑀. The following diagram shows the acting of group Ǥ is into Ǥ𝐿(𝐻𝑜𝑚𝑀(𝑊4, 𝐻𝑜𝑚(𝑊3, 𝑊2), 𝐻𝑜𝑚(𝑊2 ∗ , 𝑊1 ∗))).
- 6. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 89 Proposition (3.4): Let 𝑟𝑖, 𝑖 = 1,2,3, 𝑎𝑛𝑑 4 be for representations of Lie group affects the vector spaces 𝑊𝑖, 𝑖 = 1,2,3, 𝑎𝑛𝑑 4 respectively, and let 𝐻𝑜𝑚𝑀(𝑊4, 𝐻𝑜𝑚(𝑊3, 𝑊2 ∗)), 𝑊1 ∗ ), be M-vector space of the whole linear mapping from 𝑊3 to 𝑊2 ∗ and from 𝐻𝑜𝑚(𝑊3, 𝑊2 ∗))into 𝑊1 ∗ as well as 𝑊4 into 𝐻𝑜𝑚(𝐻𝑜𝑚(𝑊3, 𝑊2 ∗)), 𝑊1 ∗ ).Then the (𝑄𝑍𝐴) of Lie group is on 𝐻𝑜𝑚𝑀(𝑊4, 𝐻𝑜𝑚(𝑊3, 𝑊2 ∗)), 𝑊1 ∗ )). Proof: Define 𝜑: Ǥ → Ǥ𝐿(𝐻𝑜𝑚𝑀(𝑊4, 𝐻𝑜𝑚(𝑊3, 𝑊2 ∗)), 𝑊1 ∗ )). Such that Э ́ (𝑠)ℎ ́ [(𝑟1(𝑠)−1 ∘ ℎ ́1 ∘ (𝑟2(𝑠)−1 ∘ ℎ ́2 ∘ 𝑟3(𝑠))] ∘ ℎ ́3 ∘ 𝑟4(𝑠) , for all 𝑠 ∈ Ǥ , ℎ𝑖: 𝑊𝑖 → 𝑀. Proposition (3.5): Let 𝑟𝑖, 𝑖 = 1,2,3, 𝑎𝑛𝑑 4 be for representations of Lie group affects the vector spaces 𝑊𝑖, 𝑖 = 1,2,3, 𝑎𝑛𝑑 4 respectively, and let 𝐻𝑜𝑚𝑀(𝑊4, 𝐻𝑜𝑚(𝑊3 ∗ , 𝑊2)), 𝑊1 ∗ ), be M-vector space of the whole linear mapping from 𝑊3 to 𝑊2 and from 𝐻𝑜𝑚(𝑊3 ∗ , 𝑊2))into 𝑊1 as well as 𝑊4 into 𝐻𝑜𝑚(𝐻𝑜𝑚(𝑊3 ∗ , 𝑊2)).Then the (𝑄𝑍𝐴) of Lie group is on 𝐻𝑜𝑚𝑀(𝑊4, 𝐻𝑜𝑚(𝑊3 ∗ , 𝑊2)), 𝑊1 ∗ )).
- 7. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 90 Proof: Define Э ́ : Ǥ → Ǥ𝐿𝐻𝑜𝑚𝑀(𝑊4, 𝐻𝑜𝑚(𝑊3 ∗ , 𝑊2), 𝑊1 ∗ ), by proposition (3.2) such that: Э ́ (𝑠)ℎ ́ = [(𝑟1(𝑠)−1 ∘ ℎ ́1 ∘ (𝑟2(𝑠) ∘ ℎ ́2 ∘ 𝑟3(𝑠)−1 )] ∘ ℎ ́3 ∘ 𝑟4(𝑠). for all 𝑠 ∈ Ǥ and ℎ𝑖: 𝑊𝑖 → 𝑀. Proposition (3.6): Let 𝑟𝑖, 𝑖 = 1,2,3, 𝑎𝑛𝑑 4 be four representations of Lie group affects the vector spaces 𝑊𝑖, 𝑖 = 1,2,3, 𝑎𝑛𝑑 4 respectively, and let 𝐻𝑜𝑚𝑀(𝑊4 ∗ , 𝐻𝑜𝑚(𝑊3 ∗ , 𝑊2)), 𝑊1), be M-vector space of the whole linear mapping from 𝑊3 ∗ to 𝑊2 and 𝐻𝑜𝑚(𝑊3 ∗ , 𝑊2))into 𝑊1 as well as 𝑊4 ∗ into 𝐻𝑜𝑚(𝐻𝑜𝑚(𝑊3 ∗ , 𝑊2)).Then the (𝑄𝑍𝐴) of Lie group is on 𝐻𝑜𝑚𝑀(𝑊4 ∗ , 𝐻𝑜𝑚(𝑊3 ∗ , 𝑊2)), 𝑊1)). Proof: Define Э ́ : Ǥ → Ǥ𝐿𝐻𝑜𝑚𝑀(𝑊4 ∗ , 𝐻𝑜𝑚(𝑊3 ∗ , 𝑊2), 𝑊1), by proposition (3.4) such that: Э ́ (𝑠)ℎ ́ = [(𝑟1(𝑠) ∘ ℎ ́1 ∘ (𝑟2(𝑠) ∘ ℎ ́2 ∘ 𝑟3(𝑠)−1 )] ∘ ℎ ́3 ∘ 𝑟4(𝑠)−1 . for all (𝑠 ∈ Ǥ) and ℎ𝑖: 𝑊𝑖 → 𝑀.
- 8. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 91 Proposition (3.7): Let 𝑟𝑖, 𝑖 = 1,2,3,4, 𝑎𝑛𝑑 5 be five representation of Lie group affects the vector spaces 𝑊𝑖, 𝑖 = 1,2,3,4, 𝑎𝑛𝑑 5 respectively, and let 𝐻𝑜𝑚𝑀(𝑊5, 𝐻𝑜𝑚(𝑊4, 𝑊3 ∗), 𝑊2 ∗ ⊕ 𝑊1 ∗ )), be M-vector space of the whole linear mapping from 𝑊4 to 𝑊3 ∗ and 𝐻𝑜𝑚(𝑊4, 𝑊3 ∗))into 𝑊2 ∗ ⊕ 𝑊1 ∗ as well as 𝑊5 into 𝐻𝑜𝑚(𝐻𝑜𝑚(𝑊4, 𝑊3 ∗), 𝑊2 ∗ ⊕ 𝑊1 ∗ ).Then the (𝑄𝑍𝐴) of Lie group is on 𝐻𝑜𝑚𝑀(𝑊5, 𝐻𝑜𝑚(𝐻𝑜𝑚(𝑊4, 𝑊3), 𝑊2 ∗ ⊕ 𝑊1 ∗ )). Proof: Define ∅ ́ : Ǥ → Ǥ𝐿(𝐻𝑜𝑚𝑀(𝑊5, 𝐻𝑜𝑚(𝐻𝑜𝑚(𝑊4, 𝑊3 ∗), 𝑊2 ∗ ⊕ 𝑊1 ∗ ))), by proposition (3.4) such that: ∅ ́ (𝑠)ℎ ́ = [(𝑟1(𝑠)−1 ⊕ (𝑟2(𝑠)−1 ) ∘ ℎ ́1 ∘ (𝑟3(𝑠)−1 ∘ ℎ ́2 ∘ 𝑟4(𝑠))] ∘ ℎ ́3 ∘ 𝑟5(𝑠). for all 𝑠 ∈ Ǥ and ℎ𝑖: 𝑊𝑖 → 𝑀. Proposition (3.8): Let 𝑟𝑖, 𝑖 = 1,2,3,4, 𝑎𝑛𝑑 5 be five representation of Lie group affects the vector spaces 𝑊𝑖, 𝑖 = 1,2,3,4, 𝑎𝑛𝑑 5 respectively, and let 𝐻𝑜𝑚𝑀(𝑊5, 𝐻𝑜𝑚(𝑊4 ∗ , 𝑊3 ∗), 𝐻𝑜𝑚(𝑊2, 𝑊1 ∗ ), be M-vector space of the whole linear mapping from 𝑊2 to 𝑊1 ∗ and 𝐻𝑜𝑚(𝑊2 ∗ , 𝑊3 ∗))into 𝐻𝑜𝑚(𝑊2, 𝑊1 ∗ ) as well as 𝑊5 into (𝐻𝑜𝑚(𝑊4 ∗ ⊕ 𝑊3 ∗), 𝐻𝑜𝑚(𝑊2, 𝑊1 ∗)). Then the (𝑄𝑍𝐴) of Lie group is on 𝐻𝑜𝑚𝑀(𝑊5, 𝐻𝑜𝑚(𝑊4 ∗ , 𝑊3 ∗)𝐻𝑜𝑚(𝑊2, 𝑊1 ∗)).
- 9. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 92 Proof: Define 𝜑́ : Ǥ → Ǥ𝐿(𝐻𝑜𝑚𝑀(𝑊5, (𝑊4 ∗ ⊕ 𝑊3 ∗)𝐻𝑜𝑚(𝑊2, 𝑊1 ∗))),by proposition (3.4) such that: 𝜑́ (𝑠)ℎ ́ = [(𝑟1(𝑠)−1 ∘ ℎ ́1 ∘ 𝑟2(𝑠)) ∘ ℎ ́2 ∘ (𝑟3(𝑠)−1 ∘⊕ 𝑟4(𝑠)−1 )] ∘ ℎ ́3 ∘ 𝑟5(𝑠). for all 𝑠 ∈ Ǥ and ℎ𝑖: 𝑊𝑖 → 𝑀. Proposition (3.9): Let 𝑟𝑖, 𝑖 = 1,2,3,4,5 be five representation of Lie group affects the vector spaces 𝑊𝑖, 𝑖 = 1,2,3,4,5 respectively, and let 𝐻𝑜𝑚𝑀(𝐻𝑜𝑚(𝑊5, 𝑊4 ∗), 𝐻𝑜𝑚(𝑊3, 𝑊2 ⊕ 𝑊1 ∗)) be M-vector space of all linear mapping from 𝑊5 to 𝑊4 ∗ and from 𝑊3 into (𝑊2 ⊕ 𝑊1 ∗)) as well as from 𝐻𝑜𝑚(𝑊5, 𝑊4 ∗) into 𝐻𝑜𝑚(𝑊3, 𝑊2 ⊕ 𝑊1 ∗) Then the (𝑄𝑍𝐴) of Lie group on 𝐻𝑜𝑚𝑀(𝐻𝑜𝑚(𝑊5, 𝑊4 ∗), 𝐻𝑜𝑚(𝑊3, 𝑊2 ⊕ 𝑊1 ∗). Proof: Define 𝜑́ : Ǥ → Ǥ𝐿(𝐻𝑜𝑚𝑀(𝐻𝑜𝑚(𝑊5, 𝑊4 ∗), 𝐻𝑜𝑚(𝑊3, 𝑊2 ⊕ 𝑊1 ∗))),by proposition (3.4) such that: 𝜑́ (𝑠)ℎ ́ = [(𝑟1(𝑠)−1 ⊕ 𝑟2(𝑠)) ∘ ℎ ́1 ∘ 𝑟3(𝑠))] ∘ ℎ ́2 ∘ 𝑟4(𝑠)−1 ∘ ℎ ́3 ∘ 𝑟5(𝑠). for all 𝑠 ∈ Ǥ and ℎ𝑖: 𝑊𝑖 → 𝑀.
- 10. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 93 4. The dual action of representation for Lie Group Proposition (4.1): [6] Let 𝑊1and 𝑊2 are finite dimensional vector space, and (𝑛) be a natural no, then the action of Lie group action on tensor product is : 1. 𝑊 2 ∗∗,..∗ ⏟ 𝑛 ⊗ 𝑊1 ≅ 𝑊2 ⊗ 𝑊1, if 𝑛 is an even number. 2. 𝑊 2 ∗∗,..∗ ⏟ 𝑛 ⊗ 𝑊1 ≅ 𝑊2 ∗ ⊗ 𝑊1, if 𝑛 is an odd number. 3. 𝑊2 ⊗ 𝑊 2 ∗∗,..∗ ⏟ 𝑛 ≅ 𝑊2 ⊗ 𝑊1, if 𝑛 is an even number. 4. 𝑊2 ⊗ 𝑊 2 ∗∗,..∗ ⏟ 𝑛 ≅ 𝑊2 ⊗ 𝑊1 ∗ , if 𝑛 is an odd number. Corollary (4.2): If (𝑊)1and (𝑊2) are finite- dimensional vector spaces, and (𝑛) be a natural number, then the Lie group affects the tensor product and 𝐻𝑜𝑚𝑀(𝑊2, 𝑊1): 1. 𝑊2 ⊗ 𝑊1 ∗ ≅ 𝐻𝑜𝑚𝑀(𝑊2, 𝑊1 ∗) 2. 𝑊2 ⊕ 𝑊3 ⊗ 𝑊1 ∗ ≅ 𝐻𝑜𝑚𝑀(𝑊1, 𝑊2 ⊕ 𝑊3) } if 𝑛 is an odd number. Proposition (4.3): Let 𝑟1: Ǥ → Ǥ𝐿(𝑊𝑖), 𝑅𝑖 ∗ : Ǥ → Ǥ𝐿(𝑊𝑖 ∗ ) for 𝑖 = 1,2, and the Lie group Ǥ affects the 𝜑(𝑠)ℎ = 𝑟1(𝑠) ∘ ℎ ∘ 𝑟2(𝑠), for every 𝑠 ∈ Ǥ, ℎ ∈ 𝐻𝑜𝑚(𝑊2, 𝑊1). Then the Lie group Ǥ affects (𝐻𝑜𝑚(𝑊2, 𝑊1))∗ being also given provided a representation 𝜑∗ , where: 𝜑∗(𝑠) = 𝑟(𝑠)−1 ∘ ℎ∗ ∘ 𝑟1(𝑠), for all 𝑠 ∈ Ǥ and ℎ∗ ∈ (𝐻𝑜𝑚(𝑊2, 𝑊1))∗ . Proof: Let action of Lie group Ǥ affects 𝐻𝑜𝑚(𝑊2, 𝑊1) being induced via the representation
- 11. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 94 𝜑: Ǥ → Ǥ𝐿(𝐻𝑜𝑚(𝑊2, 𝑊1)), where 𝜑(𝑠) = 𝑟1(𝑠) ∘ ℎ ∘ 𝑟2(𝑠)−1 , for 𝑠 ∈ Ǥ, and ℎ ∈ 𝐻𝑜𝑚(𝑊2, 𝑊1). Such that 𝜑∗(𝑠) = 𝑟(𝑠)−1 ∘ ℎ∗ ∘ 𝑟1(𝑠) representation for all 𝑠 ∈ Ǥ and ℎ∗ ∈ (𝐻𝑜𝑚(𝑊2, 𝑊1))∗ . Since: ∅∗(𝑠) = (𝑟1(𝑠) ∘ ℎ ∘ 𝑟2(𝑠)−1 , )∗ = 𝑟2 ∗(𝑠)−1 ∘ ℎ∗ ∘ 𝑟1 ∗(𝑠) for all (𝑠 ∈ Ǥ) and ℎ∗ : 𝑊1 ∗ → 𝑊2 ∗ , we have 𝜑∗(𝑠𝑡) = (𝑟(𝑠𝑡))∗ = (𝑟(𝑡) ∘ ℎ ∘ 𝑟(𝑠))∗ = 𝑟∗ (𝑠) ∘ ℎ∗ ∘ 𝑟∗ (𝑡) Thus, 𝜑∗ is a representation from Ǥ(𝜑∗ . 𝑠 𝑔𝑟𝑜𝑢𝑝 ℎ𝑜𝑚𝑜𝑚𝑜𝑟𝑝ℎ𝑖𝑠𝑚 𝑜𝑓 Ǥ) Corollary (4.4): Let 𝑟1: Ǥ → Ǥ𝐿(𝑗, 𝑀), Ǥ𝐿(𝑊1) ≅ Ǥ𝐿(𝑗, 𝑀) And 𝑟2: Ǥ → Ǥ𝐿(ℓ, 𝑀), Ǥ𝐿(𝑊2) ≅ Ǥ𝐿(ℓ, 𝑀). Where 𝑟1 and 𝑟2 have matrix representation, if Lie group Ǥ affects 𝐻𝑜𝑚(𝑊2, 𝑊1) being a representation 𝜑: Ǥ → Ǥ𝐿(𝐻𝑜𝑚(𝑊2, 𝑊1)), where 𝜑(𝑠) = 𝑟1(𝑠) ∘ ℎ ∘ 𝑟2(𝑠)−1 , for 𝑠 ∈ Ǥ. Then Lie group Ǥ affects 𝐻𝑜𝑚(𝑊2, 𝑊1) is a representation 𝜑∗ : Ǥ → Ǥ𝐿(𝐻𝑜𝑚𝑀(𝑊2, 𝑊1))∗ . Where: 𝜑∗(𝑠) = (𝑟2 ∗(𝑠)) 𝑡𝑟 ∘ ℎ∗ ∘ (𝑟1(𝑠)−1)𝑡𝑟 , for all 𝑠 ∈ Ǥ.
- 12. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 95 Proof: Since (𝑠) = 𝑟1(𝑠) ∘ ℎ ∘ 𝑟2(𝑠)−1 , 𝜑∗(𝑠) = 𝑟(𝑠)−1 ∘ ℎ∗ ∘ 𝑟1 ∗(𝑠) = (𝑟2 ∗(𝑠)) 𝑡𝑟 ∘ ℎ∗ ∘ (𝑟1(𝑠)−1)𝑡𝑟 And 𝜑∗ is a representation a matrix representation of dimension 𝑗ℓ, then 𝑟∗(𝑠𝑡) = (𝑟(𝑠𝑡)−1)𝑡𝑟 = (𝑟(𝑡)−1)𝑡𝑟 ∘ (𝑟(𝑠)−1)𝑡𝑟 = 𝑟∗(𝑡) ∘ 𝑟∗(𝑠) Example (4.5): Let 𝑟1 = 𝑆1 → 𝑆𝑜(2) ⊂ Ǥ𝐿(2, ₵) and 𝑟2 = 𝑆1 → 𝑆𝑜(3) ⊂ Ǥ𝐿(3, ₵) Where Ǥ = 𝑆1(𝑗 = 2, ℓ = 3) and (𝑊1) is the ₵ − vector space of dimensional 2, and ( 𝑊2) being the ₵ − vector space of dimensional 3, then by collar we have 𝜑∗ (𝑒𝑖∝ ) = 𝑟2 ∗ (𝑒𝑖∝ ) ∘ ℎ∗ ∘ 𝑟1 ∗ (𝑒𝑖∝ ) = (𝑟2 ∗ (𝑒𝑖∝ )) 𝑡𝑟 ∘ ℎ∗ ∘ (𝑟1(𝑒𝑖∝ )) 𝑡𝑟 = ( 1 0 0 0 cos ∝ − sin ∝ 0 sin ∝ cos ∝ ) 𝑡𝑟 ∘ ℎ∗ ∘ ( cos ∝ sin ∝ −sin ∝ cos ∝ ) 𝑡𝑟 = ( 1 0 0 0 cos ∝ − sin ∝ 0 sin ∝ cos ∝ ) 𝑡𝑟 ∘ ℎ∗ ∘ ( cos ∝ − sin ∝ sin ∝ cos ∝ ) 𝑡𝑟 Let 𝐴 = ( 1 0 0 0 cos ∝ − sin ∝ 0 sin ∝ cos ∝ ) Thus 𝜑∗ (𝑒𝑖∝ ) = ( (cos ∝)𝐴 (− sin ∝)𝐴 (sin ∝)𝐴 (cos ∝)𝐴 ) 6×6
- 13. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 96 = ( cos ∝ 0 0 0 cos ∝2 cos ∝ sin ∝ 0 − cos ∝ sin ∝ cos ∝2 − sin ∝ 0 0 0 − sin ∝ cos ∝ − sin ∝2 0 sin ∝2 sin ∝ cos ∝ sin ∝ 0 0 0 sin ∝ cos ∝ sin ∝2 0 − sin ∝2 sin ∝ cos ∝ cos ∝2 0 0 0 cos ∝2 cos ∝ sin ∝ 0 − cos ∝ sin ∝ cos ∝2 ) Is the representation of Ǥ𝐿(6, ₵) acting on 𝐻𝑜𝑚𝑀(𝑊2, 𝑊1) of dimensional 6. Proposition (4.6): Let (𝑊1)and (𝑊2) be two n- dimensional vector spaces and 𝑛 is a natural number, then 𝐻𝑜𝑚𝑀(𝐻𝑜𝑚(𝑊3, 𝑊2, 𝑊1)) ∗∗…∗ ⏟ 𝑛 = { 𝐻𝑜𝑚𝑀(𝐻𝑜𝑚(𝑊3, 𝑊2, 𝑊1))𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛 𝐻𝑜𝑚𝑀(𝑊1 ∗ , 𝐻𝑜𝑚(𝑊2 ∗ , 𝑊2)𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑 … Proof: We can prove this proposition by mathematical induction. If 𝑛 = 1, then (𝐻𝑜𝑚𝑀(𝑊2, 𝑊1))∗ = (𝐻𝑜𝑚𝑀(𝑊1 ∗ , 𝑊2 ∗). And hence the action of group Ǥ on 𝐻𝑜𝑚𝑀(𝑊2, 𝑊1) is: 𝜑∗(𝑠) = 𝑟1 ∗(𝑠) ∘ ℎ∗ ∘ 𝑟2 ∗(𝑠)−1 (see proposition (3.2). Suppose that (3.2) is true when 𝑛 = 𝑘 it mean (𝐻𝑜𝑚𝑀(𝑊2, 𝑊1)) ∗∗…∗ ⏟ 𝑘 = { (𝐻𝑜𝑚𝑀(𝑊2, 𝑊1)), 𝑖𝑓 𝑘 𝑖𝑠 𝑒𝑣𝑒𝑛 … (1) (𝐻𝑜𝑚𝑀(𝑊1 ∗ , 𝑊2), 𝑖𝑓 𝑘 𝑖𝑠 𝑜𝑑𝑑 … (2) If 𝑘 is even then the acting of Lie group Ǥ on 𝐻𝑜𝑚𝑀(𝑊2, 𝑊1) is: 𝜑(𝑠) = 𝑟2(𝑠)−1 ∘ ℎ ∘ 𝑟1(𝑠). And if 𝑘 is odd then the action of Lie group Ǥ on 𝐻𝑜𝑚𝑀(𝑊2, 𝑊1) is: 𝜑∗(𝑠) = 𝑟1 ∗(𝑠)−1 ∘ ℎ∗ ∘ 𝑟2 ∗(𝑠) We will prove that (3.2) it is true, when 𝑛 = 𝑘 + 1 ,so it will be proven 𝐻𝑜𝑚𝑀(𝑊2, 𝑊1) ∗∗…∗ ⏟ 𝑘+1 = { 𝐻𝑜𝑚𝐺(𝑊2, 𝑊1))𝑖𝑓 𝑘 𝑖𝑠 𝑜𝑑𝑑 𝐻𝑜𝑚𝑀(𝑊2, 𝑊1))𝑖𝑓 𝑘 𝑖𝑠 𝑖𝑠 𝑒𝑣𝑒𝑛 (𝜑(𝑠)) ∗∗…∗ ⏟ 𝑘+1 = (𝑟1 ∗(𝑠) ∘ ℎ∗ 𝑟2(𝑠)) ∗∗…∗ ⏟ 𝑘+1 (𝑟1 ∗(𝑠) ∘ ℎ∗ ∘ 𝑟2 ∗(𝑠))
- 14. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 97 When 𝑘 is odd we have 𝑘 + 1 is even, then 𝐻𝑜𝑚𝑀(𝐻𝑜𝑚(𝑊3, 𝑊2), 𝑊1)) ∗∗…∗ ⏟ 𝑘+1 = 𝐻𝑜𝑚𝑀(𝐻𝑜𝑚(𝑊3, 𝑊2), 𝑊1)) by(1), Thus the Lie group Ǥ affects 𝐻𝑜𝑚𝑀(𝑊2, 𝑊1) ∗∗…∗ ⏟ 𝑘+1 is: (𝜑(𝑠)) ∗∗…∗ ⏟ 𝑘+1 = [(𝑟1(𝑠)−1 ∘ ℎ1 ∘ (𝑟2(𝑠) ∘ ℎ2 ∘ 𝑟3 −1(𝑠))] And when 𝑘 is even we have 𝑘 + 1 is odd. Then 𝐻𝑜𝑚𝑀(𝐻𝑜𝑚(𝑊3, 𝑊2), 𝑊1)) ∗∗…∗ ⏟ 𝑘+1 = 𝐻𝑜𝑚𝑀(𝑊1 ∗ , 𝐻𝑜𝑚(𝑊2 ∗ , 𝑊3)) by (2), Thus the affects of Lie group Ǥ on 𝐻𝑜𝑚𝑀(𝑊2, 𝑊1) ∗∗…∗ ⏟ 𝑘+1 is: (𝜑(𝑠)) ∗∗…∗ ⏟ 𝑘+1 = [(𝑟3(𝑠) ∘ ℎ∗ 1 ∘ 𝑟2(𝑠)−1 ) ∘ ℎ∗ 2 ∘ 𝑟1 −1(𝑠)−1] Then the proposition is true for all ∈ 𝑍∗ . 5. Conclusion: We concerned of Lie group, Lie algebra, we representation of Lie group, representation of Lie algebra, tensor product of representation of Lie group. We obtain new propositions by using four and five- representations structure as manifested, which was the starting point for the Schur's lemma. 6. Acknowledgement: The authors (Zinah Makki Kadhim and Taghreed Hur Majeed) would be grateful to thank Mustansiriyah University in Baghdad, Iraq. (www.mustansiriyah.ed.iq) for Collaboration and Support in the present work. References 1. A.K. Radhi and T.H. Majeed, Certain Types of Complex Lie Group Action (Journal of Al- Qadisiyah for Computer Science and Mathematics, Qadisiyah-Iraq, 2018), pp.54-62. 2. B. Jubin, A. Kotov, N. Poncin and V. Salnikov, Differential Graded Lie Groups and Their Differential Graded Lie Algebra (Springer Undergraduate Mathematics Series, Ukraine, 2022), pp. 27-39.
- 15. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 98 3. T.H Majeed, "Action of Topological Groupoid on Topological Space " (The international Journal of Nonlinear Analysis and Applications, vol.13,No.1,2022) p.p85-89. 4. J .Jiang, Y. Sherg and C.Zhu, Cohomologics of Relative Rota-Baxter Operators on Lie Groups and Lie Algebras (arxiv e-prints, arxiv- 2018, United States of America, 2021), pp. 302-315. 5. J. Lauret and C.E. Will, On Ricci Negative Lie Groups (spring, Ukrain, 2922), pp.171-191. 6. P.Etingof, Lie Groups and Lie Algebra, (arxiv: 2201, 0939771[math], United States of America, 2021), pp. 18-34. 7. T.T. Nguyen and V.A. Le, Representation of Real Solvable lie Algebras having 2- Dimensional Derived Ideal and Geometry of Coadjaint Orbits of Corresponding Lie Groups (Asian-European Journal of Mathematics, Singapore, 2022), pp. 193-225. 8. W.S. Gan, Lie Groups and Lie Algebras (spring, Singapore, 2021), pp. 7-13. 9. X.Zhu, C. Xu and D. Tao, Commutative Lie Group VAE for Disentanglement Learning (International Conference On Machine Learning, Japan, 2021), pp. 12924-12934.