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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 Η€. Keywords: Lie group, representation of Lie group, dual representation of Lie group, tensor product for representation of Lie group. 1. Introduction A Lie group Η€ is finite dimensional smooth manifold, together with a group structure on Η€, such that the multiplication Η€Γ—Η€β†’Η€ and attaching of an inverse Η€β†’Η€βˆΆ Η₯ β†’ Η₯βˆ’1 being smooth maps [1]. Lie group homomorphism from Η€ to β„‹ is linear map such that Η€ and β„‹ represent Lie groups , Π­: Η€ β†’ β„‹ represents the group homomorphism , and Π­ represent 𝐢∞ βˆ’map on β„‹ [6]. The direct sum of π‘Ÿπ‘– of Lie group Η€ acting on the vector spaces π‘Šπ‘– over the field M is: π‘Ÿ1(𝑠)π‘Š1, … , π‘Ÿπ‘–(𝑠)π‘Šπ‘– [4]. π»π‘œπ‘š(π‘Š1, π‘Š2) is linear map from vector space π‘Š1 to vector space π‘Š2 such that: π»π‘œπ‘š(π‘Š1, π‘Š2) β‰… π‘Š1 βˆ— βŠ— π‘Š2[2]. And, in this paper, We will symbolize for representations quadrilaterals and pentagons by (𝑄𝑍𝐴) and (𝑃𝑍𝐴) respectively. see [7] and [3].
UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 85 2. Dual representation of Lie Groups: Definition (2.1): [5] Let Lie group Η€ , be a finite dimensional real or complex representation of Η€ being a Lie group homomorphism π‘Ÿ: Η€ β†’ ǀ𝐿(𝑛, β„›) = (𝑛 β‰₯ 1). In general, a Lie group homomorphism is π‘Ÿ: Η€ β†’ ǀ𝐿(π‘Š) where π‘Š being a finite dimensional real or complex vector space with a dim π‘Š β‰₯ 1. Example (2.2): The 1-dimensional complex vector space (β‚΅). For any Lie group Η€ , one can describe the trivial representation of Η€, π‘Ÿ: Η€ β†’ ǀ𝐿(1, β‚΅), via the formular: π‘Ÿ(𝑠) = 1, for all 𝑠 ∈ Η€. Definition (2.3): [1] Let Η€ be a Lie group and π‘Ÿπ‘–, 𝑖 = 1,2, … , π‘š be representation of Η€ affects the vector spaces π‘Šπ‘–, 𝑖 = 1,2, … , π‘š, then the direct sum of π‘Ÿπ‘–, bring the representation defined by: {π‘Ÿ1 βŠ• π‘Ÿ2 βŠ• … βŠ• π‘Ÿπ‘š(𝑠)}(π‘Š1, π‘Š2, … , π‘Š π‘š) = π‘Ÿ1(𝑠)π‘Š1, π‘Ÿ2(𝑠)π‘Š2, … , π‘Ÿπ‘š(𝑠)π‘Š π‘š for all (𝑠 ∈ Η€), π‘Š1, π‘Š2, … , π‘Š π‘š ∈ π‘Š1, π‘Š2 Γ— … Γ— π‘Š π‘š. Example (2.4): Let π‘Ÿ1: β„› β†’ ǀ𝐿(2, β„›), such that π‘Ÿ1(𝑠) = ( 1 𝑠 0 1 ) for all 𝑠 ∈ β„›. And π‘Ÿ2: β„› β†’ ǀ𝐿(2, β„›), such that π‘Ÿ2(𝑑) = ( βˆ’1 0 𝑑 1 ) for all 𝑠 ∈ β„›. {π‘Ÿ1 + π‘Ÿ2(𝑠)}(π‘Š1, π‘Š2) = (π‘Ÿ1(𝑠)π‘Š1, π‘Ÿ2(𝑠)π‘Š2) = (( 1 𝑠 0 1 ) π‘Š1 , ( βˆ’1 0 𝑑 1 ) π‘Š2 ). Definition (2.5): [9] Let both Η€ and β„‹ be the Lie groups , let π‘Ÿ1 be a representation of Η€ affects the space π‘Š1 and let π‘Ÿ2 be representation of β„‹ affects the space π‘Š2, then the tensor product of π‘Ÿ1and π‘Ÿ2 being the representation on (π‘Ÿ1 βŠ— π‘Ÿ2)(𝑠, 𝑑) = π‘Ÿ(𝑠) βŠ— π‘Ÿ2(𝑑). for all 𝑠 ∈ Η€ and 𝑑 ∈ β„‹. Example (2.6): Let π‘Ÿ1: β„› β†’ ǀ𝐿(2, β„›), such that π‘Ÿ1(𝑠) = ( 1 𝑠 0 1 ) for all 𝑠 ∈ β„›.
UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 86 Let π‘Ÿ2: β„› β†’ ǀ𝐿(2, β„›), such that π‘Ÿ2(𝑑) = ( 1 0 βˆ’π‘‘ βˆ’1 ) for all 𝑠 ∈ β„›. (π‘Ÿ1 βŠ— π‘Ÿ2)(𝑠, 𝑑) = π‘Ÿ1(𝑠) βŠ— π‘Ÿ2(𝑑) = ( 1 𝑠 0 1 ) βŠ— ( 1 0 βˆ’π‘‘ βˆ’1 ) = ( ( 1 0 βˆ’π‘‘ βˆ’1 ) ( 𝑠 0 βˆ’π‘‘π‘  βˆ’π‘  ) 0 ( 1 0 βˆ’π‘‘ βˆ’1 ) ) = ( 1 0 βˆ’π‘‘ βˆ’1 𝑠 0 βˆ’π‘‘π‘  βˆ’π‘  0 0 0 0 1 0 βˆ’π‘‘ βˆ’1 ) 4Γ—4 , the matrix in ǀ𝐿(4, β„›). Proposition (2.7): [7] Let π‘Ÿ be a representation for Lie group Η€ affects the finite dimensional vector space π‘Š, then representation of dual of π‘Ÿ is representation of Η€ on π‘Šβˆ— given by: π‘Ÿβˆ—(𝑠) = [π‘Ÿ(π‘ βˆ’1)]π‘‘π‘Ÿ , the dual representation is also called contragredient representation. Example (2.8): Let π‘Ÿ: π‘ βˆ’1 β†’ π‘†π‘œ(2, β‚΅), where 𝑠1{(cos ∝ , sin ∝), 0 β‰€βˆβ‰€ 2πœ‹}, and 𝑠1 = π‘’π‘–βˆ = cos ∝ + 𝑖 sin ∝ π‘†π‘œ(2, βŠ„) = {( cos ∝ βˆ’ sin ∝ sin ∝ cos ∝ ) , 0 β‰€βˆβ‰€ 2πœ‹} such that π‘Ÿ(cos ∝ , sin ∝) = ( cos ∝ βˆ’ sin ∝ sin ∝ cos ∝ ) π‘Ÿ(π‘’π‘–βˆ ) = ( cos ∝ βˆ’ sin ∝ sin ∝ cos ∝ ) , 0 β‰€βˆβ‰€ 2πœ‹. π‘Ÿ is a representation for Lie group 𝑆1 , let 𝑆 = π‘’π‘–βˆ , π‘Ÿ(π‘’π‘–βˆ ) = ( cos ∝ sin ∝ βˆ’sin ∝ cos ∝ ) 𝑆1 = cos ∝ βˆ’ 𝑖 sin ∝ , π‘Ÿ(𝑠)βˆ’1 = ( cos ∝ βˆ’ sin ∝ sin ∝ cos ∝ ) {π‘Ÿ(𝑠)βˆ’1}π‘‘π‘Ÿ = ( cos ∝ βˆ’ 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 π‘Ÿβ€²1 and π‘Ÿβ€²2 are representation of Lie algebra (Η₯) affects the finite-dimensional spaces π‘Š1 as well as π‘Š2, correspondingly. defined as an action of (Η₯) on π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1), where 𝑀 be field, Π­: Η₯ β†’ Η₯𝐿(π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1)). By for all 𝑠 ∈ Η₯, β„Ž ∈ π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1).
UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 87 Π­(𝑠) = π‘Ÿβ€²1(𝑠)β„Ž = β„Žπ‘Ÿβ€²2(𝑠). And π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1) β‰… π‘Š2 βˆ— βŠ— π‘Š1 as equivalence of representations, see [2] . Proposition (3.1): [4] Let π‘Ÿπ‘–: Η€ β†’ ǀ𝐿(π‘Šπ‘–) be representations of Lie group Η€ on 𝑀 βˆ’ finite dimensional vector space (π‘Šπ‘–), for 𝑖 = 1,2, correspondingly, and π‘Ÿπ‘– βˆ— : Η€ β†’ ǀ𝐿(π‘Šπ‘– βˆ— ) the dual representation on (π‘Šπ‘– βˆ— ),for 𝑖 = 1,2, which is give by π‘Ÿπ‘– βˆ— (𝑠) = β„Žπ‘– ∘ π‘Ÿπ‘–(𝑠) , for all 𝑠 ∈ Η€, where β„Žπ‘–: π‘Šπ‘– β†’ 𝑀. See the diagram: Proposition (3.2): Let π‘Ÿπ‘–, 𝑖 = 1,2,3, π‘Žπ‘›π‘‘ 4 be for representations of Lie group affects the vector spaces π‘Šπ‘–, 𝑖 = 1,2,3, π‘Žπ‘›π‘‘ 4 respectively, and let π»π‘œπ‘šπ‘€(π‘Š4 βˆ— , Hom (π‘Š3 βˆ— , π‘Š2), π»π‘œπ‘š(π‘Š2, π‘Š1)), be vector space of the whole linear mapping from π‘Š2 and from Hom (π‘Š3 βˆ— , π‘Š2) to Hom (π‘Š3 βˆ— , π‘Š1) as well as from (π‘Š4, Hom (π‘Š3 βˆ— , π‘Š2) to Hom(π‘Š3, π‘Š1)), then the (𝑄𝑍𝐴) of Lie group on π»π‘œπ‘šπ‘€(π‘Š4 βˆ— , Hom (π‘Š3 βˆ— , π‘Š2), π»π‘œπ‘š(π‘Š2, π‘Š1)). Proof: Define Π­ ́ : Η€ β†’ ǀ𝐿(π»π‘œπ‘šπ‘€(π‘Š4 βˆ— , Hom (π‘Š3 βˆ— , π‘Š2), π»π‘œπ‘š(π‘Š2, π‘Š1)). Such that: Π­ ́ (𝑠)β„Ž ́ [(π‘Ÿ1(𝑠) ∘ β„Ž ́1 ∘ π‘Ÿ2(𝑠)) ∘ (π‘Ÿ2(𝑠) ∘ β„Ž ́2π‘Ÿ3(𝑠)βˆ’1 )] ∘ β„Ž ́3 ∘ π‘Ÿ4(𝑠)βˆ’1 , for all (𝑠 ∈ Η€) and β„Žπ‘–: π‘Šπ‘– β†’ 𝑀. The following diagram shows the use acting of group Η€ into ǀ𝐿(π»π‘œπ‘šπ‘€(π‘Š4 βˆ— , Hom (π‘Š3 βˆ— , π‘Š2), π»π‘œπ‘š(π‘Š2, π‘Š1)).
UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 88 Proposition (3.3): Let π‘Ÿπ‘–, 𝑖 = 1,2,3, π‘Žπ‘›π‘‘ 4 be for representations of Lie group affects the vector spaces π‘Šπ‘–, 𝑖 = 1,2,3, π‘Žπ‘›π‘‘ 4 respectively, and let π»π‘œπ‘šπ‘€(π‘Š4, π»π‘œπ‘š(π‘Š3, π‘Š2), π»π‘œπ‘š(π‘Š2 βˆ— , π‘Š1 βˆ—)), be M-vector space of the whole linear mapping from π‘Š4 to π»π‘œπ‘š(π‘Š3, π‘Š2), π»π‘œπ‘š(π‘Š2 βˆ— , π‘Š1 βˆ—). Then the (𝑄𝑍𝐴) of Lie group on π»π‘œπ‘šπ‘€(π‘Š4, π»π‘œπ‘š(π‘Š3, π‘Š2), π»π‘œπ‘š(π‘Š2 βˆ— , π‘Š1 βˆ—)). Proof: Define Π­ ́ : Η€ β†’ Η€πΏπ»π‘œπ‘šπ‘€(π‘Š4, π»π‘œπ‘š(π‘Š3, π‘Š2), π»π‘œπ‘š(π‘Š2 βˆ— , π‘Š1 βˆ—)), such that Π­ ́ (𝑠)β„Ž ́ [(π‘Ÿ1(𝑠)βˆ’1 ∘ β„Ž ́1 ∘ π‘Ÿ2(𝑠)βˆ’1 ) ∘ (π‘Ÿ2(𝑠) ∘ β„Ž ́2 ∘ π‘Ÿ3(𝑠))] ∘ β„Ž ́3 ∘ π‘Ÿ4(𝑠) , for all 𝑠 ∈ Η€ and β„Žπ‘–: π‘Šπ‘– β†’ 𝑀. The following diagram shows the acting of group Η€ is into ǀ𝐿(π»π‘œπ‘šπ‘€(π‘Š4, π»π‘œπ‘š(π‘Š3, π‘Š2), π»π‘œπ‘š(π‘Š2 βˆ— , π‘Š1 βˆ—))).
UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 89 Proposition (3.4): Let π‘Ÿπ‘–, 𝑖 = 1,2,3, π‘Žπ‘›π‘‘ 4 be for representations of Lie group affects the vector spaces π‘Šπ‘–, 𝑖 = 1,2,3, π‘Žπ‘›π‘‘ 4 respectively, and let π»π‘œπ‘šπ‘€(π‘Š4, π»π‘œπ‘š(π‘Š3, π‘Š2 βˆ—)), π‘Š1 βˆ— ), be M-vector space of the whole linear mapping from π‘Š3 to π‘Š2 βˆ— and from π»π‘œπ‘š(π‘Š3, π‘Š2 βˆ—))into π‘Š1 βˆ— as well as π‘Š4 into π»π‘œπ‘š(π»π‘œπ‘š(π‘Š3, π‘Š2 βˆ—)), π‘Š1 βˆ— ).Then the (𝑄𝑍𝐴) of Lie group is on π»π‘œπ‘šπ‘€(π‘Š4, π»π‘œπ‘š(π‘Š3, π‘Š2 βˆ—)), π‘Š1 βˆ— )). Proof: Define πœ‘: Η€ β†’ ǀ𝐿(π»π‘œπ‘šπ‘€(π‘Š4, π»π‘œπ‘š(π‘Š3, π‘Š2 βˆ—)), π‘Š1 βˆ— )). Such that Π­ ́ (𝑠)β„Ž ́ [(π‘Ÿ1(𝑠)βˆ’1 ∘ β„Ž ́1 ∘ (π‘Ÿ2(𝑠)βˆ’1 ∘ β„Ž ́2 ∘ π‘Ÿ3(𝑠))] ∘ β„Ž ́3 ∘ π‘Ÿ4(𝑠) , for all 𝑠 ∈ Η€ , β„Žπ‘–: π‘Šπ‘– β†’ 𝑀. Proposition (3.5): Let π‘Ÿπ‘–, 𝑖 = 1,2,3, π‘Žπ‘›π‘‘ 4 be for representations of Lie group affects the vector spaces π‘Šπ‘–, 𝑖 = 1,2,3, π‘Žπ‘›π‘‘ 4 respectively, and let π»π‘œπ‘šπ‘€(π‘Š4, π»π‘œπ‘š(π‘Š3 βˆ— , π‘Š2)), π‘Š1 βˆ— ), be M-vector space of the whole linear mapping from π‘Š3 to π‘Š2 and from π»π‘œπ‘š(π‘Š3 βˆ— , π‘Š2))into π‘Š1 as well as π‘Š4 into π»π‘œπ‘š(π»π‘œπ‘š(π‘Š3 βˆ— , π‘Š2)).Then the (𝑄𝑍𝐴) of Lie group is on π»π‘œπ‘šπ‘€(π‘Š4, π»π‘œπ‘š(π‘Š3 βˆ— , π‘Š2)), π‘Š1 βˆ— )).
UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 90 Proof: Define Π­ ́ : Η€ β†’ Η€πΏπ»π‘œπ‘šπ‘€(π‘Š4, π»π‘œπ‘š(π‘Š3 βˆ— , π‘Š2), π‘Š1 βˆ— ), by proposition (3.2) such that: Π­ ́ (𝑠)β„Ž ́ = [(π‘Ÿ1(𝑠)βˆ’1 ∘ β„Ž ́1 ∘ (π‘Ÿ2(𝑠) ∘ β„Ž ́2 ∘ π‘Ÿ3(𝑠)βˆ’1 )] ∘ β„Ž ́3 ∘ π‘Ÿ4(𝑠). for all 𝑠 ∈ Η€ and β„Žπ‘–: π‘Šπ‘– β†’ 𝑀. Proposition (3.6): Let π‘Ÿπ‘–, 𝑖 = 1,2,3, π‘Žπ‘›π‘‘ 4 be four representations of Lie group affects the vector spaces π‘Šπ‘–, 𝑖 = 1,2,3, π‘Žπ‘›π‘‘ 4 respectively, and let π»π‘œπ‘šπ‘€(π‘Š4 βˆ— , π»π‘œπ‘š(π‘Š3 βˆ— , π‘Š2)), π‘Š1), be M-vector space of the whole linear mapping from π‘Š3 βˆ— to π‘Š2 and π»π‘œπ‘š(π‘Š3 βˆ— , π‘Š2))into π‘Š1 as well as π‘Š4 βˆ— into π»π‘œπ‘š(π»π‘œπ‘š(π‘Š3 βˆ— , π‘Š2)).Then the (𝑄𝑍𝐴) of Lie group is on π»π‘œπ‘šπ‘€(π‘Š4 βˆ— , π»π‘œπ‘š(π‘Š3 βˆ— , π‘Š2)), π‘Š1)). Proof: Define Π­ ́ : Η€ β†’ Η€πΏπ»π‘œπ‘šπ‘€(π‘Š4 βˆ— , π»π‘œπ‘š(π‘Š3 βˆ— , π‘Š2), π‘Š1), by proposition (3.4) such that: Π­ ́ (𝑠)β„Ž ́ = [(π‘Ÿ1(𝑠) ∘ β„Ž ́1 ∘ (π‘Ÿ2(𝑠) ∘ β„Ž ́2 ∘ π‘Ÿ3(𝑠)βˆ’1 )] ∘ β„Ž ́3 ∘ π‘Ÿ4(𝑠)βˆ’1 . for all (𝑠 ∈ Η€) and β„Žπ‘–: π‘Šπ‘– β†’ 𝑀.
UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 91 Proposition (3.7): Let π‘Ÿπ‘–, 𝑖 = 1,2,3,4, π‘Žπ‘›π‘‘ 5 be five representation of Lie group affects the vector spaces π‘Šπ‘–, 𝑖 = 1,2,3,4, π‘Žπ‘›π‘‘ 5 respectively, and let π»π‘œπ‘šπ‘€(π‘Š5, π»π‘œπ‘š(π‘Š4, π‘Š3 βˆ—), π‘Š2 βˆ— βŠ• π‘Š1 βˆ— )), be M-vector space of the whole linear mapping from π‘Š4 to π‘Š3 βˆ— and π»π‘œπ‘š(π‘Š4, π‘Š3 βˆ—))into π‘Š2 βˆ— βŠ• π‘Š1 βˆ— as well as π‘Š5 into π»π‘œπ‘š(π»π‘œπ‘š(π‘Š4, π‘Š3 βˆ—), π‘Š2 βˆ— βŠ• π‘Š1 βˆ— ).Then the (𝑄𝑍𝐴) of Lie group is on π»π‘œπ‘šπ‘€(π‘Š5, π»π‘œπ‘š(π»π‘œπ‘š(π‘Š4, π‘Š3), π‘Š2 βˆ— βŠ• π‘Š1 βˆ— )). Proof: Define βˆ… ́ : Η€ β†’ ǀ𝐿(π»π‘œπ‘šπ‘€(π‘Š5, π»π‘œπ‘š(π»π‘œπ‘š(π‘Š4, π‘Š3 βˆ—), π‘Š2 βˆ— βŠ• π‘Š1 βˆ— ))), by proposition (3.4) such that: βˆ… ́ (𝑠)β„Ž ́ = [(π‘Ÿ1(𝑠)βˆ’1 βŠ• (π‘Ÿ2(𝑠)βˆ’1 ) ∘ β„Ž ́1 ∘ (π‘Ÿ3(𝑠)βˆ’1 ∘ β„Ž ́2 ∘ π‘Ÿ4(𝑠))] ∘ β„Ž ́3 ∘ π‘Ÿ5(𝑠). for all 𝑠 ∈ Η€ and β„Žπ‘–: π‘Šπ‘– β†’ 𝑀. Proposition (3.8): Let π‘Ÿπ‘–, 𝑖 = 1,2,3,4, π‘Žπ‘›π‘‘ 5 be five representation of Lie group affects the vector spaces π‘Šπ‘–, 𝑖 = 1,2,3,4, π‘Žπ‘›π‘‘ 5 respectively, and let π»π‘œπ‘šπ‘€(π‘Š5, π»π‘œπ‘š(π‘Š4 βˆ— , π‘Š3 βˆ—), π»π‘œπ‘š(π‘Š2, π‘Š1 βˆ— ), be M-vector space of the whole linear mapping from π‘Š2 to π‘Š1 βˆ— and π»π‘œπ‘š(π‘Š2 βˆ— , π‘Š3 βˆ—))into π»π‘œπ‘š(π‘Š2, π‘Š1 βˆ— ) as well as π‘Š5 into (π»π‘œπ‘š(π‘Š4 βˆ— βŠ• π‘Š3 βˆ—), π»π‘œπ‘š(π‘Š2, π‘Š1 βˆ—)). Then the (𝑄𝑍𝐴) of Lie group is on π»π‘œπ‘šπ‘€(π‘Š5, π»π‘œπ‘š(π‘Š4 βˆ— , π‘Š3 βˆ—)π»π‘œπ‘š(π‘Š2, π‘Š1 βˆ—)).
UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 92 Proof: Define πœ‘Μ : Η€ β†’ ǀ𝐿(π»π‘œπ‘šπ‘€(π‘Š5, (π‘Š4 βˆ— βŠ• π‘Š3 βˆ—)π»π‘œπ‘š(π‘Š2, π‘Š1 βˆ—))),by proposition (3.4) such that: πœ‘Μ (𝑠)β„Ž ́ = [(π‘Ÿ1(𝑠)βˆ’1 ∘ β„Ž ́1 ∘ π‘Ÿ2(𝑠)) ∘ β„Ž ́2 ∘ (π‘Ÿ3(𝑠)βˆ’1 βˆ˜βŠ• π‘Ÿ4(𝑠)βˆ’1 )] ∘ β„Ž ́3 ∘ π‘Ÿ5(𝑠). for all 𝑠 ∈ Η€ and β„Žπ‘–: π‘Šπ‘– β†’ 𝑀. Proposition (3.9): Let π‘Ÿπ‘–, 𝑖 = 1,2,3,4,5 be five representation of Lie group affects the vector spaces π‘Šπ‘–, 𝑖 = 1,2,3,4,5 respectively, and let π»π‘œπ‘šπ‘€(π»π‘œπ‘š(π‘Š5, π‘Š4 βˆ—), π»π‘œπ‘š(π‘Š3, π‘Š2 βŠ• π‘Š1 βˆ—)) be M-vector space of all linear mapping from π‘Š5 to π‘Š4 βˆ— and from π‘Š3 into (π‘Š2 βŠ• π‘Š1 βˆ—)) as well as from π»π‘œπ‘š(π‘Š5, π‘Š4 βˆ—) into π»π‘œπ‘š(π‘Š3, π‘Š2 βŠ• π‘Š1 βˆ—) Then the (𝑄𝑍𝐴) of Lie group on π»π‘œπ‘šπ‘€(π»π‘œπ‘š(π‘Š5, π‘Š4 βˆ—), π»π‘œπ‘š(π‘Š3, π‘Š2 βŠ• π‘Š1 βˆ—). Proof: Define πœ‘Μ : Η€ β†’ ǀ𝐿(π»π‘œπ‘šπ‘€(π»π‘œπ‘š(π‘Š5, π‘Š4 βˆ—), π»π‘œπ‘š(π‘Š3, π‘Š2 βŠ• π‘Š1 βˆ—))),by proposition (3.4) such that: πœ‘Μ (𝑠)β„Ž ́ = [(π‘Ÿ1(𝑠)βˆ’1 βŠ• π‘Ÿ2(𝑠)) ∘ β„Ž ́1 ∘ π‘Ÿ3(𝑠))] ∘ β„Ž ́2 ∘ π‘Ÿ4(𝑠)βˆ’1 ∘ β„Ž ́3 ∘ π‘Ÿ5(𝑠). for all 𝑠 ∈ Η€ and β„Žπ‘–: π‘Šπ‘– β†’ 𝑀.
UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 93 4. The dual action of representation for Lie Group Proposition (4.1): [6] Let π‘Š1and π‘Š2 are finite dimensional vector space, and (𝑛) be a natural no, then the action of Lie group action on tensor product is : 1. π‘Š 2 βˆ—βˆ—,..βˆ— ⏟ 𝑛 βŠ— π‘Š1 β‰… π‘Š2 βŠ— π‘Š1, if 𝑛 is an even number. 2. π‘Š 2 βˆ—βˆ—,..βˆ— ⏟ 𝑛 βŠ— π‘Š1 β‰… π‘Š2 βˆ— βŠ— π‘Š1, if 𝑛 is an odd number. 3. π‘Š2 βŠ— π‘Š 2 βˆ—βˆ—,..βˆ— ⏟ 𝑛 β‰… π‘Š2 βŠ— π‘Š1, if 𝑛 is an even number. 4. π‘Š2 βŠ— π‘Š 2 βˆ—βˆ—,..βˆ— ⏟ 𝑛 β‰… π‘Š2 βŠ— π‘Š1 βˆ— , if 𝑛 is an odd number. Corollary (4.2): If (π‘Š)1and (π‘Š2) are finite- dimensional vector spaces, and (𝑛) be a natural number, then the Lie group affects the tensor product and π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1): 1. π‘Š2 βŠ— π‘Š1 βˆ— β‰… π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1 βˆ—) 2. π‘Š2 βŠ• π‘Š3 βŠ— π‘Š1 βˆ— β‰… π»π‘œπ‘šπ‘€(π‘Š1, π‘Š2 βŠ• π‘Š3) } if 𝑛 is an odd number. Proposition (4.3): Let π‘Ÿ1: Η€ β†’ ǀ𝐿(π‘Šπ‘–), 𝑅𝑖 βˆ— : Η€ β†’ ǀ𝐿(π‘Šπ‘– βˆ— ) for 𝑖 = 1,2, and the Lie group Η€ affects the πœ‘(𝑠)β„Ž = π‘Ÿ1(𝑠) ∘ β„Ž ∘ π‘Ÿ2(𝑠), for every 𝑠 ∈ Η€, β„Ž ∈ π»π‘œπ‘š(π‘Š2, π‘Š1). Then the Lie group Η€ affects (π»π‘œπ‘š(π‘Š2, π‘Š1))βˆ— being also given provided a representation πœ‘βˆ— , where: πœ‘βˆ—(𝑠) = π‘Ÿ(𝑠)βˆ’1 ∘ β„Žβˆ— ∘ π‘Ÿ1(𝑠), for all 𝑠 ∈ Η€ and β„Žβˆ— ∈ (π»π‘œπ‘š(π‘Š2, π‘Š1))βˆ— . Proof: Let action of Lie group Η€ affects π»π‘œπ‘š(π‘Š2, π‘Š1) being induced via the representation
UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 94 πœ‘: Η€ β†’ ǀ𝐿(π»π‘œπ‘š(π‘Š2, π‘Š1)), where πœ‘(𝑠) = π‘Ÿ1(𝑠) ∘ β„Ž ∘ π‘Ÿ2(𝑠)βˆ’1 , for 𝑠 ∈ Η€, and β„Ž ∈ π»π‘œπ‘š(π‘Š2, π‘Š1). Such that πœ‘βˆ—(𝑠) = π‘Ÿ(𝑠)βˆ’1 ∘ β„Žβˆ— ∘ π‘Ÿ1(𝑠) representation for all 𝑠 ∈ Η€ and β„Žβˆ— ∈ (π»π‘œπ‘š(π‘Š2, π‘Š1))βˆ— . Since: βˆ…βˆ—(𝑠) = (π‘Ÿ1(𝑠) ∘ β„Ž ∘ π‘Ÿ2(𝑠)βˆ’1 , )βˆ— = π‘Ÿ2 βˆ—(𝑠)βˆ’1 ∘ β„Žβˆ— ∘ π‘Ÿ1 βˆ—(𝑠) for all (𝑠 ∈ Η€) and β„Žβˆ— : π‘Š1 βˆ— β†’ π‘Š2 βˆ— , we have πœ‘βˆ—(𝑠𝑑) = (π‘Ÿ(𝑠𝑑))βˆ— = (π‘Ÿ(𝑑) ∘ β„Ž ∘ π‘Ÿ(𝑠))βˆ— = π‘Ÿβˆ— (𝑠) ∘ β„Žβˆ— ∘ π‘Ÿβˆ— (𝑑) Thus, πœ‘βˆ— is a representation from Η€(πœ‘βˆ— . 𝑠 π‘”π‘Ÿπ‘œπ‘’π‘ β„Žπ‘œπ‘šπ‘œπ‘šπ‘œπ‘Ÿπ‘β„Žπ‘–π‘ π‘š π‘œπ‘“ Η€) Corollary (4.4): Let π‘Ÿ1: Η€ β†’ ǀ𝐿(𝑗, 𝑀), ǀ𝐿(π‘Š1) β‰… ǀ𝐿(𝑗, 𝑀) And π‘Ÿ2: Η€ β†’ ǀ𝐿(β„“, 𝑀), ǀ𝐿(π‘Š2) β‰… ǀ𝐿(β„“, 𝑀). Where π‘Ÿ1 and π‘Ÿ2 have matrix representation, if Lie group Η€ affects π»π‘œπ‘š(π‘Š2, π‘Š1) being a representation πœ‘: Η€ β†’ ǀ𝐿(π»π‘œπ‘š(π‘Š2, π‘Š1)), where πœ‘(𝑠) = π‘Ÿ1(𝑠) ∘ β„Ž ∘ π‘Ÿ2(𝑠)βˆ’1 , for 𝑠 ∈ Η€. Then Lie group Η€ affects π»π‘œπ‘š(π‘Š2, π‘Š1) is a representation πœ‘βˆ— : Η€ β†’ ǀ𝐿(π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1))βˆ— . Where: πœ‘βˆ—(𝑠) = (π‘Ÿ2 βˆ—(𝑠)) π‘‘π‘Ÿ ∘ β„Žβˆ— ∘ (π‘Ÿ1(𝑠)βˆ’1)π‘‘π‘Ÿ , for all 𝑠 ∈ Η€.
UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 95 Proof: Since (𝑠) = π‘Ÿ1(𝑠) ∘ β„Ž ∘ π‘Ÿ2(𝑠)βˆ’1 , πœ‘βˆ—(𝑠) = π‘Ÿ(𝑠)βˆ’1 ∘ β„Žβˆ— ∘ π‘Ÿ1 βˆ—(𝑠) = (π‘Ÿ2 βˆ—(𝑠)) π‘‘π‘Ÿ ∘ β„Žβˆ— ∘ (π‘Ÿ1(𝑠)βˆ’1)π‘‘π‘Ÿ And πœ‘βˆ— is a representation a matrix representation of dimension 𝑗ℓ, then π‘Ÿβˆ—(𝑠𝑑) = (π‘Ÿ(𝑠𝑑)βˆ’1)π‘‘π‘Ÿ = (π‘Ÿ(𝑑)βˆ’1)π‘‘π‘Ÿ ∘ (π‘Ÿ(𝑠)βˆ’1)π‘‘π‘Ÿ = π‘Ÿβˆ—(𝑑) ∘ π‘Ÿβˆ—(𝑠) Example (4.5): Let π‘Ÿ1 = 𝑆1 β†’ π‘†π‘œ(2) βŠ‚ ǀ𝐿(2, β‚΅) and π‘Ÿ2 = 𝑆1 β†’ π‘†π‘œ(3) βŠ‚ ǀ𝐿(3, β‚΅) Where Η€ = 𝑆1(𝑗 = 2, β„“ = 3) and (π‘Š1) is the β‚΅ βˆ’ vector space of dimensional 2, and ( π‘Š2) being the β‚΅ βˆ’ vector space of dimensional 3, then by collar we have πœ‘βˆ— (π‘’π‘–βˆ ) = π‘Ÿ2 βˆ— (π‘’π‘–βˆ ) ∘ β„Žβˆ— ∘ π‘Ÿ1 βˆ— (π‘’π‘–βˆ ) = (π‘Ÿ2 βˆ— (π‘’π‘–βˆ )) π‘‘π‘Ÿ ∘ β„Žβˆ— ∘ (π‘Ÿ1(π‘’π‘–βˆ )) π‘‘π‘Ÿ = ( 1 0 0 0 cos ∝ βˆ’ sin ∝ 0 sin ∝ cos ∝ ) π‘‘π‘Ÿ ∘ β„Žβˆ— ∘ ( cos ∝ sin ∝ βˆ’sin ∝ cos ∝ ) π‘‘π‘Ÿ = ( 1 0 0 0 cos ∝ βˆ’ sin ∝ 0 sin ∝ cos ∝ ) π‘‘π‘Ÿ ∘ β„Žβˆ— ∘ ( cos ∝ βˆ’ sin ∝ sin ∝ cos ∝ ) π‘‘π‘Ÿ Let 𝐴 = ( 1 0 0 0 cos ∝ βˆ’ sin ∝ 0 sin ∝ cos ∝ ) Thus πœ‘βˆ— (π‘’π‘–βˆ ) = ( (cos ∝)𝐴 (βˆ’ sin ∝)𝐴 (sin ∝)𝐴 (cos ∝)𝐴 ) 6Γ—6
UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 96 = ( cos ∝ 0 0 0 cos ∝2 cos ∝ sin ∝ 0 βˆ’ cos ∝ sin ∝ cos ∝2 βˆ’ sin ∝ 0 0 0 βˆ’ sin ∝ cos ∝ βˆ’ sin ∝2 0 sin ∝2 sin ∝ cos ∝ sin ∝ 0 0 0 sin ∝ cos ∝ sin ∝2 0 βˆ’ sin ∝2 sin ∝ cos ∝ cos ∝2 0 0 0 cos ∝2 cos ∝ sin ∝ 0 βˆ’ cos ∝ sin ∝ cos ∝2 ) Is the representation of ǀ𝐿(6, β‚΅) acting on π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1) of dimensional 6. Proposition (4.6): Let (π‘Š1)and (π‘Š2) be two n- dimensional vector spaces and 𝑛 is a natural number, then π»π‘œπ‘šπ‘€(π»π‘œπ‘š(π‘Š3, π‘Š2, π‘Š1)) βˆ—βˆ—β€¦βˆ— ⏟ 𝑛 = { π»π‘œπ‘šπ‘€(π»π‘œπ‘š(π‘Š3, π‘Š2, π‘Š1))𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛 π»π‘œπ‘šπ‘€(π‘Š1 βˆ— , π»π‘œπ‘š(π‘Š2 βˆ— , π‘Š2)𝑖𝑓 𝑛 𝑖𝑠 π‘œπ‘‘π‘‘ … Proof: We can prove this proposition by mathematical induction. If 𝑛 = 1, then (π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1))βˆ— = (π»π‘œπ‘šπ‘€(π‘Š1 βˆ— , π‘Š2 βˆ—). And hence the action of group Η€ on π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1) is: πœ‘βˆ—(𝑠) = π‘Ÿ1 βˆ—(𝑠) ∘ β„Žβˆ— ∘ π‘Ÿ2 βˆ—(𝑠)βˆ’1 (see proposition (3.2). Suppose that (3.2) is true when 𝑛 = π‘˜ it mean (π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1)) βˆ—βˆ—β€¦βˆ— ⏟ π‘˜ = { (π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1)), 𝑖𝑓 π‘˜ 𝑖𝑠 𝑒𝑣𝑒𝑛 … (1) (π»π‘œπ‘šπ‘€(π‘Š1 βˆ— , π‘Š2), 𝑖𝑓 π‘˜ 𝑖𝑠 π‘œπ‘‘π‘‘ … (2) If π‘˜ is even then the acting of Lie group Η€ on π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1) is: πœ‘(𝑠) = π‘Ÿ2(𝑠)βˆ’1 ∘ β„Ž ∘ π‘Ÿ1(𝑠). And if π‘˜ is odd then the action of Lie group Η€ on π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1) is: πœ‘βˆ—(𝑠) = π‘Ÿ1 βˆ—(𝑠)βˆ’1 ∘ β„Žβˆ— ∘ π‘Ÿ2 βˆ—(𝑠) We will prove that (3.2) it is true, when 𝑛 = π‘˜ + 1 ,so it will be proven π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1) βˆ—βˆ—β€¦βˆ— ⏟ π‘˜+1 = { π»π‘œπ‘šπΊ(π‘Š2, π‘Š1))𝑖𝑓 π‘˜ 𝑖𝑠 π‘œπ‘‘π‘‘ π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1))𝑖𝑓 π‘˜ 𝑖𝑠 𝑖𝑠 𝑒𝑣𝑒𝑛 (πœ‘(𝑠)) βˆ—βˆ—β€¦βˆ— ⏟ π‘˜+1 = (π‘Ÿ1 βˆ—(𝑠) ∘ β„Žβˆ— π‘Ÿ2(𝑠)) βˆ—βˆ—β€¦βˆ— ⏟ π‘˜+1 (π‘Ÿ1 βˆ—(𝑠) ∘ β„Žβˆ— ∘ π‘Ÿ2 βˆ—(𝑠))
UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 97 When π‘˜ is odd we have π‘˜ + 1 is even, then π»π‘œπ‘šπ‘€(π»π‘œπ‘š(π‘Š3, π‘Š2), π‘Š1)) βˆ—βˆ—β€¦βˆ— ⏟ π‘˜+1 = π»π‘œπ‘šπ‘€(π»π‘œπ‘š(π‘Š3, π‘Š2), π‘Š1)) by(1), Thus the Lie group Η€ affects π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1) βˆ—βˆ—β€¦βˆ— ⏟ π‘˜+1 is: (πœ‘(𝑠)) βˆ—βˆ—β€¦βˆ— ⏟ π‘˜+1 = [(π‘Ÿ1(𝑠)βˆ’1 ∘ β„Ž1 ∘ (π‘Ÿ2(𝑠) ∘ β„Ž2 ∘ π‘Ÿ3 βˆ’1(𝑠))] And when π‘˜ is even we have π‘˜ + 1 is odd. Then π»π‘œπ‘šπ‘€(π»π‘œπ‘š(π‘Š3, π‘Š2), π‘Š1)) βˆ—βˆ—β€¦βˆ— ⏟ π‘˜+1 = π»π‘œπ‘šπ‘€(π‘Š1 βˆ— , π»π‘œπ‘š(π‘Š2 βˆ— , π‘Š3)) by (2), Thus the affects of Lie group Η€ on π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1) βˆ—βˆ—β€¦βˆ— ⏟ π‘˜+1 is: (πœ‘(𝑠)) βˆ—βˆ—β€¦βˆ— ⏟ π‘˜+1 = [(π‘Ÿ3(𝑠) ∘ β„Žβˆ— 1 ∘ π‘Ÿ2(𝑠)βˆ’1 ) ∘ β„Žβˆ— 2 ∘ π‘Ÿ1 βˆ’1(𝑠)βˆ’1] Then the proposition is true for all ∈ π‘βˆ— . 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.
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.

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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 Η€. Keywords: Lie group, representation of Lie group, dual representation of Lie group, tensor product for representation of Lie group. 1. Introduction A Lie group Η€ is finite dimensional smooth manifold, together with a group structure on Η€, such that the multiplication Η€Γ—Η€β†’Η€ and attaching of an inverse Η€β†’Η€βˆΆ Η₯ β†’ Η₯βˆ’1 being smooth maps [1]. Lie group homomorphism from Η€ to β„‹ is linear map such that Η€ and β„‹ represent Lie groups , Π­: Η€ β†’ β„‹ represents the group homomorphism , and Π­ represent 𝐢∞ βˆ’map on β„‹ [6]. The direct sum of π‘Ÿπ‘– of Lie group Η€ acting on the vector spaces π‘Šπ‘– over the field M is: π‘Ÿ1(𝑠)π‘Š1, … , π‘Ÿπ‘–(𝑠)π‘Šπ‘– [4]. π»π‘œπ‘š(π‘Š1, π‘Š2) is linear map from vector space π‘Š1 to vector space π‘Š2 such that: π»π‘œπ‘š(π‘Š1, π‘Š2) β‰… π‘Š1 βˆ— βŠ— π‘Š2[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 Η€ , be a finite dimensional real or complex representation of Η€ being a Lie group homomorphism π‘Ÿ: Η€ β†’ ǀ𝐿(𝑛, β„›) = (𝑛 β‰₯ 1). In general, a Lie group homomorphism is π‘Ÿ: Η€ β†’ ǀ𝐿(π‘Š) where π‘Š being a finite dimensional real or complex vector space with a dim π‘Š β‰₯ 1. Example (2.2): The 1-dimensional complex vector space (β‚΅). For any Lie group Η€ , one can describe the trivial representation of Η€, π‘Ÿ: Η€ β†’ ǀ𝐿(1, β‚΅), via the formular: π‘Ÿ(𝑠) = 1, for all 𝑠 ∈ Η€. Definition (2.3): [1] Let Η€ be a Lie group and π‘Ÿπ‘–, 𝑖 = 1,2, … , π‘š be representation of Η€ affects the vector spaces π‘Šπ‘–, 𝑖 = 1,2, … , π‘š, then the direct sum of π‘Ÿπ‘–, bring the representation defined by: {π‘Ÿ1 βŠ• π‘Ÿ2 βŠ• … βŠ• π‘Ÿπ‘š(𝑠)}(π‘Š1, π‘Š2, … , π‘Š π‘š) = π‘Ÿ1(𝑠)π‘Š1, π‘Ÿ2(𝑠)π‘Š2, … , π‘Ÿπ‘š(𝑠)π‘Š π‘š for all (𝑠 ∈ Η€), π‘Š1, π‘Š2, … , π‘Š π‘š ∈ π‘Š1, π‘Š2 Γ— … Γ— π‘Š π‘š. Example (2.4): Let π‘Ÿ1: β„› β†’ ǀ𝐿(2, β„›), such that π‘Ÿ1(𝑠) = ( 1 𝑠 0 1 ) for all 𝑠 ∈ β„›. And π‘Ÿ2: β„› β†’ ǀ𝐿(2, β„›), such that π‘Ÿ2(𝑑) = ( βˆ’1 0 𝑑 1 ) for all 𝑠 ∈ β„›. {π‘Ÿ1 + π‘Ÿ2(𝑠)}(π‘Š1, π‘Š2) = (π‘Ÿ1(𝑠)π‘Š1, π‘Ÿ2(𝑠)π‘Š2) = (( 1 𝑠 0 1 ) π‘Š1 , ( βˆ’1 0 𝑑 1 ) π‘Š2 ). Definition (2.5): [9] Let both Η€ and β„‹ be the Lie groups , let π‘Ÿ1 be a representation of Η€ affects the space π‘Š1 and let π‘Ÿ2 be representation of β„‹ affects the space π‘Š2, then the tensor product of π‘Ÿ1and π‘Ÿ2 being the representation on (π‘Ÿ1 βŠ— π‘Ÿ2)(𝑠, 𝑑) = π‘Ÿ(𝑠) βŠ— π‘Ÿ2(𝑑). for all 𝑠 ∈ Η€ and 𝑑 ∈ β„‹. Example (2.6): Let π‘Ÿ1: β„› β†’ ǀ𝐿(2, β„›), such that π‘Ÿ1(𝑠) = ( 1 𝑠 0 1 ) for all 𝑠 ∈ β„›.
  • 3. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 86 Let π‘Ÿ2: β„› β†’ ǀ𝐿(2, β„›), such that π‘Ÿ2(𝑑) = ( 1 0 βˆ’π‘‘ βˆ’1 ) for all 𝑠 ∈ β„›. (π‘Ÿ1 βŠ— π‘Ÿ2)(𝑠, 𝑑) = π‘Ÿ1(𝑠) βŠ— π‘Ÿ2(𝑑) = ( 1 𝑠 0 1 ) βŠ— ( 1 0 βˆ’π‘‘ βˆ’1 ) = ( ( 1 0 βˆ’π‘‘ βˆ’1 ) ( 𝑠 0 βˆ’π‘‘π‘  βˆ’π‘  ) 0 ( 1 0 βˆ’π‘‘ βˆ’1 ) ) = ( 1 0 βˆ’π‘‘ βˆ’1 𝑠 0 βˆ’π‘‘π‘  βˆ’π‘  0 0 0 0 1 0 βˆ’π‘‘ βˆ’1 ) 4Γ—4 , the matrix in ǀ𝐿(4, β„›). Proposition (2.7): [7] Let π‘Ÿ be a representation for Lie group Η€ affects the finite dimensional vector space π‘Š, then representation of dual of π‘Ÿ is representation of Η€ on π‘Šβˆ— given by: π‘Ÿβˆ—(𝑠) = [π‘Ÿ(π‘ βˆ’1)]π‘‘π‘Ÿ , the dual representation is also called contragredient representation. Example (2.8): Let π‘Ÿ: π‘ βˆ’1 β†’ π‘†π‘œ(2, β‚΅), where 𝑠1{(cos ∝ , sin ∝), 0 β‰€βˆβ‰€ 2πœ‹}, and 𝑠1 = π‘’π‘–βˆ = cos ∝ + 𝑖 sin ∝ π‘†π‘œ(2, βŠ„) = {( cos ∝ βˆ’ sin ∝ sin ∝ cos ∝ ) , 0 β‰€βˆβ‰€ 2πœ‹} such that π‘Ÿ(cos ∝ , sin ∝) = ( cos ∝ βˆ’ sin ∝ sin ∝ cos ∝ ) π‘Ÿ(π‘’π‘–βˆ ) = ( cos ∝ βˆ’ sin ∝ sin ∝ cos ∝ ) , 0 β‰€βˆβ‰€ 2πœ‹. π‘Ÿ is a representation for Lie group 𝑆1 , let 𝑆 = π‘’π‘–βˆ , π‘Ÿ(π‘’π‘–βˆ ) = ( cos ∝ sin ∝ βˆ’sin ∝ cos ∝ ) 𝑆1 = cos ∝ βˆ’ 𝑖 sin ∝ , π‘Ÿ(𝑠)βˆ’1 = ( cos ∝ βˆ’ sin ∝ sin ∝ cos ∝ ) {π‘Ÿ(𝑠)βˆ’1}π‘‘π‘Ÿ = ( cos ∝ βˆ’ 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 π‘Ÿβ€²1 and π‘Ÿβ€²2 are representation of Lie algebra (Η₯) affects the finite-dimensional spaces π‘Š1 as well as π‘Š2, correspondingly. defined as an action of (Η₯) on π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1), where 𝑀 be field, Π­: Η₯ β†’ Η₯𝐿(π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1)). By for all 𝑠 ∈ Η₯, β„Ž ∈ π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1).
  • 4. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 87 Π­(𝑠) = π‘Ÿβ€²1(𝑠)β„Ž = β„Žπ‘Ÿβ€²2(𝑠). And π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1) β‰… π‘Š2 βˆ— βŠ— π‘Š1 as equivalence of representations, see [2] . Proposition (3.1): [4] Let π‘Ÿπ‘–: Η€ β†’ ǀ𝐿(π‘Šπ‘–) be representations of Lie group Η€ on 𝑀 βˆ’ finite dimensional vector space (π‘Šπ‘–), for 𝑖 = 1,2, correspondingly, and π‘Ÿπ‘– βˆ— : Η€ β†’ ǀ𝐿(π‘Šπ‘– βˆ— ) the dual representation on (π‘Šπ‘– βˆ— ),for 𝑖 = 1,2, which is give by π‘Ÿπ‘– βˆ— (𝑠) = β„Žπ‘– ∘ π‘Ÿπ‘–(𝑠) , for all 𝑠 ∈ Η€, where β„Žπ‘–: π‘Šπ‘– β†’ 𝑀. See the diagram: Proposition (3.2): Let π‘Ÿπ‘–, 𝑖 = 1,2,3, π‘Žπ‘›π‘‘ 4 be for representations of Lie group affects the vector spaces π‘Šπ‘–, 𝑖 = 1,2,3, π‘Žπ‘›π‘‘ 4 respectively, and let π»π‘œπ‘šπ‘€(π‘Š4 βˆ— , Hom (π‘Š3 βˆ— , π‘Š2), π»π‘œπ‘š(π‘Š2, π‘Š1)), be vector space of the whole linear mapping from π‘Š2 and from Hom (π‘Š3 βˆ— , π‘Š2) to Hom (π‘Š3 βˆ— , π‘Š1) as well as from (π‘Š4, Hom (π‘Š3 βˆ— , π‘Š2) to Hom(π‘Š3, π‘Š1)), then the (𝑄𝑍𝐴) of Lie group on π»π‘œπ‘šπ‘€(π‘Š4 βˆ— , Hom (π‘Š3 βˆ— , π‘Š2), π»π‘œπ‘š(π‘Š2, π‘Š1)). Proof: Define Π­ ́ : Η€ β†’ ǀ𝐿(π»π‘œπ‘šπ‘€(π‘Š4 βˆ— , Hom (π‘Š3 βˆ— , π‘Š2), π»π‘œπ‘š(π‘Š2, π‘Š1)). Such that: Π­ ́ (𝑠)β„Ž ́ [(π‘Ÿ1(𝑠) ∘ β„Ž ́1 ∘ π‘Ÿ2(𝑠)) ∘ (π‘Ÿ2(𝑠) ∘ β„Ž ́2π‘Ÿ3(𝑠)βˆ’1 )] ∘ β„Ž ́3 ∘ π‘Ÿ4(𝑠)βˆ’1 , for all (𝑠 ∈ Η€) and β„Žπ‘–: π‘Šπ‘– β†’ 𝑀. The following diagram shows the use acting of group Η€ into ǀ𝐿(π»π‘œπ‘šπ‘€(π‘Š4 βˆ— , Hom (π‘Š3 βˆ— , π‘Š2), π»π‘œπ‘š(π‘Š2, π‘Š1)).
  • 5. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 88 Proposition (3.3): Let π‘Ÿπ‘–, 𝑖 = 1,2,3, π‘Žπ‘›π‘‘ 4 be for representations of Lie group affects the vector spaces π‘Šπ‘–, 𝑖 = 1,2,3, π‘Žπ‘›π‘‘ 4 respectively, and let π»π‘œπ‘šπ‘€(π‘Š4, π»π‘œπ‘š(π‘Š3, π‘Š2), π»π‘œπ‘š(π‘Š2 βˆ— , π‘Š1 βˆ—)), be M-vector space of the whole linear mapping from π‘Š4 to π»π‘œπ‘š(π‘Š3, π‘Š2), π»π‘œπ‘š(π‘Š2 βˆ— , π‘Š1 βˆ—). Then the (𝑄𝑍𝐴) of Lie group on π»π‘œπ‘šπ‘€(π‘Š4, π»π‘œπ‘š(π‘Š3, π‘Š2), π»π‘œπ‘š(π‘Š2 βˆ— , π‘Š1 βˆ—)). Proof: Define Π­ ́ : Η€ β†’ Η€πΏπ»π‘œπ‘šπ‘€(π‘Š4, π»π‘œπ‘š(π‘Š3, π‘Š2), π»π‘œπ‘š(π‘Š2 βˆ— , π‘Š1 βˆ—)), such that Π­ ́ (𝑠)β„Ž ́ [(π‘Ÿ1(𝑠)βˆ’1 ∘ β„Ž ́1 ∘ π‘Ÿ2(𝑠)βˆ’1 ) ∘ (π‘Ÿ2(𝑠) ∘ β„Ž ́2 ∘ π‘Ÿ3(𝑠))] ∘ β„Ž ́3 ∘ π‘Ÿ4(𝑠) , for all 𝑠 ∈ Η€ and β„Žπ‘–: π‘Šπ‘– β†’ 𝑀. The following diagram shows the acting of group Η€ is into ǀ𝐿(π»π‘œπ‘šπ‘€(π‘Š4, π»π‘œπ‘š(π‘Š3, π‘Š2), π»π‘œπ‘š(π‘Š2 βˆ— , π‘Š1 βˆ—))).
  • 6. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 89 Proposition (3.4): Let π‘Ÿπ‘–, 𝑖 = 1,2,3, π‘Žπ‘›π‘‘ 4 be for representations of Lie group affects the vector spaces π‘Šπ‘–, 𝑖 = 1,2,3, π‘Žπ‘›π‘‘ 4 respectively, and let π»π‘œπ‘šπ‘€(π‘Š4, π»π‘œπ‘š(π‘Š3, π‘Š2 βˆ—)), π‘Š1 βˆ— ), be M-vector space of the whole linear mapping from π‘Š3 to π‘Š2 βˆ— and from π»π‘œπ‘š(π‘Š3, π‘Š2 βˆ—))into π‘Š1 βˆ— as well as π‘Š4 into π»π‘œπ‘š(π»π‘œπ‘š(π‘Š3, π‘Š2 βˆ—)), π‘Š1 βˆ— ).Then the (𝑄𝑍𝐴) of Lie group is on π»π‘œπ‘šπ‘€(π‘Š4, π»π‘œπ‘š(π‘Š3, π‘Š2 βˆ—)), π‘Š1 βˆ— )). Proof: Define πœ‘: Η€ β†’ ǀ𝐿(π»π‘œπ‘šπ‘€(π‘Š4, π»π‘œπ‘š(π‘Š3, π‘Š2 βˆ—)), π‘Š1 βˆ— )). Such that Π­ ́ (𝑠)β„Ž ́ [(π‘Ÿ1(𝑠)βˆ’1 ∘ β„Ž ́1 ∘ (π‘Ÿ2(𝑠)βˆ’1 ∘ β„Ž ́2 ∘ π‘Ÿ3(𝑠))] ∘ β„Ž ́3 ∘ π‘Ÿ4(𝑠) , for all 𝑠 ∈ Η€ , β„Žπ‘–: π‘Šπ‘– β†’ 𝑀. Proposition (3.5): Let π‘Ÿπ‘–, 𝑖 = 1,2,3, π‘Žπ‘›π‘‘ 4 be for representations of Lie group affects the vector spaces π‘Šπ‘–, 𝑖 = 1,2,3, π‘Žπ‘›π‘‘ 4 respectively, and let π»π‘œπ‘šπ‘€(π‘Š4, π»π‘œπ‘š(π‘Š3 βˆ— , π‘Š2)), π‘Š1 βˆ— ), be M-vector space of the whole linear mapping from π‘Š3 to π‘Š2 and from π»π‘œπ‘š(π‘Š3 βˆ— , π‘Š2))into π‘Š1 as well as π‘Š4 into π»π‘œπ‘š(π»π‘œπ‘š(π‘Š3 βˆ— , π‘Š2)).Then the (𝑄𝑍𝐴) of Lie group is on π»π‘œπ‘šπ‘€(π‘Š4, π»π‘œπ‘š(π‘Š3 βˆ— , π‘Š2)), π‘Š1 βˆ— )).
  • 7. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 90 Proof: Define Π­ ́ : Η€ β†’ Η€πΏπ»π‘œπ‘šπ‘€(π‘Š4, π»π‘œπ‘š(π‘Š3 βˆ— , π‘Š2), π‘Š1 βˆ— ), by proposition (3.2) such that: Π­ ́ (𝑠)β„Ž ́ = [(π‘Ÿ1(𝑠)βˆ’1 ∘ β„Ž ́1 ∘ (π‘Ÿ2(𝑠) ∘ β„Ž ́2 ∘ π‘Ÿ3(𝑠)βˆ’1 )] ∘ β„Ž ́3 ∘ π‘Ÿ4(𝑠). for all 𝑠 ∈ Η€ and β„Žπ‘–: π‘Šπ‘– β†’ 𝑀. Proposition (3.6): Let π‘Ÿπ‘–, 𝑖 = 1,2,3, π‘Žπ‘›π‘‘ 4 be four representations of Lie group affects the vector spaces π‘Šπ‘–, 𝑖 = 1,2,3, π‘Žπ‘›π‘‘ 4 respectively, and let π»π‘œπ‘šπ‘€(π‘Š4 βˆ— , π»π‘œπ‘š(π‘Š3 βˆ— , π‘Š2)), π‘Š1), be M-vector space of the whole linear mapping from π‘Š3 βˆ— to π‘Š2 and π»π‘œπ‘š(π‘Š3 βˆ— , π‘Š2))into π‘Š1 as well as π‘Š4 βˆ— into π»π‘œπ‘š(π»π‘œπ‘š(π‘Š3 βˆ— , π‘Š2)).Then the (𝑄𝑍𝐴) of Lie group is on π»π‘œπ‘šπ‘€(π‘Š4 βˆ— , π»π‘œπ‘š(π‘Š3 βˆ— , π‘Š2)), π‘Š1)). Proof: Define Π­ ́ : Η€ β†’ Η€πΏπ»π‘œπ‘šπ‘€(π‘Š4 βˆ— , π»π‘œπ‘š(π‘Š3 βˆ— , π‘Š2), π‘Š1), by proposition (3.4) such that: Π­ ́ (𝑠)β„Ž ́ = [(π‘Ÿ1(𝑠) ∘ β„Ž ́1 ∘ (π‘Ÿ2(𝑠) ∘ β„Ž ́2 ∘ π‘Ÿ3(𝑠)βˆ’1 )] ∘ β„Ž ́3 ∘ π‘Ÿ4(𝑠)βˆ’1 . for all (𝑠 ∈ Η€) and β„Žπ‘–: π‘Šπ‘– β†’ 𝑀.
  • 8. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 91 Proposition (3.7): Let π‘Ÿπ‘–, 𝑖 = 1,2,3,4, π‘Žπ‘›π‘‘ 5 be five representation of Lie group affects the vector spaces π‘Šπ‘–, 𝑖 = 1,2,3,4, π‘Žπ‘›π‘‘ 5 respectively, and let π»π‘œπ‘šπ‘€(π‘Š5, π»π‘œπ‘š(π‘Š4, π‘Š3 βˆ—), π‘Š2 βˆ— βŠ• π‘Š1 βˆ— )), be M-vector space of the whole linear mapping from π‘Š4 to π‘Š3 βˆ— and π»π‘œπ‘š(π‘Š4, π‘Š3 βˆ—))into π‘Š2 βˆ— βŠ• π‘Š1 βˆ— as well as π‘Š5 into π»π‘œπ‘š(π»π‘œπ‘š(π‘Š4, π‘Š3 βˆ—), π‘Š2 βˆ— βŠ• π‘Š1 βˆ— ).Then the (𝑄𝑍𝐴) of Lie group is on π»π‘œπ‘šπ‘€(π‘Š5, π»π‘œπ‘š(π»π‘œπ‘š(π‘Š4, π‘Š3), π‘Š2 βˆ— βŠ• π‘Š1 βˆ— )). Proof: Define βˆ… ́ : Η€ β†’ ǀ𝐿(π»π‘œπ‘šπ‘€(π‘Š5, π»π‘œπ‘š(π»π‘œπ‘š(π‘Š4, π‘Š3 βˆ—), π‘Š2 βˆ— βŠ• π‘Š1 βˆ— ))), by proposition (3.4) such that: βˆ… ́ (𝑠)β„Ž ́ = [(π‘Ÿ1(𝑠)βˆ’1 βŠ• (π‘Ÿ2(𝑠)βˆ’1 ) ∘ β„Ž ́1 ∘ (π‘Ÿ3(𝑠)βˆ’1 ∘ β„Ž ́2 ∘ π‘Ÿ4(𝑠))] ∘ β„Ž ́3 ∘ π‘Ÿ5(𝑠). for all 𝑠 ∈ Η€ and β„Žπ‘–: π‘Šπ‘– β†’ 𝑀. Proposition (3.8): Let π‘Ÿπ‘–, 𝑖 = 1,2,3,4, π‘Žπ‘›π‘‘ 5 be five representation of Lie group affects the vector spaces π‘Šπ‘–, 𝑖 = 1,2,3,4, π‘Žπ‘›π‘‘ 5 respectively, and let π»π‘œπ‘šπ‘€(π‘Š5, π»π‘œπ‘š(π‘Š4 βˆ— , π‘Š3 βˆ—), π»π‘œπ‘š(π‘Š2, π‘Š1 βˆ— ), be M-vector space of the whole linear mapping from π‘Š2 to π‘Š1 βˆ— and π»π‘œπ‘š(π‘Š2 βˆ— , π‘Š3 βˆ—))into π»π‘œπ‘š(π‘Š2, π‘Š1 βˆ— ) as well as π‘Š5 into (π»π‘œπ‘š(π‘Š4 βˆ— βŠ• π‘Š3 βˆ—), π»π‘œπ‘š(π‘Š2, π‘Š1 βˆ—)). Then the (𝑄𝑍𝐴) of Lie group is on π»π‘œπ‘šπ‘€(π‘Š5, π»π‘œπ‘š(π‘Š4 βˆ— , π‘Š3 βˆ—)π»π‘œπ‘š(π‘Š2, π‘Š1 βˆ—)).
  • 9. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 92 Proof: Define πœ‘Μ : Η€ β†’ ǀ𝐿(π»π‘œπ‘šπ‘€(π‘Š5, (π‘Š4 βˆ— βŠ• π‘Š3 βˆ—)π»π‘œπ‘š(π‘Š2, π‘Š1 βˆ—))),by proposition (3.4) such that: πœ‘Μ (𝑠)β„Ž ́ = [(π‘Ÿ1(𝑠)βˆ’1 ∘ β„Ž ́1 ∘ π‘Ÿ2(𝑠)) ∘ β„Ž ́2 ∘ (π‘Ÿ3(𝑠)βˆ’1 βˆ˜βŠ• π‘Ÿ4(𝑠)βˆ’1 )] ∘ β„Ž ́3 ∘ π‘Ÿ5(𝑠). for all 𝑠 ∈ Η€ and β„Žπ‘–: π‘Šπ‘– β†’ 𝑀. Proposition (3.9): Let π‘Ÿπ‘–, 𝑖 = 1,2,3,4,5 be five representation of Lie group affects the vector spaces π‘Šπ‘–, 𝑖 = 1,2,3,4,5 respectively, and let π»π‘œπ‘šπ‘€(π»π‘œπ‘š(π‘Š5, π‘Š4 βˆ—), π»π‘œπ‘š(π‘Š3, π‘Š2 βŠ• π‘Š1 βˆ—)) be M-vector space of all linear mapping from π‘Š5 to π‘Š4 βˆ— and from π‘Š3 into (π‘Š2 βŠ• π‘Š1 βˆ—)) as well as from π»π‘œπ‘š(π‘Š5, π‘Š4 βˆ—) into π»π‘œπ‘š(π‘Š3, π‘Š2 βŠ• π‘Š1 βˆ—) Then the (𝑄𝑍𝐴) of Lie group on π»π‘œπ‘šπ‘€(π»π‘œπ‘š(π‘Š5, π‘Š4 βˆ—), π»π‘œπ‘š(π‘Š3, π‘Š2 βŠ• π‘Š1 βˆ—). Proof: Define πœ‘Μ : Η€ β†’ ǀ𝐿(π»π‘œπ‘šπ‘€(π»π‘œπ‘š(π‘Š5, π‘Š4 βˆ—), π»π‘œπ‘š(π‘Š3, π‘Š2 βŠ• π‘Š1 βˆ—))),by proposition (3.4) such that: πœ‘Μ (𝑠)β„Ž ́ = [(π‘Ÿ1(𝑠)βˆ’1 βŠ• π‘Ÿ2(𝑠)) ∘ β„Ž ́1 ∘ π‘Ÿ3(𝑠))] ∘ β„Ž ́2 ∘ π‘Ÿ4(𝑠)βˆ’1 ∘ β„Ž ́3 ∘ π‘Ÿ5(𝑠). for all 𝑠 ∈ Η€ and β„Žπ‘–: π‘Šπ‘– β†’ 𝑀.
  • 10. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 93 4. The dual action of representation for Lie Group Proposition (4.1): [6] Let π‘Š1and π‘Š2 are finite dimensional vector space, and (𝑛) be a natural no, then the action of Lie group action on tensor product is : 1. π‘Š 2 βˆ—βˆ—,..βˆ— ⏟ 𝑛 βŠ— π‘Š1 β‰… π‘Š2 βŠ— π‘Š1, if 𝑛 is an even number. 2. π‘Š 2 βˆ—βˆ—,..βˆ— ⏟ 𝑛 βŠ— π‘Š1 β‰… π‘Š2 βˆ— βŠ— π‘Š1, if 𝑛 is an odd number. 3. π‘Š2 βŠ— π‘Š 2 βˆ—βˆ—,..βˆ— ⏟ 𝑛 β‰… π‘Š2 βŠ— π‘Š1, if 𝑛 is an even number. 4. π‘Š2 βŠ— π‘Š 2 βˆ—βˆ—,..βˆ— ⏟ 𝑛 β‰… π‘Š2 βŠ— π‘Š1 βˆ— , if 𝑛 is an odd number. Corollary (4.2): If (π‘Š)1and (π‘Š2) are finite- dimensional vector spaces, and (𝑛) be a natural number, then the Lie group affects the tensor product and π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1): 1. π‘Š2 βŠ— π‘Š1 βˆ— β‰… π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1 βˆ—) 2. π‘Š2 βŠ• π‘Š3 βŠ— π‘Š1 βˆ— β‰… π»π‘œπ‘šπ‘€(π‘Š1, π‘Š2 βŠ• π‘Š3) } if 𝑛 is an odd number. Proposition (4.3): Let π‘Ÿ1: Η€ β†’ ǀ𝐿(π‘Šπ‘–), 𝑅𝑖 βˆ— : Η€ β†’ ǀ𝐿(π‘Šπ‘– βˆ— ) for 𝑖 = 1,2, and the Lie group Η€ affects the πœ‘(𝑠)β„Ž = π‘Ÿ1(𝑠) ∘ β„Ž ∘ π‘Ÿ2(𝑠), for every 𝑠 ∈ Η€, β„Ž ∈ π»π‘œπ‘š(π‘Š2, π‘Š1). Then the Lie group Η€ affects (π»π‘œπ‘š(π‘Š2, π‘Š1))βˆ— being also given provided a representation πœ‘βˆ— , where: πœ‘βˆ—(𝑠) = π‘Ÿ(𝑠)βˆ’1 ∘ β„Žβˆ— ∘ π‘Ÿ1(𝑠), for all 𝑠 ∈ Η€ and β„Žβˆ— ∈ (π»π‘œπ‘š(π‘Š2, π‘Š1))βˆ— . Proof: Let action of Lie group Η€ affects π»π‘œπ‘š(π‘Š2, π‘Š1) being induced via the representation
  • 11. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 94 πœ‘: Η€ β†’ ǀ𝐿(π»π‘œπ‘š(π‘Š2, π‘Š1)), where πœ‘(𝑠) = π‘Ÿ1(𝑠) ∘ β„Ž ∘ π‘Ÿ2(𝑠)βˆ’1 , for 𝑠 ∈ Η€, and β„Ž ∈ π»π‘œπ‘š(π‘Š2, π‘Š1). Such that πœ‘βˆ—(𝑠) = π‘Ÿ(𝑠)βˆ’1 ∘ β„Žβˆ— ∘ π‘Ÿ1(𝑠) representation for all 𝑠 ∈ Η€ and β„Žβˆ— ∈ (π»π‘œπ‘š(π‘Š2, π‘Š1))βˆ— . Since: βˆ…βˆ—(𝑠) = (π‘Ÿ1(𝑠) ∘ β„Ž ∘ π‘Ÿ2(𝑠)βˆ’1 , )βˆ— = π‘Ÿ2 βˆ—(𝑠)βˆ’1 ∘ β„Žβˆ— ∘ π‘Ÿ1 βˆ—(𝑠) for all (𝑠 ∈ Η€) and β„Žβˆ— : π‘Š1 βˆ— β†’ π‘Š2 βˆ— , we have πœ‘βˆ—(𝑠𝑑) = (π‘Ÿ(𝑠𝑑))βˆ— = (π‘Ÿ(𝑑) ∘ β„Ž ∘ π‘Ÿ(𝑠))βˆ— = π‘Ÿβˆ— (𝑠) ∘ β„Žβˆ— ∘ π‘Ÿβˆ— (𝑑) Thus, πœ‘βˆ— is a representation from Η€(πœ‘βˆ— . 𝑠 π‘”π‘Ÿπ‘œπ‘’π‘ β„Žπ‘œπ‘šπ‘œπ‘šπ‘œπ‘Ÿπ‘β„Žπ‘–π‘ π‘š π‘œπ‘“ Η€) Corollary (4.4): Let π‘Ÿ1: Η€ β†’ ǀ𝐿(𝑗, 𝑀), ǀ𝐿(π‘Š1) β‰… ǀ𝐿(𝑗, 𝑀) And π‘Ÿ2: Η€ β†’ ǀ𝐿(β„“, 𝑀), ǀ𝐿(π‘Š2) β‰… ǀ𝐿(β„“, 𝑀). Where π‘Ÿ1 and π‘Ÿ2 have matrix representation, if Lie group Η€ affects π»π‘œπ‘š(π‘Š2, π‘Š1) being a representation πœ‘: Η€ β†’ ǀ𝐿(π»π‘œπ‘š(π‘Š2, π‘Š1)), where πœ‘(𝑠) = π‘Ÿ1(𝑠) ∘ β„Ž ∘ π‘Ÿ2(𝑠)βˆ’1 , for 𝑠 ∈ Η€. Then Lie group Η€ affects π»π‘œπ‘š(π‘Š2, π‘Š1) is a representation πœ‘βˆ— : Η€ β†’ ǀ𝐿(π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1))βˆ— . Where: πœ‘βˆ—(𝑠) = (π‘Ÿ2 βˆ—(𝑠)) π‘‘π‘Ÿ ∘ β„Žβˆ— ∘ (π‘Ÿ1(𝑠)βˆ’1)π‘‘π‘Ÿ , for all 𝑠 ∈ Η€.
  • 12. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 95 Proof: Since (𝑠) = π‘Ÿ1(𝑠) ∘ β„Ž ∘ π‘Ÿ2(𝑠)βˆ’1 , πœ‘βˆ—(𝑠) = π‘Ÿ(𝑠)βˆ’1 ∘ β„Žβˆ— ∘ π‘Ÿ1 βˆ—(𝑠) = (π‘Ÿ2 βˆ—(𝑠)) π‘‘π‘Ÿ ∘ β„Žβˆ— ∘ (π‘Ÿ1(𝑠)βˆ’1)π‘‘π‘Ÿ And πœ‘βˆ— is a representation a matrix representation of dimension 𝑗ℓ, then π‘Ÿβˆ—(𝑠𝑑) = (π‘Ÿ(𝑠𝑑)βˆ’1)π‘‘π‘Ÿ = (π‘Ÿ(𝑑)βˆ’1)π‘‘π‘Ÿ ∘ (π‘Ÿ(𝑠)βˆ’1)π‘‘π‘Ÿ = π‘Ÿβˆ—(𝑑) ∘ π‘Ÿβˆ—(𝑠) Example (4.5): Let π‘Ÿ1 = 𝑆1 β†’ π‘†π‘œ(2) βŠ‚ ǀ𝐿(2, β‚΅) and π‘Ÿ2 = 𝑆1 β†’ π‘†π‘œ(3) βŠ‚ ǀ𝐿(3, β‚΅) Where Η€ = 𝑆1(𝑗 = 2, β„“ = 3) and (π‘Š1) is the β‚΅ βˆ’ vector space of dimensional 2, and ( π‘Š2) being the β‚΅ βˆ’ vector space of dimensional 3, then by collar we have πœ‘βˆ— (π‘’π‘–βˆ ) = π‘Ÿ2 βˆ— (π‘’π‘–βˆ ) ∘ β„Žβˆ— ∘ π‘Ÿ1 βˆ— (π‘’π‘–βˆ ) = (π‘Ÿ2 βˆ— (π‘’π‘–βˆ )) π‘‘π‘Ÿ ∘ β„Žβˆ— ∘ (π‘Ÿ1(π‘’π‘–βˆ )) π‘‘π‘Ÿ = ( 1 0 0 0 cos ∝ βˆ’ sin ∝ 0 sin ∝ cos ∝ ) π‘‘π‘Ÿ ∘ β„Žβˆ— ∘ ( cos ∝ sin ∝ βˆ’sin ∝ cos ∝ ) π‘‘π‘Ÿ = ( 1 0 0 0 cos ∝ βˆ’ sin ∝ 0 sin ∝ cos ∝ ) π‘‘π‘Ÿ ∘ β„Žβˆ— ∘ ( cos ∝ βˆ’ sin ∝ sin ∝ cos ∝ ) π‘‘π‘Ÿ Let 𝐴 = ( 1 0 0 0 cos ∝ βˆ’ sin ∝ 0 sin ∝ cos ∝ ) Thus πœ‘βˆ— (π‘’π‘–βˆ ) = ( (cos ∝)𝐴 (βˆ’ sin ∝)𝐴 (sin ∝)𝐴 (cos ∝)𝐴 ) 6Γ—6
  • 13. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 96 = ( cos ∝ 0 0 0 cos ∝2 cos ∝ sin ∝ 0 βˆ’ cos ∝ sin ∝ cos ∝2 βˆ’ sin ∝ 0 0 0 βˆ’ sin ∝ cos ∝ βˆ’ sin ∝2 0 sin ∝2 sin ∝ cos ∝ sin ∝ 0 0 0 sin ∝ cos ∝ sin ∝2 0 βˆ’ sin ∝2 sin ∝ cos ∝ cos ∝2 0 0 0 cos ∝2 cos ∝ sin ∝ 0 βˆ’ cos ∝ sin ∝ cos ∝2 ) Is the representation of ǀ𝐿(6, β‚΅) acting on π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1) of dimensional 6. Proposition (4.6): Let (π‘Š1)and (π‘Š2) be two n- dimensional vector spaces and 𝑛 is a natural number, then π»π‘œπ‘šπ‘€(π»π‘œπ‘š(π‘Š3, π‘Š2, π‘Š1)) βˆ—βˆ—β€¦βˆ— ⏟ 𝑛 = { π»π‘œπ‘šπ‘€(π»π‘œπ‘š(π‘Š3, π‘Š2, π‘Š1))𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛 π»π‘œπ‘šπ‘€(π‘Š1 βˆ— , π»π‘œπ‘š(π‘Š2 βˆ— , π‘Š2)𝑖𝑓 𝑛 𝑖𝑠 π‘œπ‘‘π‘‘ … Proof: We can prove this proposition by mathematical induction. If 𝑛 = 1, then (π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1))βˆ— = (π»π‘œπ‘šπ‘€(π‘Š1 βˆ— , π‘Š2 βˆ—). And hence the action of group Η€ on π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1) is: πœ‘βˆ—(𝑠) = π‘Ÿ1 βˆ—(𝑠) ∘ β„Žβˆ— ∘ π‘Ÿ2 βˆ—(𝑠)βˆ’1 (see proposition (3.2). Suppose that (3.2) is true when 𝑛 = π‘˜ it mean (π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1)) βˆ—βˆ—β€¦βˆ— ⏟ π‘˜ = { (π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1)), 𝑖𝑓 π‘˜ 𝑖𝑠 𝑒𝑣𝑒𝑛 … (1) (π»π‘œπ‘šπ‘€(π‘Š1 βˆ— , π‘Š2), 𝑖𝑓 π‘˜ 𝑖𝑠 π‘œπ‘‘π‘‘ … (2) If π‘˜ is even then the acting of Lie group Η€ on π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1) is: πœ‘(𝑠) = π‘Ÿ2(𝑠)βˆ’1 ∘ β„Ž ∘ π‘Ÿ1(𝑠). And if π‘˜ is odd then the action of Lie group Η€ on π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1) is: πœ‘βˆ—(𝑠) = π‘Ÿ1 βˆ—(𝑠)βˆ’1 ∘ β„Žβˆ— ∘ π‘Ÿ2 βˆ—(𝑠) We will prove that (3.2) it is true, when 𝑛 = π‘˜ + 1 ,so it will be proven π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1) βˆ—βˆ—β€¦βˆ— ⏟ π‘˜+1 = { π»π‘œπ‘šπΊ(π‘Š2, π‘Š1))𝑖𝑓 π‘˜ 𝑖𝑠 π‘œπ‘‘π‘‘ π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1))𝑖𝑓 π‘˜ 𝑖𝑠 𝑖𝑠 𝑒𝑣𝑒𝑛 (πœ‘(𝑠)) βˆ—βˆ—β€¦βˆ— ⏟ π‘˜+1 = (π‘Ÿ1 βˆ—(𝑠) ∘ β„Žβˆ— π‘Ÿ2(𝑠)) βˆ—βˆ—β€¦βˆ— ⏟ π‘˜+1 (π‘Ÿ1 βˆ—(𝑠) ∘ β„Žβˆ— ∘ π‘Ÿ2 βˆ—(𝑠))
  • 14. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 97 When π‘˜ is odd we have π‘˜ + 1 is even, then π»π‘œπ‘šπ‘€(π»π‘œπ‘š(π‘Š3, π‘Š2), π‘Š1)) βˆ—βˆ—β€¦βˆ— ⏟ π‘˜+1 = π»π‘œπ‘šπ‘€(π»π‘œπ‘š(π‘Š3, π‘Š2), π‘Š1)) by(1), Thus the Lie group Η€ affects π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1) βˆ—βˆ—β€¦βˆ— ⏟ π‘˜+1 is: (πœ‘(𝑠)) βˆ—βˆ—β€¦βˆ— ⏟ π‘˜+1 = [(π‘Ÿ1(𝑠)βˆ’1 ∘ β„Ž1 ∘ (π‘Ÿ2(𝑠) ∘ β„Ž2 ∘ π‘Ÿ3 βˆ’1(𝑠))] And when π‘˜ is even we have π‘˜ + 1 is odd. Then π»π‘œπ‘šπ‘€(π»π‘œπ‘š(π‘Š3, π‘Š2), π‘Š1)) βˆ—βˆ—β€¦βˆ— ⏟ π‘˜+1 = π»π‘œπ‘šπ‘€(π‘Š1 βˆ— , π»π‘œπ‘š(π‘Š2 βˆ— , π‘Š3)) by (2), Thus the affects of Lie group Η€ on π»π‘œπ‘šπ‘€(π‘Š2, π‘Š1) βˆ—βˆ—β€¦βˆ— ⏟ π‘˜+1 is: (πœ‘(𝑠)) βˆ—βˆ—β€¦βˆ— ⏟ π‘˜+1 = [(π‘Ÿ3(𝑠) ∘ β„Žβˆ— 1 ∘ π‘Ÿ2(𝑠)βˆ’1 ) ∘ β„Žβˆ— 2 ∘ π‘Ÿ1 βˆ’1(𝑠)βˆ’1] Then the proposition is true for all ∈ π‘βˆ— . 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.