aa r X i v : . [ m a t h . R T ] A p r A Structure Theorem for Leibniz Homology
Jerry M. Lodder
Mathematical Sciences, Dept. 3MBBox 30001New Mexico State UniversityLas Cruces NM, 88003, U.S.A. e-mail: [email protected]
Abstract.
Presented is a structure theorem for the Leibniz homology, HL ∗ ,of an Abelian extension of a simple real Lie algebra g . As applications, resultsare stated for affine extensions of the classical Lie algebras sl n ( R ), so n ( R ),and sp n ( R ). Furthermore, HL ∗ ( h ) is calculated when h is the Lie algebraof the Poincar´e group as well as the Lie algebra of the affine Lorentz group.The general theorem identifies all of these in terms of g -invariants. Mathematics Subject Classifications (2000):
Key Words:
Leibniz Homology, Extensions of Lie Algebras, Invariant The-ory.
For a (semi-)simple Lie algebra g over R , the Milnor-Moore theorem identifiesthe Lie algebra homology, H Lie ∗ ( g ; R ), as a graded exterior algebra on theprimitive elements of H Lie ∗ ( g ; R ), i.e., H Lie ∗ ( g ) ≃ Λ ∗ (Prim(H ∗ ( g ))) , where the coefficients are understood to be in the field R . The algebrastructure on H Lie ∗ ( g ) can be deduced from the exterior product of g -invariantcycles on the chain level, and agrees with the corresponding Pontrajagin1roduct induced from the Lie group [4]. For Leibniz homology, however, wehave HL n ( g ) = 0, n ≥
1, for g simple with R coefficients [8]. Now, let g bea simple real Lie algebra, h an extension of g by an Abelian ideal I :0 −−−→ I −−−→ h −−−→ g −−−→ g acts on I , and this extends to a g action on Λ ∗ ( I ) by derivations. Forany g -module M , let M g = { m ∈ M | [ g, m ] = 0 ∀ g ∈ g } denote the submodule of g -invariants. Under a mild hypothesis, we provethat HL ∗ ( h ) ≃ [Λ ∗ ( I )] g ⊗ T ( K ∗ ) , where K ∗ = Ker (cid:0) H Lie ∗ ( I ; h ) g → H Lie ∗ +1 ( h ) (cid:1) , and T ( K ∗ ) = P n ≥ K ⊗ n ∗ is the tensor algebra over R . Above, H Lie ∗ ( I ; h )denotes the Lie algebra homology of I with coefficients in h , and the map H Lie ∗ ( I ; h ) → H Lie ∗ +1 ( h ) is the composition H Lie ∗ ( I ; h ) j ∗ −−−→ H Lie ∗ ( h ; h ) π ∗ −−−→ H Lie ∗ +1 ( h ) , where j ∗ is induced by the inclusion of Lie algebras j : I ֒ → h , and π ∗ isinduced by the projection of chain complexes π : h ⊗ h ∧ n → h ∧ ( n +1) π ( h ⊗ h ∧ h ∧ . . . ∧ h n ) = h ∧ h ∧ h ∧ . . . ∧ h n . Of course, K ∗ is computed from the module of g -invariants, beginning with H Lie ∗ ( I ; h ) g as indicated above.The main theorem is easily applied when g is a classical Lie algebra and h is an affine extension of g . In the final section we state the results for g being sl n ( R ), so n ( R ), n odd or even, and I = R n , whereby g acts on I via matrixmultiplication on vectors, which is often called the standard representation.For the (special) orthogonal Lie algebra, so n ( R ), the general theorem agreeswith calculations of Biyogmam [1]. When g = sp n ( R ) and I = R n , werecover the author’s previous result [7]. Additionally, HL ∗ ( h ) is computedwhen h is the Lie algebra of the Poincar´e group and the Lie algebra of theaffine Lorentz group. 2 Preliminaries on Lie Algebra Homology
For any Lie algebra g over a ring k , the Lie algebra homology of g , written H Lie ∗ ( g ; k ), is the homology of the chain complex Λ ∗ ( g ), namely k ←−−− g [ , ] ←−−− g ∧ ←−−− . . . ←−−− g ∧ ( n − d ←−−− g ∧ n ←−−− . . . , where d ( g ∧ g ∧ . . . ∧ g n ) = X ≤ i Let g be a simple Lie algebra over R , and let −−−→ I j −−−→ h ρ −−−→ g −−−→ , be an Abelian extension of g . There are natural vector space isomorphisms H Lie ∗ ( h ) ≃ [Λ ∗ ( I )] g ⊗ H Lie ∗ ( g ) (2.1) H Lie ∗ ( h ; h ) ≃ [ H Lie ∗ ( I ; h )] g ⊗ H Lie ∗ ( g ) . (2.2) Proof. Apply the homological version of the Hochschild-Serre spectral se-quence to the subalgebra g of h [7]. Then H Lie ∗ ( h ) ≃ H Lie ∗ ( I ) g ⊗ H Lie ∗ ( g ) . Since I is Abelian, H Lie ∗ ( I ) g = [Λ ∗ ( I )] g , and isomorphism (2.1) follows. Thespectral sequence yields isomorphism (2.2) directly. Note that, since I actstrivially on H Lie ∗ ( I ; h ), we have H Lie ∗ ( I ; h ) g = H Lie ∗ ( I ; h ) h as well, yielding H Lie ∗ ( h ; h ) ≃ [ H Lie ∗ ( I ; h )] h ⊗ H Lie ∗ ( g ) . Compare with Hochschild and Serre [3].4he natural inclusion g ֒ → h of Lie algebras leads to a map of long exactsequences δ Lie −−−→ HR n − ( g ) −−−→ H Lie n − ( g ; g ) −−−→ H Lie n ( g ) δ Lie −−−→ y y y δ Lie −−−→ HR n − ( h ) −−−→ H Lie n − ( h ; h ) −−−→ H Lie n ( h ) δ Lie −−−→ , where δ Lie is the connecting homomorphism. For g simple, H Lie n − ( g ; g ) = 0, n ≥ δ Lie : H Lie n ( g ) → HR n − ( g )is an isomorphism for n ≥ 3. Note that H Lie1 ( g ) ≃ H Lie2 ( g ) ≃ 0. Theinclusion j : I ֒ → h is g -equivariant and induces an endormorphism H Lie ∗ ( I ; h ) g j ∗ −→ H Lie ∗ ( h ; h ) g = H Lie ∗ ( h ; h ) . Recall that every element of H Lie ∗ ( h ; h ) can in fact be represented by a g -invariant cycle at the chain level. Additionally, all elements of H Lie ∗ ( I ; h ) g can be be represented by g -invariant cycles, although in general H Lie ∗ ( I ; h ) g is not isomorphic to H Lie ∗ ( I ; h ). Let K ∗ be the kernel of the composition π ∗ ◦ j ∗ : H Lie n ( I ; h ) g j ∗ −→ H Lie n ( h ; h ) π ∗ −→ H Lie n +1 ( h ) ,K n = Ker[ H Lie n ( I ; h ) g −→ H Lie n +1 ( h )] , n ≥ . Theorem 2.2. With I , h and g as in Lemma (2.1) , we have HR n ( h ) ≃ δ Lie [ H Lie n +3 ( g )] ⊕ n +1 X i =0 K n +1 − i ⊗ H Lie i ( g ) , n ≥ . Proof. The proof follows from the long exact sequence relating HR ∗ ( h ), H Lie ∗ +1 ( h ; h ) and H Lie ∗ +2 ( h ) together with a specific knowledge of the genera-tors of the latter two homology groups gleaned from Lemma (2.1).Note that H Lie n ( I ; h ) g contains Λ n +1 ( I ) g as a direct summand, induced bya g -equivariant chain map ζ : Λ n +1 ( I ) → h ⊗ I ∧ n ζ ( a ∧ a ∧ . . . ∧ a n ) = 1 n + 1 n X i =0 ( − i a i ⊗ a ∧ a ∧ . . . ˆ a i . . . ∧ a n , a i ∈ I . Then ζ ∗ : H Lie n +1 ( I ) g = Λ n +1 ( I ) g → H Lie n ( I ; h ) g is an inclusion, since the composition π ◦ ζ : Λ n +1 ( I ) → h ⊗ I ∧ n → h ∧ ( n +1) is the identity on Λ n +1 ( I ). Let ¯Λ ∗ ( I ) g = P k ≥ Λ k ( I ) g . Thus, the morphism π ∗ : H Lie ∗ ( h ; h ) → H Lie ∗ +1 ( h )induces a surjection H Lie ∗ ( h ; h ) → ¯Λ ∗ ( I ) g ⊗ H Lie ∗ ( g )with kernel X n ≥ n +1 X i =0 K n +1 − i ⊗ H Lie i ( g ) . There is also an inclusion i ∗ : H Lie ∗ ( g ) → H Lie ∗ ( h ), and it follows that H Lie n +3 ( g ) maps isomorphically to i ∗ ◦ δ Lie [ H Lie n +3 ( g )] in the commutative square −−−→ H Lie n +3 ( g ) δ Lie −−−→ HR n ( g ) −−−→ i ∗ y y i ∗ −−−→ H Lie n +3 ( h ) δ Lie −−−→ HR n ( h ) −−−→ Recall that H Lie ∗ ( g ; g ) = 0, ∗ ≥ 0, for g simple. Returning to the general setting of any Lie algebra g over a ring k , we recallthat the Leibniz homology [5] of g , written HL ∗ ( g ; k ), is the homology ofthe chain complex T ( g ): k ←−−− g [ , ] ←−−− g ⊗ ←−−− . . . ←−−− g ⊗ ( n − d ←−−− g ⊗ n ←−−− . . . , d ( g , g , . . . , g n ) = X ≤ i 0, is a map of chaincomplexes, T ( g ) → Λ ∗ ( g ), and induces a k -linear map on homology HL ∗ ( g ; k ) → H Lie ∗ ( g ; k ) . Letting (Ker π ′ ) n [2] = Ker [ g ⊗ ( n +2) → g ∧ ( n +2) ] , n ≥ , Pirashvili [9] defines the relative theory H rel ( g ) as the homology of the com-plex C rel n ( g ) = (Ker π ′ ) n [2] , and studies the resulting long exact sequence relating Lie and Leibniz ho-mology: · · · δ −−−→ H rel n − ( g ) −−−→ HL n ( g ) −−−→ H Lie n ( g ) δ −−−→ H rel n − ( g ) −−−→· · · δ −−−→ H rel0 ( g ) −−−→ HL ( g ) −−−→ H Lie2 ( g ) −−−→ −−−→ HL ( g ) −−−→ H Lie1 ( g ) −−−→ −−−→ HL ( g ) −−−→ H Lie0 ( g ) −−−→ . The projection π ′ : g ⊗ ( n +1) → g ∧ ( n +1) can be factored as the compositionof projections g ⊗ ( n +1) −→ g ⊗ g ∧ n −→ g ∧ ( n +1) , which leads to a natural map between exact sequences H rel n − ( g ) −−−→ HL n +1 ( g ) −−−→ H Lie n +1 ( g ) δ −−−→ H rel n − ( g ) y y y y HR n − ( g ) −−−→ H Lie n ( g ; g ) −−−→ H Lie n +1 ( g ) δ Lie −−−→ HR n − ( g )7 key technique in the calculation of Leibniz homology is the Pirashvilispectral sequence [9], which converges to the relative groups H rel ∗ . Considerthe filtration of C rel n ( g ) = Ker( g ⊗ ( n +2) → g ∧ ( n +2) ) , n ≥ , given by F km ( g ) = g ⊗ k ⊗ Ker( g ⊗ ( m +2) → g ∧ ( m +2) ) , m ≥ , k ≥ . Then F ∗ m − is a subcomplex of F ∗ m , and E m, k = F km / F k +1 m − ≃ g k ⊗ Ker( g ⊗ g ∧ ( m +1) → g ∧ ( m +2) )= g k ⊗ CR m ( g ) . From [9], we have E m, k ≃ HL k ( g ) ⊗ HR m ( g ) , m ≥ , k ≥ . Lemma 3.1. Let → I → h → g → be an Abelian extension of a simplereal Lie algebra g . Then there is a natural injection ǫ ∗ : H Lie ∗ ( I ; h ) g → HL ∗ +1 ( h ) induced by a g -equivariant chain map ǫ n : h ⊗ Λ n ( I ) → h ⊗ ( n +1) , n ≥ . Proof. For b ∈ h , a i ∈ I , i = 1 , , . . . n , define ǫ n ( b ⊗ a ∧ a ∧ . . . ∧ a n ) = 1 n ! X σ ∈ S n sgn( σ ) b ⊗ a σ (1) ⊗ a σ (2) ⊗ . . . ⊗ a σ ( n ) . Since [ a i , a j ] = 0 for a i , a j ∈ I , it follows that d Lieb ◦ ǫ n = ǫ n − ◦ d Lie . Also, ǫ n is g -equivariant, since g acts by derivations on both h ⊗ Λ n ( I ) and h ⊗ ( n +1) . Thus, there is an induced map ǫ ∗ : H Lie ∗ ( I ; h ) g → HL ∗ +1 ( h ) g = HL ∗ +1 ( h ) . π ′ ◦ ǫ n : h ⊗ Λ n ( I ) → h ⊗ ( n +1) → h ⊗ h ∧ n is the identity on h ⊗ Λ n ( I ). Since H Lie ∗ ( h ; h ) contains H Lie ∗ ( I ; h ) g as a directsummand via H Lie ∗ ( I ; h ) g ⊗ H Lie0 ( g ) (see Lemma (2.1)), it follows that ( π ′ ◦ ǫ ) ∗ and ǫ ∗ are injective. Lemma 3.2. With I , h , g as in Lemma (3.1) , there is a vector space splitting(that is a splitting of trivial g -modules) H Lie n ( I ; h ) g ≃ [Λ n +1 ( I )] g ⊕ K n , where K n = Ker[ H Lie n ( I ; h ) g → H Lie n +1 ( h )] . Proof. The proof begins with the g -equivariant chain map ζ : Λ n +1 ( I ) → h ⊗ I ∧ n constructed in Theorem (2.2). Recall that K n is defined as the kernel of thecomposition H Lie n ( I ; h ) g → H Lie n ( h ; h ) → H Lie n +1 ( h ) . Note that H Lie n ( h ; h ) contains H Lie n ( I ; h ) g as a summand from Lemma (2.1).To begin the calculation of the differentials in the Pirashvili spectral se-quence converging to H rel ( h ), first consider the spectral sequence convergingto H rel ( g ), where g is simple. We have H Lie n ( g ; g ) = 0 for n ≥ HL n ( g ) = 0 for n ≥ δ Lie : H Lie n +3 ( g ) → HR n ( g ) and δ : H Lie n +3 ( g ) → H rel n ( g ), n ≥ 0, are isomorphisms in the square −−−→ H Lie n +3 ( g ) δ −−−→ H rel n ( g ) −−−→ y y −−−→ H Lie n +3 ( g ) δ Lie −−−→ HR n ( g ) −−−→ Lemma 3.3. In the Pirashvili spectral sequence converging to H rel ∗ ( g ) for asimple real Lie algebra g , all higher differentials d r : E rm, k → E rm − r, k + r − , r ≥ , are zero. roof. Since E m, k ≃ HL k ( g ) ⊗ HR m ( g ), and HL k ( g ) = 0 for k ≥ 1, it isenough to consider E m, ≃ R ⊗ HR m ( g ) ≃ δ Lie [ H Lie m +3 ( g )] , for which d r [ E rm, ] = 0, r ≥ F ∗ m ( g ) ֒ → F ∗ m ( h ) . Thus, in the spectral sequence converging to H rel ∗ ( h ), we also have d r [ δ Lie [ H Lie m +3 ( g )] = 0 , r ≥ . Due to the recursive nature of H rel ∗ ( h ) with E m, k ≃ HL k ( h ) ⊗ HR m ( h ),we calculate HL n ( h ) by induction on n . We have immediately HL ( h ) ≃ R HL ( h ) ≃ H Lie1 ( h ) ≃ I g ≃ H ( I ; h ) g HL ( h ) ≃ H Lie1 ( h ; h ) ≃ H Lie1 ( I ; h ) g . Elements in HL ( h ) ⊗ HR ( h ) + HL ( h ) ⊗ HR ( h )determine H rel1 ( h ), which then maps to HL ( h ), etc. We now construct thedifferentials in the Pirashvili spectral sequence. Lemma 3.4. In the Pirashvili spectral sequence converging to H rel ∗ ( h ) , thedifferential d r [ E m, ] = d r [ HL ( h ) ⊗ HR m ( h )] = d r [ R ⊗ HR m ( h )] is given by d r [ δ Lie ( H Lie m +3 ( g ))] = 0 , r ≥ ,d r [ K m +1 ] = 0 , r ≥ ,d m +3 − p : K m +1 − p ⊗ H Lie p ( g ) → K m +1 − p ⊗ δ Lie [ H Lie p ( g )] , where K m +1 − p ⊗ δ Lie [ H Lie p ( g )] ⊆ HL m +2 − p ( h ) ⊗ HR p − ( h ) ⊆ E p − , m +2 − p ,d m +3 − p ( ω ⊗ ω ) = ω ⊗ δ ( ω ) . roof. Since elements in K m +1 are represented by cycles in H Lie m +1 ( h ; I ) g thatmap via ǫ ∗ to cycles in HL m +2 ( h ), it follows d r [ K m +1 ] = 0, r ≥ ω ] ∈ H Lie p ( g ). Consider ω = X i ,i ,...,i p g i ∧ g i ∧ . . . ∧ g i p ∈ g ∧ p . Now, using the homological algebra of the long exact sequence relating Leib-niz and Lie-algebra homology,˜ ω = X i ,i ,...,i p g i ⊗ g i ⊗ . . . ⊗ g i p ∈ g ⊗ p is a chain with π ′ (˜ ω ) = ω and d (˜ ω ) = 0 in the Leibniz complex. Moreover, d (˜ ω ) ∈ Ker (cid:0) g ⊗ ( p − → g ∧ ( p − (cid:1) and d (˜ ω ) represents the class δ ( ω ) in H rel p − ( g ) ≃ HR p − ( g ). Since w ∈ K m +1 − p is g -invariant, [ ω , g ] = 0 ∀ g ∈ g . Also, ǫ ( ω ) is a g -invariant cyclerepresenting ω in T ( g ). Thus, d m +3 − p ( ω ⊗ ω ) = ω ⊗ d (˜ ω ) = ω ⊗ δ ( ω ) . By a similar argument to that in Lemma (3.4), we have that d r [Λ ∗ ( I ) g ⊗ δ Lie ( H Lie ∗ +3 ( g ))] = 0 , r ≥ , where Λ k ( I ) g ⊗ δ Lie ( H Lie m +3 ( g )) ⊆ HL k ( h ) ⊗ HR m ( h ) ⊆ E m, k . Thus, the elements in M = Λ ∗ ( I ) g ⊗ δ Lie ( H Lie ∗ +3 ( g )) represent absolute cyclesin H rel ∗ ( h ). We claim that M ⊆ Ker( H rel ∗ ( h ) → HL ∗ +2 ( h )) , which follows from: Lemma 3.5. The boundary map δ : H Lie ∗ +3 ( h ) → H rel ∗ ( h ) satisfies δ [Λ k ( I ) g ⊗ H Lie m +3 ( g )] = Λ k ( I ) g ⊗ δ Lie [ H Lie m +3 ( g )] , for k ≥ , m ≥ . roof. The proof follows from the long exact sequence relating Lie and Leib-niz homology by choosing representatives for the Lie classes Λ k ( I ) g ⊗ H Lie m +3 ( g )at the chain level. Also, note that for g simple, H Lie1 ( g ) = 0 and H Lie2 ( g ) =0. Now let ω ⊗ ( ω ⊗ ω ) ∈ Λ k ( I ) g ⊗ ( K m +1 − p ⊗ H Lie p ( g )) ⊆ E m, k . Then, using the g -invariance of elements in Λ ∗ ( I ) g and K ∗ , as well as the h -invariance of Λ ∗ ( I ) g , we have d m +3 − p ( ω ⊗ ω ⊗ ω ) = ( ω ⊗ ω ) ⊗ δ ( ω ) ∈ (Λ k ( I ) g ⊗ K m +1 − p ) ⊗ δ Lie [ H Lie p ( g )] ⊆ HL k + m +2 − p ( h ) ⊗ HR p − ( g ) ⊆ E p − , k + m +2 − p . For the remainder of this section, we suppose: Hypothesis A : Every element of K n has an h -invariant representative in HL n +1 ( h ) at the chain level. Since I acts trivially on H Lie ∗ ( I ; h ), we have H Lie ∗ ( I ; h ) g = H Lie ∗ ( I ; h ) h ,and Hypothesis A is reasonable. The end of this section offers a canonicalconstruction of h -invariants. Theorem 3.6. Let → I → h → g → be an Abelian extension of a simplereal Lie algebra g . Then, under Hypothesis A, we have HL ∗ ( h ) ≃ Λ ∗ ( I ) g ⊗ T ( K ∗ ) , where T ( K ∗ ) = P n ≥ K ⊗ n ∗ denotes the tensor algebra, and K n = Ker[ H Lie n ( I ; h ) g → H Lie n +1 ( h )] . Proof. It follows by induction on ℓ that for certain rd r [ K ⊗ ℓ ∗ ⊗ H Lie ∗ ( g )] → K ⊗ ℓ ∗ ⊗ δ Lie [ H Lie ∗ ( g )]is given by d r ( ω ⊗ ω ⊗ . . . ⊗ ω ℓ ⊗ v ) = ω ⊗ ω ⊗ . . . ⊗ ω ℓ ⊗ δ ( v ) , ω i ∈ K ∗ and v ∈ H Lie ∗ ( g ). By a similar induction argument, we have d r [Λ ∗ ( I ) g ⊗ K ⊗ ℓ ∗ ⊗ H Lie ∗ ( g )] → Λ ∗ ( I ) g ⊗ K ⊗ ℓ ∗ ⊗ δ Lie [ H Lie ∗ ( g )]is given by d r ( u ⊗ ω ⊗ ω ⊗ . . . ⊗ ω ℓ ⊗ v ) = u ⊗ ω ⊗ . . . ⊗ ω ℓ ⊗ δ ( v ) , where u ∈ Λ ∗ ( I ) g , ω i ∈ K ∗ and v ∈ H Lie ∗ ( g ). The only absolute cycles in thePirashvili spectral sequence are elements ofΛ ∗ ( I ) g ⊗ K ⊗ ℓ ∗ , which are not in Im δ : H Lie ∗ +3 ( h ) → H rel ∗ ( h ). By induction on ℓ , HL ∗ ( h ) ≃ Λ ∗ ( I ) g ⊗ T ( K ∗ ) . We now study elements in K n and outline certain canonical constructionsto produce h -invariants. Recall that H Lie ∗ ( I ; h ) g is the homology of h g [ , ] ←−−− ( h ⊗ I ) g d ←−−− ( h ⊗ I ∧ ) g d ←−−− . . . d ←−−− ( h ⊗ I ∧ n ) g d ←−−− . Note that as g -modules, h ≃ g ⊕ I and h ⊗ I ∧ n ≃ ( g ⊗ I ∧ n ) ⊕ ( I ⊗ I ∧ n ) . Thus, ( h ⊗ I ∧ n ) g ≃ ( g ⊗ I ∧ n ) g ⊕ ( I ⊗ I ∧ n ) g . Any element in K n having arepresentative in ( I ⊗ I ∧ n ) g is necessarily an h -invariant at the chain level,since I acts trivially on ( I ⊗ I ∧ n ) g . Of course, all elements of ( I ⊗ I ∧ n ) g arecycles. Lemma 3.7. Any non-zero element in K n having a representative in ( g ⊗ I ∧ n ) g is determined by an injective map of g -modules, α : g → I ∧ n , where g actson itself via the adjoint action. roof. Let B : g ≃ −→ g ∗ = Hom R ( g , R ) be the isomorphism from a simpleLie algebra to its dual induced by the Killing form. Then the composition g ⊗ I ∧ n B ⊗ −→ g ∗ ⊗ I ∧ n ≃ −→ Hom R ( g , I ∧ n )is g -equivariant, and induces an isomorphism( g ⊗ I ∧ n ) g ≃ −→ Hom g ( g , I ∧ n ) . Since g is simple, a non-zero map of g -modules α : g → I ∧ n has no kernel,and g ≃ Im ( α ).Consider the special case where I ≃ g as g -modules, although I remainsan Abelian Lie algebra. Let α : g → I be a g -module isomorphism and let B − : g ∗ → g be the inverse of B : g → g ∗ in the proof of Lemma (3.7). Fora vector space basis { b i } ni =1 of g , let { b ∗ i } ni =1 denote the dual basis. Lemma 3.8. With α : g ≃ I as above, the balanced tensor ω = n X i =1 B − ( b ∗ i ) ⊗ α ( b i ) + α ( B − ( b ∗ i )) ⊗ b i ∈ h ⊗ h is h -invariant.Proof. By construction, n X i =1 B − ( b ∗ i ) ⊗ α ( b i ) ∈ g ⊗ I ֒ → h ⊗ h is g -invariant. Since α is an isomorphism of g -modules, it follows that n X i =1 α ( B − ( b ∗ i )) ⊗ b i ∈ I ⊗ g ֒ → h ⊗ h is also a g -invariant. Now, let a ∈ I . There is some g ∈ g with α ( g ) = a .14hus, [ ω, a ] = [ ω, α ( g )]= n X i =1 [ B − ( b ∗ i ) ⊗ α ( b i ) , α ( g )] + [ α ( B − ( b ∗ i )) ⊗ b i , α ( g )]= n X i =1 α (cid:0) [ B − ( b ∗ i ) , g ] (cid:1) ⊗ α ( b i ) + α ( B − ( b ∗ i )) ⊗ α (cid:0) [ b i , g ] (cid:1) = n X i =1 ( α ⊗ α ) (cid:0) [ B − ( b ∗ i ) ⊗ b i , g ] (cid:1) = 0 . Since α : g → I is an isomorphism of g -modules, it follows that if n X i =1 B ( b ∗ i ) ⊗ α ( b i ) ∈ g ⊗ I is a g -invariant, then P ni =1 B ( b ∗ i ) ⊗ b i ∈ g ⊗ g is also a g -invariant. We compute the Leibniz homology for extensions of the classical Lie alge-bras sl n ( R ), so n ( R ), and sp n ( R ). Additionally, HL ∗ is calculated for theLie algebra of the Poincar´e group R ⋊ SL ( C ) and the Lie algebra of theaffine Lorentz group R ⋊ SO (3 , g be a (semi-)simple real Lie algebra, and consider g ⊆ gl n ( R ).Then g acts on I = R n via matrix multiplication on vectors in R n , which isoften called the standard representation. Consider ∂∂x , ∂∂x , . . . , ∂∂x n as a vector space basis for R n . Then the elementary matrix with 1 in row i , column j , and 0s everywhere else becomes x i ∂∂x j . In the sequel, h denotesthe real Lie algebra formed via the extension0 −−−→ I −−−→ h −−−→ g −−−→ . Also, the element ∂∂x ∧ ∂∂x ∧ . . . ∧ ∂∂x n ∈ I ∧ n is the volume form, and often occurs as a g -invariant.15 orollary 4.1. Let g be a simple Lie algebra and I an Abelian Lie algebra,both over R . If h ≃ g ⊕ I as Lie algebras, i.e., h is reductive, then HL ∗ ( h ) ≃ Λ ∗ ( I ) ⊗ T ( K ∗ ) ,K ∗ = Ker( I ⊗ Λ ∗ ( I ) → Λ ∗ +1 ( I )) Proof. Since g acts trivially on I , it follow that[Λ ∗ ( I )] g = Λ ∗ ( I ) ,H Lie ∗ ( h ) ≃ H Lie ∗ ( g ) ⊗ Λ ∗ ( I ) ,H Lie ∗ ( I ; h ) g ≃ I ⊗ Λ ∗ ( I ) . Thus, K n = Ker( I ⊗ Λ n ( I ) → Λ n +1 ( I )).By way of comparison, from [6], under the hypotheses of Corollary (4.1),we have HL ∗ ( h ) ≃ HL ∗ ( g ) ∗ HL ∗ ( I ) ≃ HL ∗ ( I ) ≃ T ( I ) . Thus, as vector spaces, Λ ∗ ( I ) ⊗ T ( K ∗ ) ≃ T ( I ). For tensors of degree two,the above isomorphism becomesΛ ( I ) ⊕ S ( I ) ≃ I ⊗ , where S ( I ) denotes the second symmetric power of I . Corollary 4.2. For g = sl n ( R ) and I = R n the standard representation of sl n ( R ) , we have HL ∗ ( h ) ≃ [Λ ∗ ( I )] sl n ( R ) = (cid:10) ∂∂x ∧ ∂∂x ∧ . . . ∧ ∂∂x n (cid:11) . Proof. In this case there are no non-trivial sl n ( R )-module maps from sl n ( R )to I ∧ k . Using Lemma (3.7), we have H Lie ∗ ( I ; h ) sl n ( R ) ≃ H Lie ∗ ( I ; I ) sl n ( R ) ≃ [ I ⊗ Λ ∗ ( I )] sl n ( R ) = (cid:10) ∂∂x ∧ ∂∂x ∧ . . . ∧ ∂∂x n (cid:11) . Also, H Lie ∗ ( h ) ≃ H Lie ∗ ( sl n ( R )) ⊗ [Λ ∗ ( I )] sl n ( R ) ≃ H Lie ∗ ( sl n ( R )) ⊗ (cid:10) ∂∂x ∧ ∂∂x ∧ . . . ∧ ∂∂x n (cid:11) . Thus, T ( K ∗ ) = P m ≥ K ⊗ m ∗ = R , and the corollary follows from Theorem(3.6). 16 orollary 4.3. Consider g = sl ( C ) as a real Lie algebra with real vectorspace basis: v = x ∂∂x + x ∂∂x − x ∂∂x − x ∂∂x ,v = x ∂∂x − x ∂∂x − x ∂∂x + x ∂∂x v = x ∂∂x + x ∂∂x v = x ∂∂x − x ∂∂x v = x ∂∂x + x ∂∂x v = x ∂∂x − x ∂∂x . For I = R the standard representation of sl ( C ) ⊆ sl ( R ) , we have HL ∗ ( h ) ≃ [Λ ∗ ( I )] sl ( C ) . Proof. Again, there are no non-trivial sl ( C )-module maps from sl ( C ) to I ∧ k . Thus, T ( K ∗ ) = R in this case as well. The reader may check that[Λ ( I )] sl ( C ) has a real vector space basis given by the two elements: ∂∂x ∧ ∂∂x − ∂∂x ∧ ∂∂x , ∂∂x ∧ ∂∂x + ∂∂x ∧ ∂∂x . Furthermore, [Λ ( I )] sl ( C ) is a one-dimensional (real) vector space on thevolume element ∂∂x ∧ ∂∂x ∧ ∂∂x ∧ ∂∂x . For k = 1, 3, we have [Λ k ( I )] sl ( C ) = 0. Corollary 4.4. [1] Let g = so n ( R ) , n ≥ , and α ij = x i ∂∂x j − x j ∂∂x i ∈ so n ( R ) , ≤ i < j ≤ n. For I = R n the standard representation of so n ( R ) , we have HL ∗ ( h ) ≃ [Λ ∗ ( I )] so n ( R ) ⊗ T ( W ) , here W is the one-dimensional vector space with h -invariant basis element ω = X σ ∈ Sh , n − sgn( σ ) α σ (1) σ (2) ⊗ ǫ (cid:16) ∂∂x σ (3) ∧ ∂∂x σ (4) ∧ . . . ∧ ∂∂x σ ( n ) (cid:17) + ( − n +1 X σ ∈ Sh n − , sgn( σ ) ǫ (cid:16) ∂∂x σ (1) ∧ ∂∂x σ (2) ∧ . . . ∧ ∂∂x σ ( n − (cid:17) ⊗ α σ ( n − σ ( n ) , and ǫ : I ∧ ( n − → I ⊗ ( n − ֒ → h ⊗ ( n − is the skew-symmetrization map. Above, Sh p, q denotes the set of p , q shuffles in the symmetric group S ( p + q ) .Proof. There are two non-trivial so n ( R )-module maps so n ( R ) → I ∧ k to con-sider ρ : so n ( R ) → I ∧ , ρ : so n ( R ) → I ∧ ( n − , given by ρ ( α ij ) = ∂∂x i ∧ ∂∂x j ,ρ ( α ij ) = sgn( τ ) ∂∂x ∧ ∂∂x ∧ . . . ˆ ∂∂x i . . . ˆ ∂∂x j . . . ∧ ∂∂x n , where τ is the permutation sending1 , , . . . , i, . . . , j, . . . , n to i, j, , , . . . , n. Now, P i Let g = so (3 , and α ij = x i ∂∂x j − x j ∂∂x i ∈ so (3 , , ≤ i < j ≤ ,β ij = x i ∂∂x j + x j ∂∂x i ∈ so (3 , , i = 1 , , , j = 4 . For I = R the standard representation of so (3 , , we have HL ∗ ( h ) ≃ [Λ ∗ ( I )] so (3 , ⊗ T ( W ) , where W is the one-dimensional vector space with h -invariant basis element ω = α ⊗ (cid:16) ∂∂x ∧ ∂∂x (cid:17) − α ⊗ (cid:16) ∂∂x ∧ ∂∂x (cid:17) + α ⊗ (cid:16) ∂∂x ∧ ∂∂x (cid:17) + β ⊗ (cid:16) ∂∂x ∧ ∂∂x (cid:17) − β ⊗ (cid:16) ∂∂x ∧ ∂∂x (cid:17) + β ⊗ (cid:16) ∂∂x ∧ ∂∂x (cid:17) − (cid:16) ∂∂x ∧ ∂∂x (cid:17) ⊗ α + (cid:16) ∂∂x ∧ ∂∂x (cid:17) ⊗ α − (cid:16) ∂∂x ∧ ∂∂x (cid:17) ⊗ α − (cid:16) ∂∂x ∧ ∂∂x (cid:17) ⊗ β + (cid:16) ∂∂x ∧ ∂∂x (cid:17) ⊗ β − (cid:16) ∂∂x ∧ ∂∂x (cid:17) ⊗ β . Proof. The proof follows from identifying so (3 , 1) module maps ρ : so (3 , → I k and constructing h -invariants via balanced tensors. Note that the Killingform to establish g ≃ g ∗ is different for so ( R ) and so (3 , Corollary 4.6. [7] Let g = sp n ( R ) be the real symplectic Lie algebra withvector space basis given by the families: x k ∂∂y k , k = 1 , , , . . . , n ,(2) y k ∂∂x k , k = 1 , , , . . . , n ,(3) x i ∂∂y j + x j ∂∂y i , ≤ i < j ≤ n ,(4) y i ∂∂x j + y j ∂∂x i , ≤ i < j ≤ n ,(5) y j ∂∂y i − x i ∂∂x j , i = 1 , , , . . . , n , j = 1 , , , . . . , n .Let I = R n have basis ∂∂x , ∂∂x , . . . , ∂∂x n , ∂∂y , ∂∂y , . . . , ∂∂y n . Then HL ∗ ( h ) ≃ [Λ ∗ ( I )] sp n = Λ ∗ ( ω n ) , where ω n = P ni =1 ∂∂x i ∧ ∂∂y i .Proof. Since there are no non-trivial sp n ( R )-module maps sp n ( R ) → I ∧ k , wehave T ( K ∗ ) = R . The algebra of symplectic invariants [Λ ∗ ( I )] sp n is identifiedin another paper [7]. References [1] Biyogmam, G. 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