On the cohomology of soluble Lie algebras

Barnes, Donald W.

doi:10.1007/bf01109799

Cited by 93 publications

(61 citation statements)

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“…Thus L(H)j=H, R{H) = 0 and #/L(//) is isomorphic to G. It can also be proved easily that H (2) = L{H) and that L(H) (1) …”

Section: Now Let I X = {Aehaim a N = 0) By G We Shall Denote The Twomentioning

confidence: 85%

“…Let R be a normed solvable algebra from X and let N be its nil-radical. It follows from Theorem 2.6 that R (2) = 0. Hence R (1) is a commutative ideal of R. Therefore i?…”

Section: The Structure Of Normed Solvable Algebras From Xmentioning

confidence: 93%

“…From this it follows easily that / is the maximal ideal of L{H) contained in M t , that is, 3 (L(H)) and the proof is complete. (2) …”

Section: Let M L = Mr\l(h) Since Liel(h) and H-£m And Since Codimm mentioning

confidence: 99%

“…(H) (2) =0. If H is finite-dimensional, then it was proved in [4] that H = L{H) + R{H) and that L{H) = L l + ••• +L n , where all L t are Lie ideals in H and isomorphic to s/(2, C).…”

Section: (H)i(m)mentioning

confidence: 99%

“…Barnes [2] and Towers [5] considered the case when H is a finite-dimensional Lie algebra over an arbitrary field, and all maximal subalgebras of H have codimension 1. Barnes, using the cohomology theory of Lie algebras, investigated solvable algebras, and Towers extended Barnes's results to include all Lie algebras.…”

Section: Introductionmentioning

confidence: 99%

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On normed Lie algebras with sufficiently many subalgebras of codimension I

Kissin

1986

Proceedings of the Edinburgh Mathematical Society

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Let H be a finite or infinite dimensional Lie algebra. Barnes [2] and Towers [5] considered the case when H is a finite-dimensional Lie algebra over an arbitrary field, and all maximal subalgebras of H have codimension 1. Barnes, using the cohomology theory of Lie algebras, investigated solvable algebras, and Towers extended Barnes's results to include all Lie algebras. In [4] complex finite-dimensional Lie algebras were considered for the case when all the maximal subalgebras of H are not necessarily of codimension 1 but when S(H)where S(H) is the set of all Lie subalgebras in H of codimension 1. Amayo [1] investigated the finite-dimensional Lie algebras with core-free subalgebras of codimension 1 and also obtained some interesting results about the structure of infinite dimensional Lie algebras with subalgebras of codimension 1.By X we shall denote the class of complex finite or infinite dimensional normed Lie algebras for which (1) holds. In Section 2 the results of Amayo will be applied in order to prove that for every complex normed Lie algebra H and for every subalgebra M e S(H) the largest Lie ideal /(M) of H contained in M has codimension less or equal to 3. Using this result for the case when HeX we shall show that, if S k (H) = {MeS(H):codim/(M) = /c}, for fc = l,2,3, thenis a semisimple ideal in H and R(H) = f) M es 3 (H)I(M) is the radical of H. We shall also prove that H (2) £L(H), so that i?(H) (2) =0. If H is finite-dimensional, then it was proved in [4] that H = L{H) + R{H) and that L{H) = L l + ••• +Ln , where all L t are Lie ideals in H and isomorphic to s/(2, C). If H is infinite dimensional, then this does not necessarily hold any longer. We shall consider an example of a normed Lie algebra H from 3E such that R(H) = 0 but L(H)^H. We shall also show that the property of belonging to X is inherited by all closed subalgebras of H and by all quotient algebras H/Sr s where S is any subset of S(H) and where y s = f) Ms sl(M)-Finally, we shall consider the set T of all ideals /(M) such that codim /(M) = 3 and shall introduce a Jacobson's topology on T.In Section 3 the. structure of solvable algebras from X is investigated. For every R e X we consider a special set £ of functionals on R from R°l } (Rf t) is the polar of R (1) ) and 199

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“…Thus L(H)j=H, R{H) = 0 and #/L(//) is isomorphic to G. It can also be proved easily that H (2) = L{H) and that L(H) (1) …”

Section: Now Let I X = {Aehaim a N = 0) By G We Shall Denote The Twomentioning

confidence: 85%

“…Let R be a normed solvable algebra from X and let N be its nil-radical. It follows from Theorem 2.6 that R (2) = 0. Hence R (1) is a commutative ideal of R. Therefore i?…”

Section: The Structure Of Normed Solvable Algebras From Xmentioning

confidence: 93%

“…From this it follows easily that / is the maximal ideal of L{H) contained in M t , that is, 3 (L(H)) and the proof is complete. (2) …”

Section: Let M L = Mr\l(h) Since Liel(h) and H-£m And Since Codimm mentioning

confidence: 99%

“…(H) (2) =0. If H is finite-dimensional, then it was proved in [4] that H = L{H) + R{H) and that L{H) = L l + ••• +L n , where all L t are Lie ideals in H and isomorphic to s/(2, C).…”

Section: (H)i(m)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On normed Lie algebras with sufficiently many subalgebras of codimension I

Kissin

1986

Proceedings of the Edinburgh Mathematical Society

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Left-Invariant Affine Structures on Reductive Lie Groups

Burde

1996

Journal of Algebra

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ŽWe describe left-invariant affine structures that is, left-invariant flat torsion-free . affine connections ٌ on reductive linear Lie groups G. They correspond bijectively to LSA-structures on the Lie algebra ᒄ of G. Here LSA stands for left-symmetric algebra. If ᒄ has trivial or one-dimensional center ᒗ then the affine representation ␣ s [ 1 of ᒄ, induced by any LSA-structure ᒄ on ᒄ is radiant, i.e., thewhere ᒐ is split simple, admits LSA-structures if and only if ᒐ is of type A , that l is, ᒄ s ᒄ ᒉ . Here we have the associative LSA-structure given by ordinary matrix n Ž . multiplication corresponding to the bi-invariant affine structure on GL n , which was believed to be essentially the only possible LSA-structure on ᒄ ᒉ . We exhibit n interesting LSA-structures different from the associative one. They arise as certain deformations of the matrix algebra. Then we classify all LSA-structures on ᒄ ᒉ n using a result of Baues. For n s 2 we compute all structures explicitly over the complex numbers.

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On Modular* Subalgebras of a Lie Algebra

Towers

1997

Journal of Algebra

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On the cohomology of soluble Lie algebras

Cited by 93 publications

References 5 publications

On normed Lie algebras with sufficiently many subalgebras of codimension I

On normed Lie algebras with sufficiently many subalgebras of codimension I

Left-Invariant Affine Structures on Reductive Lie Groups

On Modular* Subalgebras of a Lie Algebra

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