Lie elements in the tensor algebra

Klyachko, Anton A.

doi:10.1007/bf00966559

Cited by 72 publications

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“…The Poincare-Birkhoff-Witt Theorem states that T is the symmetric algebra operator S = 52 ^* o v e r Lie. So, the following result of Klyachko (see [4]) appears as an immediate consequence of Theorem 1: In fact, when we speak about T(V) we have to point out the difference between two cases: the "non-graded" tensor algebra, whose coproduct A is cocommutative, that is T o A = A where r(x (8) y) = y ( g> x, for x, y £ T(V). the "graded" tensor algebra; whose coproducts A is graded cocommutative, that is T o A = A where T{X ®y) = (-l)' 1 ""^ <g> x, for x, y G T(V) and |x| the degree of x.…”

Section: The A-ring Of Representations Of the Symmetric Groupmentioning

confidence: 98%

“…Our goal is to find analogous decompositions in the group of formal power series 1 -|-.R [[i]] + , so we extend formula (2) by mimicking, in a certain way, the Dress-Siebeneicher construction. The idea is to replace the powers in the classical construction of the universal ring of Witt vectors by the Adams operations of the A-ring, to get the following: [3] Free Lie algebra and lambda-ring structure 375 THEOREM M. Ronco [4] and on the other hand:…”

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Free Lie algebra and lambda-ring structure

Ronco¹

1994

Bull. Austral. Math. Soc.

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Let R be a graded A-ring. We extend a well-known formula in the universal ring of Witt vectors by replacing the power operations by the Adams operations. Our method provides us an easy way to compute the inverse image by the symmetric power operators of certain elements of R. As a corollary we get identities, found by Klyachko and Hanlon, in the rings 1 + Z [[t]] + and 1 + R [[t]] + , where R is the representation ring of the symmetric groups.In their work [1] Dress and Siebeneicher proved the Cyclotomic Identity (see [7]), which takes place in the ring 1 + Z[[*]] + , by using the Burnside ring and the universal ring of Witt vectors. The main result of this paper is to generalise this identity by proving a similar identity, which takes place in 1 + -R[[<]] + , where R is a A-ring. In this new setting the n-th power operation is replaced by the Adams operation \& n .This new formula permits us to give a quick proof of two results, one by Hanlon [3] and one by Klyachko [4], about the representations of the symmetric group associated to the free Lie algebra. Here the A-ring R is the representation ring of the symmetric groups © R(S n ) • In Section 1 we construct a diagram associated to any graded A-ring R, that leads us to find the inverse image by S = £) S* (S* denotes the symmetric power operator) of certain elements of R. We use elementary concepts and results of A-ring theory and some ideas of a nice construction of Dress-Siebeneicher (see [1]). In Section 2 we apply the results of Section 1 to the representation ring R = © -R(S n ). PRIMITIVE ELEMENTS IN A A-RINGWe recall some classical notation and results. For complete definitions and proofs we refer to Knutson (see [5]).Let R be a torsion free graded A-ring, that is R = 0 R n is such that its Aoperations satisfy jR, n D X t (R n ), for i, n ^ 0. The ring R := ]\ R n has a natural A-ring structure, inherited from the one of R.

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Section: The A-ring Of Representations Of the Symmetric Groupmentioning

confidence: 98%

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confidence: 99%

Free Lie algebra and lambda-ring structure

Ronco¹

1994

Bull. Austral. Math. Soc.

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“…In this example we describe the soul of the Chevalley-Eilenberg cohomology of Lie algebras which is, due to the obvious self-duality of Proposition 9, the same as the soul of the Harrison cohomology of commutative associative algebras. In both cases [18]) and one may identify ((Com ⊗ Lie) Σ , δ Σ χ ) with the acyclic complex (6). Therefore the souls of both the Chevalley-Eilenberg cohomology and the Harrison cohomology are trivial.…”

Section: There Clearly Exists a Uniquementioning

confidence: 99%

Cohomology operations and the deligne conjecture

Markl

2007

Czech Math J

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“…n was extensively studied in the series of papers [Thrall 1942;Wever 1949;Klyachko 1974] and some others. There exists an interesting formula for the multiplicity of each irreducible representation (corresponding to some Young diagram µ) in L i n analogous to the one for the dimension of L i n .…”

Section: The Natural Representation Of Glmentioning

confidence: 99%