Hopf co-addition for free magma algebras and the non-associative Hausdorff series

Gerritzen, Lothar; Holtkamp, Ralf

doi:10.1016/s0021-8693(03)00157-1

Cited by 17 publications

(22 citation statements)

References 9 publications

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“…Similar study of the algebra of constants in a very large class of (not only associative) algebras was performed by Gerritzen and Holtkamp [41] and Drensky and Holtkamp [27]. We shall finish the survey section with the following, probably folklore known lemma.…”

Section: Derivations Of Free Algebrasmentioning

confidence: 68%

Constants of Weitzenböck derivations and invariants of unipotent transformations acting on relatively free algebras

Drensky

Gupta

2005

Journal of Algebra

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In commutative algebra, a Weitzenböck derivation is a nonzero triangular linear derivation of the polynomial algebra K[x 1 , . . . , x m ] in several variables over a field K of characteristic 0. The classical theorem of Weitzenböck states that the algebra of constants is finitely generated. (This algebra coincides with the algebra of invariants of a single unipotent transformation.) In this paper we study the problem of finite generation of the algebras of constants of triangular linear derivations of finitely generated (not necessarily commutative or associative) algebras over K assuming that the algebras are free in some sense (in most of the cases relatively free algebras in varieties of associative or Lie algebras). In this case the algebra of constants also coincides with the algebra of invariants of some unipotent transformation.The main results are the following: (1) We show that the subalgebra of constants of a factor algebra can be lifted to the subalgebra of constants.(2) For all varieties of associative algebras which are not nilpotent in Lie sense the subalgebras of constants of the relatively free algebras of rank 2 are not finitely generated. (3) We describe the generators of the subalgebra of constants for all factor algebras K x, y /I modulo a GL 2 (K)-invariant ideal I . (4) Applying known results from commutative algebra, we construct classes of automorphisms of the algebra generated by two generic 2 × 2 matrices. We obtain also some partial results on relatively free Lie algebras.

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Section: Derivations Of Free Algebrasmentioning

confidence: 68%

Constants of Weitzenböck derivations and invariants of unipotent transformations acting on relatively free algebras

Drensky

Gupta

2005

Journal of Algebra

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“…An analogous formula for non-associative algebras has appeared in the literature based on a particular definition of the exponential function so that e A e A = e 2A [37] (but see also [38] for an update of the recent developments in the subject). We are not going to worry about it here, 4 because any deviations from e A e B = exp(A + B + [A, B]/2 + · · · ) appear at the level of triple commutators or higher, which are not contributing to (3.11).…”

Section: -Cocycles In Lie Group Cohomologymentioning

confidence: 99%

3-cocycles, non-associative star-products and the magnetic paradigm of R-flux string vacua

Bakas

Lüst

2014

J. High Energ. Phys.

134

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Abstract:We consider the geometric and non-geometric faces of closed string vacua arising by T-duality from principal torus bundles with constant H-flux and pay attention to their double phase space description encompassing all toroidal coordinates, momenta and their dual on equal footing. We construct a star-product algebra on functions in phase space that is manifestly duality invariant and substitutes for canonical quantization. The 3-cocycles of the Abelian group of translations in double phase space are seen to account for non-associativity of the star-product. We also provide alternative cohomological descriptions of non-associativity and draw analogies with the quantization of point-particles in the field of a Dirac monopole or other distributions of magnetic charge. The magnetic field analogue of the R-flux string model is provided by a constant uniform distribution of magnetic charge in space and non-associativity manifests as breaking of angular symmetry. The Poincaré vector comes to rescue angular symmetry as well as associativity and also allow for quantization in terms of operators and Hilbert space only in the case of charged particles moving in the field of a single magnetic monopole.

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“…If exp is a non-associative power series in one variable that sends primitive elements bijectively onto the the group-like elements in the algebra of non-associative power series in one variable, the same reasoning as before can be used to show that the expression log(exp x exp y) can be written as an infinite linear combination of primitive operations in x and y and, hence, gives a primitive series. A primitive series of this type was studied in [6].…”

Section: 2mentioning

confidence: 99%

“…It is not immediately clear, however, that the primitive series defined in this way gives an integration functor. An interesting point is that the power series exp and log are not uniquely determined by the condition that they interchange primitive and group-like elements [6]. Our strategy will be to use a different primitive series whose definition is based on the geometry of the Mikheev-Sabinin brackets.…”

Section: 2mentioning

confidence: 99%

Nilpotent Sabinin algebras

2014

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In this paper we establish several basic properties of nilpotent Sabinin algebras. Namely, we show that nilpotent Sabinin algebras (1) can be integrated to produce nilpotent loops, (2) satisfy an analogue of the Ado theorem, (3) have nilpotent Lie envelopes. We also give a new set of axioms for Sabinin algebras. These axioms reflect the fact that a complementary subspace to a Lie subalgebra in a Lie algebra is a Sabinin algebra. Finally, we note that the non-associative version of the Jennings theorem produces a version of the Ado theorem for loops whose commutator-associator filtration is of finite length.

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Hopf co-addition for free magma algebras and the non-associative Hausdorff series

Cited by 17 publications

References 9 publications

Constants of Weitzenböck derivations and invariants of unipotent transformations acting on relatively free algebras

Constants of Weitzenböck derivations and invariants of unipotent transformations acting on relatively free algebras

3-cocycles, non-associative star-products and the magnetic paradigm of R-flux string vacua

Nilpotent Sabinin algebras

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