Theorie der Transformationsgruppen I

Lie, Sophus

doi:10.1007/bf01446218

Cited by 416 publications

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“…. , 4} they are given by the operators H 0 = µ x µ ∂/∂x µ and G µ = 2x µ ν x ν ∂/∂x ν − ν x 2 ν (∂/∂x ν ), respectively [28]. Since we are deforming with respect to the two-torus coming from the Cartan subalgebra H 1 , H 2 of so (5), we write the Lie algebra so(5, 1) as so (5) together with five extra generators, H 0 , G r , the latter labeled by the corresponding roots with respect to H 1 and H 2 , that is r = (±1, 0), (0, ±1).…”

Section: Twisted Conformal Transformationsmentioning

confidence: 99%

Noncommutative Instantons from Twisted Conformal Symmetries

Landi

Suijlekom

2007

Commun. Math. Phys.

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We construct a five-parameter family of gauge-nonequivalent SU (2) instantons on a noncommutative four sphere S 4 θ and of topological charge equal to 1. These instantons are critical points of a gauge functional and satisfy self-duality equations with respect to a Hodge star operator on forms on S 4 θ . They are obtained by acting with a twisted conformal symmetry on a basic instanton canonically associated with a noncommutative instanton bundle on the sphere. A completeness argument for this family is obtained by means of index theorems. The dimension of the "tangent space" to the moduli space is computed as the index of a twisted Dirac operator and turns out to be equal to five, a number that survives deformation.

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Section: Twisted Conformal Transformationsmentioning

confidence: 99%

Noncommutative Instantons from Twisted Conformal Symmetries

Landi

Suijlekom

2007

Commun. Math. Phys.

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“…Moreover, for all these three types we construct an explicit realization in some L. Applying obtained results to the Lie algebra W 1 (K) we give a description of all finite dimensional subalgebras of W 1 (K) (Proposition 3). In case K = C this description can be easily deduced from classical results of S. Lie (see [5]) about realizations (up to local diffeomorphisms) of finite dimensional Lie algebras by vector fields on the complex line. In [5], S. Lie has also classified analogous realizations on the complex plane and on the real line.…”

Section: Introductionmentioning

confidence: 94%

Finite-Dimensional Subalgebras In Polynomial Lie Algebras Of Rank One

2011

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“…An example of this, already considered by Lie in [29], is α ij (x) = f ij k x k , where the f ij k 's are the structure constants of some Lie algebra. Example 1.3.…”

Section: Graded Algebras and Graded Lie Algebrasmentioning

confidence: 99%

Graded Poisson Algebras

Cattaneo

Fiorenza

Longoni

2006

Encyclopedia of Mathematical Physics

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1. Definitions 1.1. Graded vector spaces. By a Z-graded vector space (or simply, graded vector space) we mean a direct sum A = ⊕ i∈Z A i of vector spaces over a field k of characteristic zero. The A i are called the components of A of degree i and the degree of a homogeneous element a ∈ A is denoted by |a|. We also denote by A[n] the graded vector space with degree shifted by n, namely,The tensor product of two graded vector spaces A and B is again a graded vector space whose degree r component is given byThe symmetric and exterior algebra of a graded vector space A are defined respectively as S(A) = T (A)/I S and (A) = T (A)/I ∧ , where T (A) = ⊕ n≥0 A ⊗n is the tensor algebra of A and I S (resp. I ∧ ) is the two-sided ideal generated by elements of the form a ⊗ b − (−1) |a| |b| b ⊗ a (resp. a ⊗ b + (−1) |a| |b| b ⊗ a), with a and b homogeneous elements of A. The images of A ⊗n in S(A) and (A) are denoted by S n (A) and n (A) respectively. Notice that there is a canonical decalage isomorphism1.2. Graded algebras and graded Lie algebras. We say that A is a graded algebra (of degree zero) if A is a graded vector space endowed with a degree zero bilinear associative product · : A⊗A → A. A graded algebra is graded commutative if the product satisfies the conditionfor any two homogeneous elements a, b ∈ A of degree |a| and |b| respectively. A graded Lie algebra of degree n is a graded vector space A endowed with a graded Lie bracket on A[n]. Such a bracket can be seen as a degree −n Lie bracket on A, i.e., as bilinear operation {·, ·} : A ⊗ A → A[−n] satisfying graded antisymmetry and graded Jacobi relations: {a, b} = −(−1) (|a|+n)(|b|+n) {b, a} {a, {b, c}} = {{a, b}, c} + (−1) (|a|+n)(|b|+n) {a{b, c}}

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Theorie der Transformationsgruppen I

Cited by 416 publications

References 0 publications

Noncommutative Instantons from Twisted Conformal Symmetries

Noncommutative Instantons from Twisted Conformal Symmetries

Finite-Dimensional Subalgebras In Polynomial Lie Algebras Of Rank One

Graded Poisson Algebras

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