Notes on Lie Algebras

Samelson, Hans

doi:10.1007/978-1-4613-9014-5

Cited by 146 publications

(117 citation statements)

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“…We quote some notation and results from [10,18] as follows. For k ∈ N with 1 k r + 1, let e k be the canonical unit vector which has components 0 except for its kth component equal to 1.…”

Section: Meromorphic Continuation Of ζ Sl(r+1) (S)mentioning

confidence: 99%

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On Witten multiple zeta-functions associated with semisimple Lie algebras I

Matsumoto¹,

Tsumura²

2006

Ann. inst. Fourier

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Section: Meromorphic Continuation Of ζ Sl(r+1) (S)mentioning

confidence: 99%

“…It follows from the Cartan-Weyl theory of highest weights (see [10] Chapter 4 §7, [18] §3.6) that any highest weight λ for sl(r + 1) can be parameterized by λ = r+1 ν=1 n ν e ν with n 1 n 2 . .…”

Section: Meromorphic Continuation Of ζ Sl(r+1) (S)mentioning

confidence: 99%

On Witten multiple zeta-functions associated with semisimple Lie algebras I

Matsumoto¹,

Tsumura²

2006

Ann. inst. Fourier

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“…Theorem 7.2 does leave out non-integral lattices, but we will further defend this shortly. Fortunately, indecomposable root systems in finite dimensions have been fully enumerated: Theorem 7.3 [40,42] There exists exactly the following indecomposable root systems:…”

Section: Identifying the Ideal Latticementioning

confidence: 99%

Lattice-Based High-Dimensional Gaussian Filtering and the Permutohedral Lattice

2012

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High-dimensional Gaussian filtering is a popular technique in image processing, geometry processing and computer graphics for smoothing data while preserving important features. For instance, the bilateral filter, cross bilateral filter and non-local means filter fall under the broad umbrella of high-dimensional Gaussian filters. Recent algorithmic advances therein have demonstrated that by relying on a sampled representation of the underlying space, one can obtain speed-ups of orders of magnitude over the naïve approach. The simplest such sampled representation is a lattice, and it has been used successfully in the bilateral grid and the permutohedral lattice algorithms. In this paper, we analyze these lattice-based algorithms, developing a general theory of lattice-based high-dimensional Gaussian filtering. We consider the set of criteria for an optimal lattice for filtering, as it offers a good tradeoff of quality for computational efficiency, and evaluate the existing lattices under the criteria. In particular, we give a rigorous exposition of the properties of the permutohedral lattice and argue that it is the optimal lattice for Gaussian filtering. Lastly, we explore further uses of the permutohedral-lattice-based Gaussian filtering framework, showing that it can be easily adapted to perform mean shift filtering and yield improvement over the traditional approach based on a Cartesian grid.

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“…In this section we provide for reference a summary of the necessary facts of representation theory [7,8]. Let G be a compact semisimple Lie group with Lie algebra g. G ç U(n) for some n, and so setting gR = Ag we define the complexification g = gR © /gR in terms of matrices.…”

Section: Appendix a Representation Theorymentioning

confidence: 99%

“…In order to make the paper reasonably self-contained, a summary of the representation theory of semisimple Lie groups [7,8] is given, while an outline of the theory of large deviations is given in Appendix B. Some technical proofs in §2 are deferred to Appendix C.…”

Section: Introductionmentioning

confidence: 99%