Casimir Operators for Semisimple Lie Groups

Perelomov, A. M.; Попов, В. С.

doi:10.1070/im1968v002n06abeh000731

Cited by 65 publications

(84 citation statements)

References 15 publications

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“…Finally, our analysis of Perelomov-Popov measures also uses the formulas of [PP68,Pop76,Pop77] relating the moments of these measures to the moments of counting measures. It is convenient for us to encode signatures by probability measures on R. We will use two different sets of measures.…”

Section: Our Methodsmentioning

confidence: 99%

“…In Sect. 1 (in order to keep it short) we present their construction only for the unitary groups, but parallel stories exist in [PP68] for orthogonal and symplectic groups as well. In Sect.…”

Section: Perelomov-popov Measuresmentioning

confidence: 99%

“…The "correct" definition of the measure comes from the work of Perelomov and Popov [PP68] on the centers of universal enveloping algebras of classical Lie groups. In Sect.…”

Section: Perelomov-popov Measuresmentioning

confidence: 99%

“…(3) We show in Theorem 2.8 that if one replaces the counting measure (1.1) by another remarkable probability measure, then an analogue of Theorem 1.1 would involve precisely the (additive) free convolution. The definition of the probability measure that we use, is inspired by the work of Perelomov and Popov [PP68] on Casimir elements in the enveloping algebras of classical Lie groups. (4) We study another analogue of Theorem 1.1 for classical Lie groups in which tensor products are replaced by the restrictions to smaller subgroups.…”

Section: Gafa Lie Groups and Quantized Free Convolution 765mentioning

confidence: 99%

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Representations of classical Lie groups and quantized free convolution

Bufetov

Gorin

2015

Geom. Funct. Anal.

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Abstract. We study the decompositions into irreducible components of tensor products and restrictions of irreducible representations for all series of classical Lie groups as the rank of the group goes to infinity. We prove the Law of Large Numbers for the random counting measures describing the decomposition. This leads to two operations on measures which are deformations of the notions of the free convolution and the free projection. We further prove that if one replaces counting measures with others coming from the work of Perelomov and Popov on the higher order Casimir operators for classical groups, then the operations on the measures turn into the free convolution and projection themselves. We also explain the relation between our results and limit shape theorems for uniformly random lozenge tilings with and without axial symmetry.

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Section: Our Methodsmentioning

confidence: 99%

Section: Perelomov-popov Measuresmentioning

confidence: 99%

“…The "correct" definition of the measure comes from the work of Perelomov and Popov [PP68] on the centers of universal enveloping algebras of classical Lie groups. In Sect.…”

Section: Perelomov-popov Measuresmentioning

confidence: 99%

Section: Gafa Lie Groups and Quantized Free Convolution 765mentioning

confidence: 99%

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Representations of classical Lie groups and quantized free convolution

Bufetov

Gorin

2015

Geom. Funct. Anal.

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“…where C(p, q) is the value of the quadratic Casimir operator in (p, q), given by [14] C(p, q) = (p + q + 1)…”

Section: Performing the Averagementioning

confidence: 99%

Eigenvalue density of Wilson loops in 2DSU(N) YM

Lohmayer

Neuberger

Wettig

2009

J. High Energy Phys.

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In 1981 Durhuus and Olesen (DO) showed that at infinite N the eigenvalue density of a Wilson loop matrix W associated with a simple loop in two-dimensional Euclidean SU(N ) Yang-Mills theory undergoes a phase transition at a critical size. The averages of det(z − W ), det(z − W ) −1 , and det(1 + uW )/(1 − vW ) at finite N lead to three different smoothed out expressions, all tending to the DO singular result at infinite N . These smooth extensions are obtained and compared to each other.

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Iwasawa nilpotency degree of non compact symmetric cosets in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \mathcal{N} $\end{document}‐extended supergravity

Cacciatori

Cerchiai

Ferrara

et al. 2014

Fortschritte der Physik

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We analyze the polynomial part of the Iwasawa realization of the coset representative of non compact symmetric Riemannian spaces. We start by studying the role of Kostant's principal SU (2) P subalgebra of simple Lie algebras, and how it determines the structure of the nilpotent subalgebras. This allows us to compute the maximal degree of the polynomials for all faithful representations of Lie algebras. In particular the metric coefficients are related to the scalar kinetic terms while the representation of electric and magnetic charges is related to the coupling of scalars to vector field strengths as they appear in the Lagrangian.We consider symmetric scalar manifolds in N−extended supergravity in various space-time dimensions, elucidating various relations with the underlying Jordan algebras and normed Hurwitz algebras. For magic supergravity theories, our results are consistent with the Tits-Satake projection of symmetric spaces and the nilpotency degree turns out to depend only on the space-time dimension of the theory.These results should be helpful within a deeper investigation of the corresponding supergravity theory, e.g. in studying ultraviolet properties of maximal supergravity in various dimensions.

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Casimir Operators for Semisimple Lie Groups

Cited by 65 publications

References 15 publications

Representations of classical Lie groups and quantized free convolution

Representations of classical Lie groups and quantized free convolution

Eigenvalue density of Wilson loops in 2DSU(N) YM

Iwasawa nilpotency degree of non compact symmetric cosets in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \mathcal{N} $\end{document}‐extended supergravity

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