VALIDITY OF THE GELL-MANN FORMULA FOR sl(n, ℝ) AND su(n) ALGEBRAS

2013 ◽  
Vol 10 (05) ◽  
pp. 1350017 ◽  
Author(s):  
IGOR SALOM ◽  
DJORDJE ŠIJAČKI
Keyword(s):  
Closed Form ◽  
Lie Algebra ◽  
Complete List ◽  
Matrix Elements ◽  
Operator Expression ◽  
General Validity ◽  
Potential Value ◽  
Explicit Expressions

The so-called Gell-Mann formula, a prescription designed to provide an inverse to the Inönü–Wigner Lie algebra contraction, has a great versatility and potential value. This formula has no general validity as an operator expression. The question of applicability of Gell-Mann's formula to various algebras and their representations was only partially treated. The validity constraints of the Gell-Mann formula for the case of sl(n, ℝ) and su(n) algebras are clarified, and the complete list of representations spaces for which this formula applies is given. Explicit expressions of the sl(n, ℝ) generators matrix elements are obtained for all these cases in a closed form by making use of the Gell-Mann formula.

scholarly journals GENERALIZATION OF THE GELL–MANN DECONTRACTION FORMULA FOR sl(n,ℝ) AND su(n) ALGEBRAS

2011 ◽  
Vol 08 (02) ◽  
pp. 395-410 ◽  
Author(s):  
IGOR SALOM ◽  
DJORDJE ŠIJAČKI
Keyword(s):  
Closed Form ◽  
Lie Algebra ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Algebraic Expression ◽  
Matrix Elements ◽  
Group Manifold ◽  
The Matrix

The so-called Gell–Mann or decontraction formula is proposed as an algebraic expression inverse to the Inönü–Wigner Lie algebra contraction. It is tailored to express the Lie algebra elements in terms of the corresponding contracted ones. In the case of sl (n,ℝ) and su (n) algebras, contracted w.r.t. so (n) subalgebras, this formula is generally not valid, and applies only in the cases of some algebra representations. A generalization of the Gell–Mann formula for sl (n,ℝ) and su (n) algebras, that is valid for all tensorial, spinorial, (non)unitary representations, is obtained in a group manifold framework of the SO(n) and/or Spin (n) group. The generalized formula is simple, concise and of ample application potentiality. The matrix elements of the [Formula: see text], i.e. SU(n)/SO(n), generators are determined, by making use of the generalized formula, in a closed form for all irreducible representations.

The use of Riemann problems in solving a class of transcendental equations

1973 ◽  
Vol 73 (1) ◽  
pp. 111-118 ◽  
Author(s):  
E. E. Burniston ◽  
C. E. Siewert
Keyword(s):  
Boundary Value Problem ◽  
Closed Form ◽  
Boundary Value ◽  
Explicit Solutions ◽  
Riemann Boundary Value Problem ◽  
Riemann Problems ◽  
Riemann Boundary ◽  
Canonical Solution ◽  
Transcendental Equations ◽  
Explicit Expressions

AbstractA method of finding explicit expressions for the roots of a certain class of transcendental equations is discussed. In particular it is shown by determining a canonical solution of an associated Riemann boundary-value problem that expressions for the roots may be derived in closed form. The explicit solutions to two transcendental equations, tan β = ωβ and β tan β = ω, are discussed in detail, and additional specific results are given.

High-resolution autoradiographic localization of DNA-containing sites and RNA synthesis in developing nucleoli of human preimplantation embryos: a new concept of embryonic nucleologenesis

Development ◽  
1987 ◽  
Vol 101 (4) ◽  
pp. 777-791 ◽  
Author(s):  
J. Tesarik ◽  
V. Kopecny ◽  
M. Plachot ◽  
J. Mandelbaum
Keyword(s):  
Rna Synthesis ◽  
Preimplantation Embryos ◽  
Synthetic Activity ◽  
Rrna Synthesis ◽  
Nucleolar Organizer Regions ◽  
Careful Study ◽  
Matrix Elements ◽  
Electron Microscopic Autoradiography ◽  
General Validity ◽  
Granular Component

Human embryos from the 2-cell to the morula stage, obtained by in vitro fertilization, were incubated with [3H]thymidine or [3H]uridine so as to achieve labelling of all replicating nuclear DNA and the newly synthesized RNA, respectively. The label was localized in different structural components of developing nucleoli using electron microscopic autoradiography. Careful study of the relationship between the structural pattern and nucleic acid distribution made it possible to define four stages of embryonic nucleologenesis. Homogeneous nuclear precursors (i) consist of nucleolar matrix elements appearing as filaments of 3 nm thickness, (ii) do not contain recently replicated DNA and (iii) lack RNA synthetic activity. Penetration of DNA into these bodies is a key event leading to their transformation into heterogeneous nucleolar precursors. In addition to the 3 nm matrix filaments, two types of 5 nm fibrillar components can be recognized in them. The denser type contains DNA and is the site of nucleolar RNA synthesis, while the more loosely arranged 5 nm fibrils are not labelled with [3H]thymidine and apparently represent the newly produced pre-rRNA detached from the transcribing rDNA filament. Compact fibrillogranular nucleoli are characterized by the first appearance of the granular component and reduction of the nontranscribing part of the fibrillar component, both indicating the activation of the machinery for rRNA processing. Finally, the granular component is most evident in reticulated nucleoli, occupying mostly the inner parts of their nucleolonema, while the transcription sites tend to be located at the nucleolar periphery. Our findings advocate a unique concept of embryonic nucleologenesis, different from any other nucleolar event during the cell cycle of differentiated cells. This developmental pattern is characterized by a gradual activation of rRNA synthesis and processing, mediated by progressive association of rDNA and, later on, the newly formed pre-rRNA with pre-existing nucleolar matrix elements that are originally topically separated from nucleolar organizer regions. This model may have a general validity in early animal embryos despite some interspecies variability in the timing of individual steps and resulting structural peculiarities.

scholarly journals THE UNIVERSAL LINK POLYNOMIAL

1992 ◽  
Vol 07 (05) ◽  
pp. 877-945 ◽  
Author(s):  
E. GUADAGNINI
Keyword(s):  
Field Theory ◽  
Closed Form ◽  
Lie Algebra ◽  
Wilson Line ◽  
Expectation Values ◽  
Simple Lie Algebra ◽  
Chern Simons

The solution of the non-Abelian SU (N) quantum Chern–Simons field theory defined in R3 is presented. It is shown how to compute the expectation values of the Wilson line operators, associated with oriented framed links, in closed form. The main properties of the universal link polynomial, defined by these expectation values, are derived in the case of a generic real simple Lie algebra. The resulting polynomials for some simple examples of links are reported.

scholarly journals Baker–Campbell–Hausdorff–Dynkin Formula for the Lie Algebra of Rigid Body Displacements

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1185
Author(s):  
Daniel Condurache ◽  
Ioan-Adrian Ciureanu
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Lie Algebra ◽  
Lie Group ◽  
First Time

The paper proposes, for the first time, a closed form of the Baker–Campbell–Hausdorff–Dynkin (BCHD) formula in the particular case of the Lie algebra of rigid body displacements. For this purpose, the structure of the Lie group of the rigid body displacements S E ( 3 ) and the properties of its Lie algebra s e ( 3 ) are used. In addition, a new solution to this problem in dual Lie algebra of dual vectors is delivered using the isomorphism between the Lie group S E ( 3 ) and the Lie group of the orthogonal dual tensors.

scholarly journals The EllipticGL(n)Dynamical Quantum Group as an𝔥-Hopf Algebroid

2009 ◽  
Vol 2009 ◽  
pp. 1-41 ◽  
Author(s):  
Jonas T. Hartwig
Keyword(s):  
Quantum Group ◽  
Lie Algebra ◽  
Spectral Parameter ◽  
Exterior Algebra ◽  
Baxter Equation ◽  
Matrix Elements ◽  
Elliptic Solution ◽  
Hopf Algebroid ◽  
Quantum Dynamical ◽  
Hopf Algebroids

Using the language of𝔥-Hopf algebroids which was introduced by Etingof and Varchenko, we construct a dynamical quantum group,ℱell(GL(n)), from the elliptic solution of the quantum dynamical Yang-Baxter equation with spectral parameter associated to the Lie algebra𝔰𝔩n. We apply the generalized FRST construction and obtain an𝔥-bialgebroidℱell(M(n)). Natural analogs of the exterior algebra and their matrix elements, elliptic minors, are defined and studied. We show how to use the cobraiding to prove that the elliptic determinant is central. Localizing at this determinant and constructing an antipode we obtain the𝔥-Hopf algebroidℱell(GL(n)).

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