The Experts below are selected from a list of 3153 Experts worldwide ranked by ideXlab platform
Giovanni Satta - One of the best experts on this subject based on the ideXlab platform.
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Analytical Bethe ansatz for closed and open gl(Script M|Script N) super-spin chains in arbitrary representations and for any Dynkin Diagram
Journal of High Energy Physics, 2007Co-Authors: Eric Ragoucy, Giovanni SattaAbstract:We present the analytical Bethe ansatz for spin chains based on the superalgebras gl(|), ≠, with at each site an arbitrary representation (and including inhomogeneities). The calculation is done for closed and open spin chains. In this latter case, the boundary matrices K±(λ) are of general type, provided they commute. We compute the Bethe ansatz equations in full generality, and for any type of Dynkin Diagram. Examples are worked out to illustrate the techniques.
Eric Ragoucy - One of the best experts on this subject based on the ideXlab platform.
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Analytical Bethe ansatz for closed and open gl(Script M|Script N) super-spin chains in arbitrary representations and for any Dynkin Diagram
Journal of High Energy Physics, 2007Co-Authors: Eric Ragoucy, Giovanni SattaAbstract:We present the analytical Bethe ansatz for spin chains based on the superalgebras gl(|), ≠, with at each site an arbitrary representation (and including inhomogeneities). The calculation is done for closed and open spin chains. In this latter case, the boundary matrices K±(λ) are of general type, provided they commute. We compute the Bethe ansatz equations in full generality, and for any type of Dynkin Diagram. Examples are worked out to illustrate the techniques.
Satta G. - One of the best experts on this subject based on the ideXlab platform.
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Analytical Bethe Ansatz for closed and open gl(M|N) super-spin chains in arbitrary representations and for any Dynkin Diagram
'IOP Publishing', 2015Co-Authors: Ragoucy E., Satta G.Abstract:We present the analytical Bethe ansatz for spin chains based on the superalgebras gl(M|N), $M\neq N$, with at each site an arbitrary representation (and including inhomogeneities). The calculation is done for closed and open spin chains. In this latter case, the boundary matrices $K_{\pm}(u)$ are of general type, provided they commute. We compute the Bethe ansatz equations in full generality, and for any type of Dynkin Diagram. Examples are worked out to illustrate the techniques.Comment: 40 pages; misprints and some references corrected; some mistakes in the expression of the vacuum eigenvalue for the open case corrected (we are grateful to R. Nepomechie who pointed them out
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Analytical Bethe Ansatz for closed and open gl(M|N) super-spin chains in arbitrary representations and for any Dynkin Diagram
'Springer Fachmedien Wiesbaden GmbH', 2007Co-Authors: Ragoucy E., Satta G.Abstract:40 pages;We present the analytical Bethe ansatz for spin chains based on the superalgebras gl(M|N), $M\neq N$, with at each site an arbitrary representation (and including inhomogeneities). The calculation is done for closed and open spin chains. In this latter case, the boundary matrices $K_{\pm}(u)$ are of general type, provided they commute. We compute the Bethe ansatz equations in full generality, and for any type of Dynkin Diagram. Examples are worked out to illustrate the techniques
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Analytical Bethe Ansatz for closed and open gl(M|N) super-spin chains in arbitrary representations and for any Dynkin Diagrams
'IOP Publishing', 2007Co-Authors: Ragoucy E., Satta G.Abstract:We present the analytical Bethe ansatz for spin chains based on the superalgebras gl(M|N), $M\neq N$, with at each site an arbitrary representation (and including inhomogeneities). The calculation is done for closed and open spin chains. In this latter case, the boundary matrices $K_{\pm}(u)$ are of general type, provided they commute. We compute the Bethe ansatz equations in full generality, and for any type of Dynkin Diagram. Examples are worked out to illustrate the techniques
Mehmet Koca - One of the best experts on this subject based on the ideXlab platform.
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su 5 grand unified theory its polytopes and 5 fold symmetric aperiodic tiling
International Journal of Geometric Methods in Modern Physics, 2017Co-Authors: Mehmet Koca, Nazife Ozdes Koca, Abeer AlsiyabiAbstract:We associate the lepton–quark families with the vertices of the 4D polytopes 5-cell (0001)A4 and the rectified 5-cell (0100)A4 derived from the SU(5) Coxeter–Dynkin Diagram. The off-diagonal gauge bosons are associated with the root polytope (1001)A4 whose facets are tetrahedra and the triangular prisms. The edge-vertex relations are interpreted as the SU(5) charge conservation. The Dynkin Diagram symmetry of the SU(5) Diagram can be interpreted as a kind of particle-antiparticle symmetry. The Voronoi cell of the root lattice consists of the union of the polytopes (1000)A4 + (0100)A4 + (0010)A4 + (0001)A4 whose facets are 20 rhombohedra. We construct the Delone (Delaunay) cells of the root lattice as the alternating 5-cell and the rectified 5-cell, a kind of dual to the Voronoi cell. The vertices of the Delone cells closest to the origin consist of the root vectors representing the gauge bosons. The faces of the rhombohedra project onto the Coxeter plane as thick and thin rhombs leading to Penrose-like ti...
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su 5 grand unified theory its polytopes and 5 fold symmetric aperiodic tiling
arXiv: General Physics, 2016Co-Authors: Mehmet Koca, Nazife Ozdes Koca, Abeer Al SiyabiAbstract:We associate the lepton-quark families with the vertices of the 4D polytopes 5-cell and the rectified 5-cell derived from the SU(5) Coxeter-Dynkin Diagram. The off-diagonal gauge bosons are associated with the root poytope (1000)A4 whose facets are tetrahedra and the triangular prisms. The edge-vertex relations are interpreted as the SU(5) charge conservation. The Dynkin Diagram symmetry of the SU(5) Diagram can be interpreted as a kind of particle-antiparticle symmetry. The Voronoi cell of the root lattice consists of the union of the polytopes (1000)A4 + (0100)A4 + (0010)A4 + (0001)A4 whose facets are 20 rhombohedra. We construct the Delone (Delaunay) cells of the root lattice as the alternating 5-cell and the rectified 5-cell, a kind of dual to the Voronoi cell. The vertices of the Delone cells closest to the origin consists of the root vectors representing the gauge bosons. The faces of the rhombohedra project onto the Coxeter plane as thick and thin rhombs leading to Penrose-like tiling of the plane which can be used for the description of the 5-fold symmetric quasicrystallography. The model can be extended to SO(10) and even to SO(11) by noting the Coxeter-Dynkin Diagram embedding in A4 in D5 in B5. Another embedding can be made through the relation A4 in D5 in E6 for more popular GUT's.
Dmitri I. Panyushev - One of the best experts on this subject based on the ideXlab platform.
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Glorious pairs of roots and Abelian ideals of a Borel subalgebra
Journal of Algebraic Combinatorics, 2019Co-Authors: Dmitri I. PanyushevAbstract:Let $${{\mathfrak {g}}}$$ g be a simple Lie algebra with a Borel subalgebra $${{\mathfrak {b}}}$$ b . Let $$\Delta ^+$$ Δ + be the corresponding (po)set of positive roots and $$\theta $$ θ the highest root. A pair $$\{\eta ,\eta '\}\subset \Delta ^+$$ { η , η ′ } ⊂ Δ + is said to be glorious, if $$\eta ,\eta '$$ η , η ′ are incomparable and $$\eta +\eta '=\theta $$ η + η ′ = θ . Using the theory of abelian ideals of $${{\mathfrak {b}}}$$ b , we (1) establish a relationship of $$\eta ,\eta '$$ η , η ′ to certain abelian ideals associated with long simple roots, (2) provide a natural bijection between the glorious pairs and the pairs of adjacent long simple roots (i.e., some edges of the Dynkin Diagram), and (3) point out a simple transform connecting two glorious pairs corresponding to the incident edges in the Dynkin Diagram. In types $${{\mathbf {\mathsf{{{DE}}}}}}_{}$$ DE , we prove that if $$\{\eta ,\eta '\}$$ { η , η ′ } corresponds to the edge through the branching node of the Dynkin Diagram, then the meet $$\eta \wedge \eta '$$ η ∧ η ′ is the unique maximal non-commutative root. There is also an analogue of this property for all other types except type $${{\mathbf {\mathsf{{{A}}}}}}_{}$$ A . As an application, we describe the minimal non-abelian ideals of $${{\mathfrak {b}}}$$ b .
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on seaweed subalgebras and meander graphs in type c
Pacific Journal of Mathematics, 2016Co-Authors: Dmitri I. Panyushev, Oksana S YakimovaAbstract:Abstract In 2000, Dergachev and Kirillov introduced subalgebras of “seaweed type” in gl n and computed their index using certain graphs, which we call type- A meander graphs. Then the subalgebras of seaweed type, or just “seaweeds”, have been defined by Panyushev (2001) [9] for arbitrary reductive Lie algebras. Recently, a meander graph approach to computing the index in types B and C has been developed by the authors. In this article, we consider the most difficult and interesting case of type D . Some new phenomena occurring here are related to the fact that the Dynkin Diagram has a branching node.
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Abelian ideals of a Borel subalgebra and subsets of the Dynkin Diagram
Journal of Algebra, 2011Co-Authors: Dmitri I. PanyushevAbstract:Let g be a simple Lie algebra and Ab(g) the set of abelian ideals of a Borel subalgebra of g. In this note, an interesting connection between Ab(g) and the subsets of the Dynkin Diagram of g is discussed. We notice that the number of abelian ideals with k generators equals the number of subsets of the Dynkin Diagram with k connected components. For g of type An or Cn, we provide a combinatorial explanation of this coincidence by constructing a suitable bijection. We also construct a general bijection between Ab(g) and the subsets of the Dynkin Diagram, which is based on the theory developed by Peterson and Kostant.
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Abelian ideals of a Borel subalgebra and subsets of the Dynkin Diagram
arXiv: Combinatorics, 2010Co-Authors: Dmitri I. PanyushevAbstract:Let $g$ be a simple Lie algebra and $Ab(g)$ the set of Abelian ideals of a Borel subalgebra of $g$. In this note, an interesting connection between $Ab(g)$ and the subsets of the Dynkin Diagram of $g$ is discussed. We notice that the number of abelian ideals with $k$ generators equals the number of subsets of the Dynkin Diagram with $k$ connected components. For $g$ of type $A_n$ or $C_n$, we provide a combinatorial explanation of this coincidence by constructing a suitable bijection. We also construct another general bijection between $Ab(g)$ and the subsets of the Dynkin Diagram, which is based on the theory developed by Peterson and Kostant.
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On divisible weighted Dynkin Diagrams and reachable elements
Transformation Groups, 2010Co-Authors: Dmitri I. PanyushevAbstract:Let e be a nilpotent element of a complex simple Lie algebra \( \mathfrak{g} \). The weighted Dynkin Diagram of e, \( \mathcal{D}(e) \), is said to be divisible if \( {{{\mathcal{D}(e)}} \left/ {2} \right.} \) is again a weighted Dynkin Diagram. The corresponding pair of nilpotent orbits is said to be friendly. In this paper we classify the friendly pairs and describe some of their properties. Any subalgebra \( \mathfrak{s}{\mathfrak{l}_3} \) in \( \mathfrak{g} \) gives rise to a friendly pair; such pairs are called A2-pairs. If Gx is the lower orbit in an A2-pair, then \( x \in \left[ {{\mathfrak{g}^x},{\mathfrak{g}^x}} \right] \), i.e., x is reachable. We also show that \( {\mathfrak{g}^x} \) has other interesting properties. Let \( {\mathfrak{g}^x} = { \oplus_{i \geqslant 0}}{\mathfrak{g}^x}(i) \) be the \( \mathbb{Z} - {\text{grading}} \) determined by a characteristic of x. We prove that \( {\mathfrak{g}^x} \) is generated by the Levi subalgebra \( {\mathfrak{g}^x}(0) \) and two elements of \( {\mathfrak{g}^x}(1) \). In particular, the nilpotent radical of \( {\mathfrak{g}^x} \) is generated by the subspace \( {\mathfrak{g}^x}(1) \).
