scholarly journals On the Gaudin model of type G2

2019 ◽  
Vol 21 (03) ◽  
pp. 1850012
Author(s):  
Kang Lu ◽  
Evgeny Mukhin
Keyword(s):  
Tensor Product ◽  
Bethe Ansatz ◽  
Irreducible Module ◽  
The Self ◽  
Gaudin Model ◽  
Bijective Correspondence ◽  
Dual Spaces ◽  
Finite Dimensional ◽  
Vector Representations ◽  
Evaluation Parameters

We derive a number of results related to the Gaudin model associated to the simple Lie algebra of type G2. We compute explicit formulas for solutions of the Bethe ansatz equations associated to the tensor product of an arbitrary finite-dimensional irreducible module and the vector representation. We use this result to show that the Bethe ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We show that the points of the spectrum of the Gaudin model in type G2 are in a bijective correspondence with self-self-dual spaces of polynomials. We study the set of all self-self-dual spaces — the self-self-dual Grassmannian. We establish a stratification of the self-self-dual Grassmannian with the strata labeled by unordered sets of dominant integral weights and unordered sets of nonnegative integers, satisfying certain explicit conditions. We describe closures of the strata in terms of representation theory.

scholarly journals Tensor Products and Bimorphisms

1976 ◽  
Vol 19 (4) ◽  
pp. 385-402 ◽  
Author(s):  
Bernhard Banaschewski ◽  
Evelyn Nelson
Keyword(s):  
Tensor Product ◽  
Commutative Ring ◽  
Tensor Products ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Bilinear Maps ◽  
Special Situation ◽  
Dual Spaces ◽  
Finite Dimensional

The binary tensor product, for modules over a commutative ring, has two different aspects: its connection with universal bilinear maps and its adjointness to the internal hom-functor. Furthermore, in the special situation of finite-dimensional vector spaces, the tensor product can also be described in terms of dual spaces and the internal hom-functor. The aim of this paper is to investigate these relationships in the setting of arbitrary concrete categories.

scholarly journals ON SEPARATION OF VARIABLES AND COMPLETENESS OF THE BETHE ANSATZ FOR QUANTUM N GAUDIN MODEL

2009 ◽  
Vol 51 (A) ◽  
pp. 137-145 ◽  
Author(s):  
E. MUKHIN ◽  
V. TARASOV ◽  
A. VARCHENKO
Keyword(s):  
Differential Operator ◽  
Bethe Ansatz ◽  
Differential Operators ◽  
Separation Of Variables ◽  
Polynomial Kernel ◽  
Simple Spectrum ◽  
Gaudin Model ◽  
Bethe Algebra ◽  
One To One ◽  
Evaluation Parameters

AbstractIn this paper, we discuss implications of the results obtained in [5]. It was shown there that eigenvectors of the Bethe algebra of the quantum N Gaudin model are in a one-to-one correspondence with Fuchsian differential operators with polynomial kernel. Here, we interpret this fact as a separation of variables in the N Gaudin model. Having a Fuchsian differential operator with polynomial kernel, we construct the corresponding eigenvector of the Bethe algebra. It was shown in [5] that the Bethe algebra has simple spectrum if the evaluation parameters of the Gaudin model are generic. In that case, our Bethe ansatz construction produces an eigenbasis of the Bethe algebra.

scholarly journals An Algorithm to Compute the Canonical Basis of an Irreducible Module Over a Quantized Enveloping Algebra

2003 ◽  
Vol 6 ◽  
pp. 105-118 ◽  
Author(s):  
Willem A. de Graaf
Keyword(s):  
Tensor Product ◽  
Lie Algebra ◽  
Canonical Basis ◽  
Irreducible Module ◽  
Semisimple Lie Algebra ◽  
Canonical Bases ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Quantized Enveloping Algebra

AbstractThe paper describes an algorithm to compute the canonical basis of an irreducible module over a quantized enveloping algebra of a finite-dimensional semisimple Lie algebra. The algorithm works for any module that is constructed as a submodule of a tensor product of modules with known canonical bases.

scholarly journals Nuclear subalgebras of UHF C*-algebras

1986 ◽  
Vol 29 (1) ◽  
pp. 97-100 ◽  
Author(s):  
R. J. Archbold ◽  
Alexander Kumjian
Keyword(s):  
Tensor Product ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
The Family ◽  
Algebraic Tensor Product

A C*-algebra A is said to be approximately finite dimensional (AF) if it is the inductive limit of a sequence of finite dimensional C*-algebras(see [2], [5]). It is said to be nuclear if, for each C*-algebra B, there is a unique C*-norm on the *-algebraic tensor product A ⊗B [11]. Since finite dimensional C*-algebras are nuclear, and inductive limits of nuclear C*-algebras are nuclear [16];,every AF C*-algebra is nuclear. The family of nuclear C*-algebras is a large and well-behaved class (see [12]). The AF C*-algebras for a particularly tractable sub-class which has been completely classified in terms of the invariant K0 [7], [5].

scholarly journals Quantum relativistic Toda chain at root of unity: isospectrality, modifiedQ-operator, and functional Bethe ansatz

2002 ◽  
Vol 31 (9) ◽  
pp. 513-553 ◽  
Author(s):  
Stanislav Pakuliak ◽  
Sergei Sergeev
Keyword(s):  
Bethe Ansatz ◽  
Weyl Algebra ◽  
Separation Of Variables ◽  
Toda Chain ◽  
Special Choice ◽  
Finite Dimensional Representation ◽  
Classical Counterpart ◽  
Discrete Integrable System ◽  
Root Of Unity ◽  
Finite Dimensional

We investigate anN-state spin model called quantum relativistic Toda chain and based on the unitary finite-dimensional representations of the Weyl algebra withqbeingNth primitive root of unity. Parameters of the finite-dimensional representation of the local Weyl algebra form the classical discrete integrable system. Nontrivial dynamics of the classical counterpart corresponds to isospectral transformations of the spin system. Similarity operators are constructed with the help of modified Baxter'sQ-operators. The classical counterpart of the modifiedQ-operator for the initial homogeneous spin chain is a Bäcklund transformation. This transformation creates an extra Hirota-type soliton in a parameterization of the chain structure. Special choice of values of solitonic amplitudes yields a degeneration of spin eigenstates, leading to the quantum separation of variables, or the functional Bethe ansatz. A projector to the separated eigenstates is constructed explicitly as a product of modifiedQ-operators.

scholarly journals Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1737
Author(s):  
Mariia Myronova ◽  
Jiří Patera ◽  
Marzena Szajewska
Keyword(s):  
Tensor Product ◽  
Lie Algebras ◽  
Coxeter Groups ◽  
Reflection Groups ◽  
Geometrical Structures ◽  
Simple Lie Algebras ◽  
Finite Dimensional ◽  
Branching Rules ◽  
Definition Of ◽  
Representation Orbit

The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups H2, H3 and H4. Using a representation-orbit replacement, the definitions and properties of the indices are formulated for individual orbits of the examined groups. The indices of orders two and four of the tensor product of k orbits are determined. Using the branching rules for the non-crystallographic Coxeter groups, the embedding index is defined similarly to the Dynkin index of a representation. Moreover, since the definition of the indices can be applied to any orbit of non-crystallographic type, the algorithm allowing to search for the orbits of smaller radii contained within any considered one is presented for the Coxeter groups H2 and H3. The geometrical structures of nested polytopes are exemplified.

scholarly journals Combined Self-Attention Mechanism for Chinese Named Entity Recognition in Military

2019 ◽  
Vol 11 (8) ◽  
pp. 180
Author(s):  
Fei Liao ◽  
Liangli Ma ◽  
Jingjing Pei ◽  
Linshan Tan
Keyword(s):  
Domain Knowledge ◽  
Short Term Memory ◽  
Contextual Information ◽  
Named Entity Recognition ◽  
Attention Mechanism ◽  
The Self ◽  
Entity Recognition ◽  
Named Entity ◽  
Vector Representations ◽  
The Military

Military named entity recognition (MNER) is one of the key technologies in military information extraction. Traditional methods for the MNER task rely on cumbersome feature engineering and specialized domain knowledge. In order to solve this problem, we propose a method employing a bidirectional long short-term memory (BiLSTM) neural network with a self-attention mechanism to identify the military entities automatically. We obtain distributed vector representations of the military corpus by unsupervised learning and the BiLSTM model combined with the self-attention mechanism is adopted to capture contextual information fully carried by the character vector sequence. The experimental results show that the self-attention mechanism can improve effectively the performance of MNER task. The F-score of the military documents and network military texts identification was 90.15% and 89.34%, respectively, which was better than other models.

scholarly journals THE SET OF QUANTUM CORRELATIONS IS NOT CLOSED

2019 ◽  
Vol 7 ◽  
Author(s):  
WILLIAM SLOFSTRA
Keyword(s):  
Hilbert Space ◽  
Linear System ◽  
Tensor Product ◽  
Quantum Correlations ◽  
Embedding Theorem ◽  
Product Strategy ◽  
Finite Dimensional ◽  
Universal Embedding ◽  
Quantum Strategies ◽  
Finite Dimensional Hilbert Space

We construct a linear system nonlocal game which can be played perfectly using a limit of finite-dimensional quantum strategies, but which cannot be played perfectly on any finite-dimensional Hilbert space, or even with any tensor-product strategy. In particular, this shows that the set of (tensor-product) quantum correlations is not closed. The constructed nonlocal game provides another counterexample to the ‘middle’ Tsirelson problem, with a shorter proof than our previous paper (though at the loss of the universal embedding theorem). We also show that it is undecidable to determine if a linear system game can be played perfectly with a finite-dimensional strategy, or a limit of finite-dimensional quantum strategies.

scholarly journals Projective geometry in characteristic one and the epicyclic category

2015 ◽  
Vol 217 ◽  
pp. 95-132
Author(s):  
Alain Connes ◽  
Caterina Consani
Keyword(s):  
Projective Geometry ◽  
Free Module ◽  
The Self ◽  
Vector Spaces ◽  
Geometric Meaning ◽  
Absolute Point ◽  
One Dimensional ◽  
Self Duality ◽  
Finite Dimensional ◽  
The One

AbstractWe show that the cyclic and epicyclic categories which play a key role in the encoding of cyclic homology and the lambda operations, are obtained from projective geometry in characteristic one over the infinite semifield ofmax-plus integersℤmax. Finite-dimensional vector spaces are replaced by modules defined by restriction of scalars from the one-dimensional free module, using the Frobenius endomorphisms of ℤmax. The associated projective spaces arefiniteand provide a mathematically consistent interpretation of Tits's original idea of a geometry over the absolute point. The self-duality of the cyclic category and the cyclic descent number of permutations both acquire a geometric meaning.

Sign in / Sign up

Export Citation Format

Share Document