Tensor powers for non-simply laced Lie algebras<i>B</i><sub>2</sub>-case

Kulish, P. P.; Lyakhovsky, V. D.; Postnova, Olga

doi:10.1088/1742-6596/346/1/012012

Cited by 10 publications

(12 citation statements)

References 13 publications

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“…It is clear that it cannot be factored into monomials, and we therefore believe that it cannot be obtained by the methods we previously developed in [1], [2], [7], [8]. The methods proposed by other authors (see [9]- [11] in particular) are based on summing over partitions of numbers, which makes the calculations more complicated and does not permit obtaining an explicit expression.…”

Section: Vector Fundamental So(5) Modulementioning

confidence: 99%

Generalized Pascal’s triangles and singular elements of modules of Lie algebras

Lyakhovsky

Postnova

2015

Theor Math Phys

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We consider the problem of determining the multiplicity function m ⊗ p ω ξ in the tensor power decomposition of a module of a semisimple algebra g into irreducible submodules. For this, we propose to pass to the corresponding decomposition of a singular element Ψ((L ω g ) ⊗p ) of the module tensor power into singular elements of irreducible submodules and formulate the problem of determining the function M ⊗ p ω ξ . This function satisfies a system of recurrence relations that corresponds to the procedure for multiplying modules. To solve this problem, we introduce a special combinatorial object, a generalized (g, ω) pyramid, i.e., a set of numbers (p, {mi})g,ω satisfying the same system of recurrence relations. We prove that M ⊗ p ω ξ can be represented as a linear combination of the corresponding (p, {mi})g,ω. We illustrate the obtained solution with several examples of modules of the algebras sl(3) and so(5).

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Section: Vector Fundamental So(5) Modulementioning

confidence: 99%

Generalized Pascal’s triangles and singular elements of modules of Lie algebras

Lyakhovsky

Postnova

2015

Theor Math Phys

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“…После упрощения окончательно имеем Известно, что оно не распадается на мономы, поэтому получить это выражение методами, сформулированными нами в работах [1], [2], [7], [8], представлялось невозможным. Методы других авторов (см., в частности, [9]- [11]) предполагают суммирования по разбиениям чисел, что затрудняет вычисления и не позволяет получить явного выражения.…”

Section: алгебра So(5)unclassified

“…Можно уточнить пределы суммирования, тогда окончательно получим Это выражение совпадает с полиномом, полученным в работе [8] рекуррентным образом.…”

Section: алгебра So(5)unclassified

Обобщенные Треугольники Паскаля И Сингулярные Элементы Модулей Алгебр Ли

Ляховский¹,

Lyakhovsky²,

Postnova³

2015

ТМФ

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Рассматривается задача нахождения функции кратности m ⊗ p ω ξ в разложении тензорной степени модуля полупростой алгебры g на неприводимые подмодули. Предложен переход к соответствующему разложению сингулярного элемента тензорной степени модуля Ψ((L ω g) ⊗p) на сингулярные элементы неприводимых подмодулей и сформулирована задача нахождения M ⊗ p ω ξ. Функция M ⊗ p ω ξ удовлетворяет системе рекуррентных соотношений, которые соответствуют процедуре перемножения модулей. Для решения поставленной задачи вводится специальный комбинаторный объект-oбобщенная (g, ω)-пирамида-массив чисел (p, {mi})g,ω, удовлетворяющих той же системе рекуррентных соотношений. Доказано, что M ⊗ p ω ξ представляется в виде линейной комбинации соответствующих (p, {mi})g,ω. Полученное решение иллюстрируется на примерах модулей алгебр sl(3) и so(5). Ключевые слова: теория представлений алгебр Ли, тензорное произведение модулей, формула Вейля.

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“…For the fundamental representations of simple Lie algebras it is sometimes possible to get an analytic formula for the dependence of the decomposition coefficients on N (See [32]). Our code provides the numerical values and can be used to check the analytic results.…”

Section: Tensor Product Decompositon For Finite-dimensional Lie Algebrasmentioning

confidence: 99%

Affine.m—Mathematica package for computations in representation theory of finite-dimensional and affine Lie algebras

Nazarov¹

2012

Computer Physics Communications

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In this paper we present Affine.m -a program for computations in representation theory of finite-dimensional and affine Lie algebras and describe implemented algorithms. The algorithms are based on the properties of weights and Weyl symmetry. Computation of weight multiplicities in irreducible and Verma modules, branching of representations and tensor product decomposition are the most important problems for us. These problems have numerous applications in physics and we provide some examples of these applications. The program is implemented in the popular computer algebra system Mathematica and works with finite-dimensional and affine Lie algebras. Keywords:Mathematica; Lie algebra; affine Lie algebra; Kac-Moody algebra; root system; weights; irreducible modules, CFT, Integrable systems Representation theory of finite-dimensional Lie algebras has many applications in different branches of physics, including elementary particle physics, molecular physics, nuclear physics. Representations of affine Lie algebras appear in string theories and two-dimensional conformal field theory used for the description of critical phenomena in two-dimensional systems. Also Lie symmetries play major role in a study of quantum integrable systems. PROGRAM SUMMARY Solution method:We work with weights and roots of finite-dimensional and affine Lie algebras and use Weyl symmetry extensively. Central problems which are the computations of weight multiplicities, branching and fusion coefficients are solved using one general recurrent algorithm based on generalization of Weyl character formula. We also offer alternative implementation based on the Freudenthal multiplicity formula which can be faster in some cases. Restrictions:Computational complexity grows fast with the rank of an algebra, so computations for algebras of ranks greater than 8 are not practical. Unusual features:We offer the possibility to use a traditional mathematical notation for the objects in representation theory of Lie algebras in computations if Affine.m is used in the Mathematica notebook interface. Running time:From seconds to days depending on the rank of the algebra and the complexity of the representation.2

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Tensor powers for non-simply laced Lie algebrasB₂-case

Cited by 10 publications

References 13 publications

Generalized Pascal’s triangles and singular elements of modules of Lie algebras

Generalized Pascal’s triangles and singular elements of modules of Lie algebras

Обобщенные Треугольники Паскаля И Сингулярные Элементы Модулей Алгебр Ли

Affine.m—Mathematica package for computations in representation theory of finite-dimensional and affine Lie algebras

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Tensor powers for non-simply laced Lie algebrasB2-case

Cited by 10 publications

References 13 publications

Generalized Pascal’s triangles and singular elements of modules of Lie algebras

Generalized Pascal’s triangles and singular elements of modules of Lie algebras

Обобщенные Треугольники Паскаля И Сингулярные Элементы Модулей Алгебр Ли

Affine.m—Mathematica package for computations in representation theory of finite-dimensional and affine Lie algebras

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Product

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Tensor powers for non-simply laced Lie algebrasB₂-case