Yangian of the Strange Lie Superalgebra of Q<sub>n-1</sub> Type, Drinfel'd Approach

Stukopin, Vladimir

doi:10.3842/sigma.2007.069

Cited by 6 publications

(4 citation statements)

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“…За рамками работы остался вопрос о геометризации теории представлений суперянгианов и квантовых аффинных супералгебр (см. [20]), а также вопрос о распространении результатов настоящей работы на случай скрученных янгианов [21]- [23] и скрученных квантовых петлевых супералгебр.…”

Section: Introductionunclassified

Relation between categories of representations of the super-Yangian of a special linear Lie superalgebra and quantum loop superalgebra

Stukopin¹

2020

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С использованием подхода Гаутама и Толедано-Ларедо вводятся аналоги категории $\mathfrak O$ для представлений янгиана $Y_\hbar(A(m,n))$ специальной линейной супералгебры Ли и квантовой петлевой супералгебры $U_q(LA(m,n))$. Исследуется связь между ними и сформулирована гипотеза об эквивалентности этих категорий.

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Section: Introductionunclassified

Relation between categories of representations of the super-Yangian of a special linear Lie superalgebra and quantum loop superalgebra

Stukopin¹

2020

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“…При определении янгиана в соответствии с подходом Дринфельда требуется наличие на исходной алгебре (или супералгебре) Ли невырожденной инвариантной билинейной формы Киллинга, которая отсутствует на некоторых супералгебрах Ли, что мешает буквальному перенесению конструкции Дринфельда на этот случай (см. статьи [14], [17], [18]). С другой стороны, по этой же причине для скрученных алгебр токов не существуют янгианные деформации в классе алгебр Хопфа, но они существуют для скрученных петлевых алгебр, что приводит к скрученным квантовым петлевым алгебрам или скрученным квантовым аффинным алгебрам.…”

Section: Introductionunclassified

Об изоморфизме янгиана $Y_{\hbar}(A(m,n))$ специальной линейной супералгебры Ли и квантовой петлевой супералгебры $U_{\hbar}(LA(m,n))$

Stukopin¹

2019

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“…Gorelik [6] obtained the Shapovalov determinants of Q-type Lie superalgebras. Stukopin [22] studied the Yangians of the strange Lie superalgebra of type Q by Drinfel'd approach. Gorelik and Serganova [7] investigated the structure of Verma modules over the twisted affine Lie superalgebra q(n) (2) .…”

Section: Introductionmentioning

confidence: 99%

On polynomial representations of strange Lie superalgebras of Q-type

Luo

2016

Acta. Math. Sin.-English Ser.

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In this paper, various polynomial representations of strange classical Lie superalgebras are investigated. It turns out that the representations for the algebras of type P are indecomposable, and we obtain the composition series of the underlying modules. As modules of the algebras of type Q , the polynomial algebras are decomposed into a direct sum of irreducible submodules. superalgebras have attracted a number of mathematicians' attention. Javis and Murray [9] obtained the Casimir invariants, characteristic identities, and tensor operators for the strange Lie superalgebras. Moreover, Nazarov [11] found Yangians of the superalgebras. In [2], Frappat, Sciarrino and Sorba studied Dynkin-like diagrams and a certain representation of the strange superalgebra P (n). In addition, they [3] gave a certain oscillator realization of the strange superalgebras. Medak [13] generalized the Baker-Campbell-Hausdorff formula and used it to examine the so-called BCH-Lie and BCH-invertible subalgebras in the Lie superalgebra P (n). Penkov and Serganova [20] discovered a surprising phenomena that the multiplicity of the highest weight in the finite-dimensional irreducible representations of q(n) is in general greater than 1. Gruson [8] computed the cohomology with trivial coefficients for the strange Lie superalgebras. Palev and Van der Jeugt [19] found a family of nongraded Fock representations of the Lie superalgebra Q(n). Gorelik [5] obtained the center of the universal enveloping algebra of the Lie superalgebra of type P . Serganova [21] determined the center of the quotient algebra of the universal enveloping algebra of the Lie superalgebra of type P by its Jacobson radical and used it to study the typical highest weight modules of the algebra. Medak [14] proved that each maximal invertible subalgebra of P (n) is Z-graded.Moon [15] obtained a "Schur-Weyl duality" for the algebras of type P . Martinez and Zelmanov [12] classified Lie superalgebras graded by P (n) and Q(n). Brundan [1] found a connection between Kazhdan-Lusztig polynomials and character formulas for the Lie superalgebra q(n). Gorelik [6] obtained the Shapovalov determinants of Q-type Lie superalgebras. Stukopin [22] studied the Yangians of the strange Lie superalgebra of type Q by Drinfel'd approach. Gorelik and Serganova [7] investigated the structure of Verma modules over the twisted affine Lie superalgebra q(n) (2) .One way of understanding simple Lie algebras and simple Lie superalgebras is to determine the structure of their natural representations. Canonical polynomial irreducible representations (also known as oscillator representations in physics (e.g., cf. [4])) of finitedimensional simple Lie algebras are very important from application point of view, where both the representation formulas and bases are clear. In [10], we determined the structure of certain noncanonical polynomial representations of classical simple Lie algebras, in particular, their irreducible submodules. In this paper, we want to generalize the above results to the strange simple ...

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Yangian of the Strange Lie Superalgebra of Q_n-1 Type, Drinfel'd Approach

Abstract: Abstract. The Yangian of the strange Lie superalgebras in Drinfel'd realization is defined. The current system generators and defining relations are described.

Cited by 6 publications

References 15 publications

Relation between categories of representations of the super-Yangian of a special linear Lie superalgebra and quantum loop superalgebra

Relation between categories of representations of the super-Yangian of a special linear Lie superalgebra and quantum loop superalgebra

Об изоморфизме янгиана $Y_{\hbar}(A(m,n))$ специальной линейной супералгебры Ли и квантовой петлевой супералгебры $U_{\hbar}(LA(m,n))$

On polynomial representations of strange Lie superalgebras of Q-type

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Yangian of the Strange Lie Superalgebra of Qn-1 Type, Drinfel'd Approach

Abstract: Abstract. The Yangian of the strange Lie superalgebras in Drinfel'd realization is defined. The current system generators and defining relations are described.

Cited by 6 publications

References 15 publications

Relation between categories of representations of the super-Yangian of a special linear Lie superalgebra and quantum loop superalgebra

Relation between categories of representations of the super-Yangian of a special linear Lie superalgebra and quantum loop superalgebra

Об изоморфизме янгиана $Y_{\hbar}(A(m,n))$ специальной линейной супералгебры Ли и квантовой петлевой супералгебры $U_{\hbar}(LA(m,n))$

On polynomial representations of strange Lie superalgebras of Q-type

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Yangian of the Strange Lie Superalgebra of Q_n-1 Type, Drinfel'd Approach