1,860 results on '"Lie superalgebra"'
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2. S-Embedding of Lie Superalgebras and Its Implications for Fuzzy Lie Algebras
- Author
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Abdullah Assiry, Sabeur Mansour, and Amir Baklouti
- Subjects
Lie superalgebra ,s-embedding ,central charge ,fuzzy Lie algebras ,canonical central extension ,Mathematics ,QA1-939 - Abstract
This paper performed an investigation into the s-embedding of the Lie superalgebra (→S1∣1), a representation of smooth vector fields on a (1,1)-dimensional super-circle. Our primary objective was to establish a precise definition of the s-embedding, effectively dissecting the Lie superalgebra into the superalgebra of super-pseudodifferential operators ( SψD⊙) residing on the super-circle S1|1. We also introduce and rigorously define the central charge within the framework of (→S1∣1), leveraging the canonical central extension of SψD⊙. Moreover, we expanded the scope of our inquiry to encompass the domain of fuzzy Lie algebras, seeking to elucidate potential connections and parallels between these ostensibly distinct mathematical constructs. Our exploration spanned various facets, including non-commutative structures, representation theory, central extensions, and central charges, as we aimed to bridge the gap between Lie superalgebras and fuzzy Lie algebras. To summarize, this paper is a pioneering work with two pivotal contributions. Initially, a meticulous definition of the s-embedding of the Lie superalgebra (→S1|1) is provided, emphasizing the representationof smooth vector fields on the (1,1)-dimensional super-circle, thereby enriching a fundamental comprehension of the topic. Moreover, an investigation of the realm of fuzzy Lie algebras was undertaken, probing associations with conventional Lie superalgebras. Capitalizing on these discoveries, we expound upon the nexus between central extensions and provide a novel deformed representation of the central charge.
- Published
- 2023
- Full Text
- View/download PDF
3. aff(1|1)-trivial deformations of aff(2|1)-modules of weighted densities on the superspace
- Author
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Ismail Laraiedh
- Subjects
Relative cohomology ,trivial deformation ,Lie superalgebra ,symbol ,Mathematics ,QA1-939 - Abstract
Over the (1|2)-dimensional real superspace, we study aff(1|1)-trivial deformations of the action of the affine Lie superalgebra aff(2|1) on the direct sum of the superspaces of weighted densities. We compute the necessary and sufficient integrability conditions of a given infinitesimal deformation of this action and we prove that any formal deformation is equivalent to its infinitisemal part.
- Published
- 2019
4. Some studies on central derivation of nilpotent Lie superalgebra.
- Author
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Padhan, Rudra Narayan and Pati, K. C.
- Subjects
LIE algebras ,SUPERALGEBRAS ,LIE superalgebras ,ALGEBRA ,MATHEMATICS - Abstract
Many theorems and formulas of Lie superalgebras run quite parallel to Lie algebras, sometimes giving interesting results. So it is quite natural to extend the new concepts of Lie algebra immediately to Lie superalgebra case as the later type of algebras have wide applications in physics and related theories. Using the concept of isoclinism, Saeedi and Sheikh-Mohseni [A characterization of stem algebras in terms of central derivations, Algebr. Represent. Theory20 (2017) 1143–1150; On ID ∗ -derivations of Filippov algebra, to appear in Asian-Eur. J. Math.; S. Sheikh-Mohseni, F. Saeedi and M. Badrkhani Asl, On special subalgebras of derivations of Lie algebras, Asian-Eur. J. Math.8(2) (2015) 1550032] recently studied the central derivation of nilpotent Lie algebra with nilindex 2. The purpose of the present paper is to continue and extend the investigation to obtain some similar results for Lie superalgebras, as isoclinism in Lie superalgebra is being recently introduced. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
5. A family of simple modules over the Rueda's algebras
- Author
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Longhui Wang and Hongjia Chen
- Subjects
Pure mathematics ,Class (set theory) ,Algebra and Number Theory ,Rank (linear algebra) ,Simple (abstract algebra) ,Lie algebra ,Universal enveloping algebra ,Lie superalgebra ,Isomorphism ,Simple module ,Mathematics - Abstract
The Rueda's algebras R ( f , ς ) (simply R) are a class of algebras similar to the universal enveloping algebra of sl 2 . We study the category H of R-modules whose objects are free of rank 1 when restricted to R 0 = C [ H ] . We classify the isomorphism classes of objects in H and determine the simplicity of these modules. As a result, we also give an explicit description of submodule structures and obtain new simple non-weight modules over R. In particular, we recover some results about U ( h ) -free modules over the Lie algebra sl 2 obtained by J. Nilsson and over the Lie superalgebra osp ( 1 | 2 ) obtained by Y. Cai and K. Zhao.
- Published
- 2022
6. Kirillov–Reshetikhin Modules of Generalized Quantum Groups of Type $A$
- Author
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Jae-Hoon Kwon and Masato Okado
- Subjects
Pure mathematics ,Quantum group ,General Mathematics ,Lie superalgebra ,Type (model theory) ,Quantum ,Crystal base ,Mathematics - Published
- 2021
7. Classification of Simple Lie Superalgebras in Characteristic 2
- Author
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Dimitry Leites, Alexei Lebedev, Sofiane Bouarroudj, and Irina Shchepochkina
- Subjects
symbols.namesake ,Pure mathematics ,Series (mathematics) ,Simple (abstract algebra) ,General Mathematics ,Modulo ,Lie algebra ,symbols ,Lie superalgebra ,Vector field ,Hamiltonian (quantum mechanics) ,Vector space ,Mathematics - Abstract
All results concern characteristic 2. Two procedures that to every simple Lie algebra assign simple Lie superalgebras, most of the latter new, are offered. We prove that every simple finite-dimensional Lie superalgebra is obtained as the result of one of these procedures, so we classified all simple finite-dimensional Lie superalgebras modulo non-existing at the moment classification of simple finite-dimensional Lie algebras. This result concerns Lie superalgebras considered naively, as vector spaces. To obtain classification of simple Lie superalgebras in the category of supervarieties, one should list the non-isomorphic deforms (results of deformations) with odd parameter. This problem is open bar several examples described in arXiv~0807.3054. For Lie algebras, in addition to the known ---"classical" --- restrictedness, we introduce a (2,4)-structure on the two non-alternating series: orthogonal and of Hamiltonian vector fields. For Lie superalgebras, the classical restrictedness of Lie algebras has two analogs: $(2|4)$- and $(2|2)$-structures, one more analog --- a $(2,4)|4$-structure on Lie superalgebras is the analog of (2,4)-structure on Lie algebras.
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- 2021
8. Constructions and Properties for a Finite-Dimensional Modular Lie Superalgebra <math xmlns='http://www.w3.org/1998/Math/MathML' id='M1'> <mi>K</mi> <mfenced open='(' close=')' separators='|'> <mrow> <mi>n</mi> <mo>,</mo> <mi>m</mi> </mrow> </mfenced> </math>
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Keli Zheng and Dan Mao
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Pure mathematics ,Article Subject ,business.industry ,General Mathematics ,Mathematics::Rings and Algebras ,Subalgebra ,MathematicsofComputing_GENERAL ,Lie superalgebra ,Field (mathematics) ,Extension (predicate logic) ,Modular design ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Simple (abstract algebra) ,Mathematics::Quantum Algebra ,QA1-939 ,Prime characteristic ,Mathematics::Representation Theory ,business ,Mathematics - Abstract
In this paper, a finite-dimensional Lie superalgebra K n , m over a field of prime characteristic is constructed. Then, we study some properties of K n , m . Moreover, we prove that K n , m is an extension of a simple Lie superalgebra, and if m = n − 1 , then it is isomorphic to a subalgebra of a restricted Lie superalgebra.
- Published
- 2021
9. Superderivations of direct and semidirect sum of Lie superalgebras
- Author
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R. N. Padhan, N. Nandi, and K. C. Pati
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Pure mathematics ,Algebra and Number Theory ,Direct sum ,Generalization ,Mathematics::Rings and Algebras ,Dimension (graph theory) ,Lie algebra ,Structure (category theory) ,Lie superalgebra ,Mathematics - Abstract
It is well known that superderivation of a Lie superalgebra is certain generalization of derivation of a Lie algebra. This paper is devoted to investigate the structure and dimension of superderiva...
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- 2021
10. Projectively equivariant quantization and symbol on supercircle $S^{1|3}$
- Author
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Taher Bichr
- Subjects
Combinatorics ,Degree (graph theory) ,Tensor (intrinsic definition) ,Equivariant map ,Lie superalgebra ,Uniqueness ,Mathematics::Representation Theory ,Space (mathematics) ,Differential operator ,Lambda ,Mathematics - Abstract
Let $${{\cal D}_{\lambda ,\mu }}$$ be the space of linear differential operators on weighted densities from $${{\cal F}_\lambda }$$ to $${{\cal F}_\mu }$$ as module over the orthosymplectic Lie superalgebra $$\mathfrak{osp}(3\left| 2 \right.)$$ , where $${{\cal F}_\lambda }$$ , $$\lambda \in \mathbb{C}$$ is the space of tensor densities of degree λ on the supercircle S1∣3. We prove the existence and uniqueness of projectively equivariant quantization map from the space of symbols to the space of differential operators. An explicite expression of this map is also given.
- Published
- 2021
11. 2-Local superderivations on the Lie superalgebra of Block type
- Author
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Xiaoyu Zhu, Xiaoqing Yue, and Yucai Su
- Subjects
Pure mathematics ,Algebra and Number Theory ,Mathematics::Quantum Algebra ,Mathematics::Rings and Algebras ,Lie superalgebra ,Block type ,Mathematics::Representation Theory ,Mathematics - Abstract
In this paper, we study 2-local superderivations on the Lie superalgebra of Block type S(q), which is an infinite dimensional Lie superalgebra with an outer derivation. We prove that all 2-local su...
- Published
- 2021
12. 2-Local superderivations on the super Virasoro algebra and the super W(2,2) algebra
- Author
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Dong Liu, Munayim Dilxat, and Shoulan Gao
- Subjects
Algebra ,High Energy Physics::Theory ,Algebra and Number Theory ,Mathematics::Quantum Algebra ,Mathematics::Rings and Algebras ,Lie superalgebra ,Super Virasoro algebra ,Algebra over a field ,Mathematics - Abstract
The present paper is devoted to studying 2-local superderivations on the super Virasoro algebra and the super W(2,2) algebra. It is proved that all 2-local superderivations on the super Virasoro al...
- Published
- 2021
13. CAPELLI OPERATORS FOR SPHERICAL SUPERHARMONICS AND THE DOUGALL–RAMANUJAN IDENTITY
- Author
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Hadi Salmasian, Siddhartha Sahi, and Vera Serganova
- Subjects
Algebra and Number Theory ,010102 general mathematics ,Lie superalgebra ,Basis (universal algebra) ,01 natural sciences ,Ramanujan's sum ,Combinatorics ,symbols.namesake ,Hypergeometric identity ,Mathematics::Quantum Algebra ,0103 physical sciences ,symbols ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Invariant (mathematics) ,Mathematics::Representation Theory ,Indecomposable module ,Eigenvalues and eigenvectors ,Vector space ,Mathematics - Abstract
Let (V, ω) be an orthosymplectic ℤ2-graded vector space and let 𝔤:= 𝔤𝔬𝔰𝔭 (V, ω) denote the Lie superalgebra of similitudes of (V, ω). It is known that as a 𝔤-module, the space (V ) of superpolynomials on V is completely reducible, unless dim $$ {V}_{\overline{\mathrm{o}}} $$ and dim $$ {V}_{\overline{1}} $$ are positive even integers and dim $$ {V}_{\overline{\mathrm{O}}}\le \dim\ {V}_{\overline{1}} $$ . When (V ) is not a completely reducible 𝔤-module, we construct a natural basis $$ {\left\{{D}_{\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$ of “Capelli operators” for the algebra (V ) 𝔤 of 𝔤 -invariant superpolynomial superdifferential operators on V , where the index set 𝒯 is the set of integer partitions of length at most two. We compute the action of the operators $$ {\left\{{D}_{\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$ on maximal indecomposable components of (V ) explicitly, in terms of Knop–Sahi interpolation polynomials. Our results show that, unlike the cases where (V ) is completely reducible, the eigenvalues of a subfamily of the {D⋋} are not given by specializing the Knop–Sahi polynomials. Rather, the formulas for these eigenvalues involve suitably regularized forms of these polynomials. This is in contrast with what occurs for previously studied Capelli operators. In addition, we demonstrate a close relationship between our eigenvalue formulas for this subfamily of Capelli operators and the Dougall–Ramanujan hypergeometric identity. We also transcend our results on the eigenvalues of Capelli operators to the Deligne category Rep (Ot). More precisely, we define categorical Capelli operators $$ {\left\{{D}_{t,\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$ that induce morphisms of indecomposable components of symmetric powers of Vt, where Vt is the generating object of Rep (Ot). We obtain formulas for the eigenvalue polynomials associated to the $$ {\left\{{D}_{t,\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$ that are analogous to our results for the operators $$ {\left\{{D}_{\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$ .
- Published
- 2021
14. Fractal nil graded Lie, associative, Poisson, and Jordan superalgebras
- Author
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Ivan P. Shestakov and V. M. Petrogradsky
- Subjects
Poisson superalgebra ,Pure mathematics ,Algebra and Number Theory ,Mathematics::Rings and Algebras ,010102 general mathematics ,Graded ring ,SUPERÁLGEBRAS DE LIE ,Lie superalgebra ,Grigorchuk group ,01 natural sciences ,Superalgebra ,Restricted Lie algebra ,0103 physical sciences ,Lie algebra ,010307 mathematical physics ,0101 mathematics ,Group theory ,Mathematics - Abstract
The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The first author constructed their analogue in case of restricted Lie algebras of characteristic 2 [50] , Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic [68] . Thus, we have examples of finitely generated restricted Lie algebras with a nil p-mapping. In characteristic zero, similar examples of Lie and Jordan algebras do not exist by results of Martinez and Zelmanov [43] and [78] . The first author constructed analogues of the Grigorchuk and Gupta-Sidki groups in the world of Lie superalgebras of arbitrary characteristic, the virtue of that construction is that Lie superalgebras have clear monomial bases [51] , they have slow polynomial growth. As an analogue of periodicity, Z 2 -homogeneous elements are ad-nilpotent. A recent example of a Lie superalgebra is of linear growth, of finite width 4, just infinite but not hereditary just infinite [13] . By that examples, an extension of the result of Martinez and Zelmanov [43] for Lie superalgebras of characteristic zero is not valid. Now, we construct a just infinite fractal 3-generated Lie superalgebra Q over arbitrary field, which gives rise to an associative hull A, a Poisson superalgebra P, and two Jordan superalgebras J and K, the latter being a factor algebra of J. In case char K ≠ 2 , A has a natural filtration, which associated graded algebra has a structure of a Poisson superalgebra such that gr A ≅ P , also P admits an algebraic quantization using a deformed superalgebra A ( t ) . The Lie superalgebra Q is finely Z 3 -graded by multidegree in the generators, A, P are also Z 3 -graded, while J and K are Z 4 -graded by multidegree in four generators. By virtue of our construction, these five superalgebras have clear monomial bases and slow polynomial growth. We describe multihomogeneous coordinates of bases of Q, A, P in space as bounded by “almost cubic paraboloids”. We determine a similar hypersurface in R 4 that bounds monomials of J and K. Constructions of the paper can be applied to Lie (super)algebras studied before to obtain Poisson and Jordan superalgebras as well. The algebras Q, A, and the algebras without unit P o , J o , K o are direct sums of two locally nilpotent subalgebras and there are continuum such decompositions. Also, Q = Q 0 ¯ ⊕ Q 1 ¯ is a nil graded Lie superalgebra, so, Q again shows that an extension of the result of Martinez and Zelmanov for Lie superalgebras of characteristic zero is not valid. In case char K = 2 , Q has a structure of a restricted Lie algebra with a nil p-mapping. The Jordan superalgebra K is nil finely Z 4 -graded, in contrast with non-existence of such examples (roughly speaking, analogues of the Grigorchuk group) of Jordan algebras in characteristic distinct from 2 [78] . Also, K is of slow polynomial growth, just infinite, but not hereditary just infinite. We call the superalgebras Q, A, P, J, K fractal because they contain infinitely many copies of themselves.
- Published
- 2021
15. Simple weight modules with finite-dimensional weight spaces over Witt superalgebras
- Author
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Yaohui Xue and Rencai Lu
- Subjects
Pure mathematics ,Algebra and Number Theory ,Laurent polynomial ,Mathematics::Rings and Algebras ,010102 general mathematics ,Cartan subalgebra ,Lie superalgebra ,01 natural sciences ,Superalgebra ,Tensor product ,Mathematics::Quantum Algebra ,Tensor (intrinsic definition) ,17B10, 17B20, 17B65, 17B66, 17B68 ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics::Representation Theory ,Exterior algebra ,Mathematics - Representation Theory ,Quotient ,Mathematics - Abstract
Let A m , n be the tensor product of the Laurent polynomial algebra in m even variables and the exterior algebra in n odd variables over the complex field C , and the Witt superalgebra W m , n be the Lie superalgebra of superderivations of A m , n . In this paper, we classify the simple weight W m , n modules with finite-dimensional weight spaces with respect to the standard Cartan subalgebra of W m , 0 . Every such module is either a simple quotient of a tensor module or a module of highest weight type.
- Published
- 2021
16. Nilpotent superderivations in prime superalgebras
- Author
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Esther García, Guillermo Vera de Salas, and Miguel Gómez Lozano
- Subjects
Pure mathematics ,Algebra and Number Theory ,Mathematics::Rings and Algebras ,Lie superalgebra ,010103 numerical & computational mathematics ,01 natural sciences ,Prime (order theory) ,Mathematics::Group Theory ,Nilpotent ,Superinvolution ,Homogeneous ,0101 mathematics ,Mathematics::Representation Theory ,Associative property ,Mathematics - Abstract
In this paper we give an in-deph analysis of the nilpotency index of nilpotent homogeneous inner superderivations in associative prime superalgebras with and without superinvolution. We also presen...
- Published
- 2021
17. Annihilators of Irreducible Representations of the Lie Superalgebra of Contact Vector Fields on the Superline
- Author
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Goode, William M.
- Subjects
- primitive spectrum, annihilators, Lie superalgebra, contact vector fields, superline, intermediate series modules, Mathematics
- Abstract
The superline has one even and one odd coordinate. We consider the Lie superalgebra of contact vector fields on the superline. Its tensor density modules are a one-parameter family of deformations of the natural action on the ring of polynomials on the superline. They are parameterized by a complex number, and they are irreducible when this parameter is not zero. In this dissertation, we describe the annihilating ideals of these representations in the universal enveloping algebra of this Lie superalgebra by providing their generators. We also describe the intersection of all such ideals: the annihilator of the direct sum of the tensor density modules. The annihilating ideal of an irreducible non-zero left module is called a primitive ideal, and the space of all such ideals in the universal enveloping algebra is its primitive spectrum. The primitive spectrum is endowed with the Jacobson topology, which induces a topology on the annihilators of the tensor density modules. We conclude our discussion with a description of the annihilators as a topological space.
- Published
- 2023
18. Exact sequences in the cohomology of a Lie superalgebra extension
- Author
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Amber Habib and Samir Kumar Hazra
- Subjects
Pure mathematics ,Algebra and Number Theory ,Endomorphism ,Mathematics::Operator Algebras ,Mathematics::Rings and Algebras ,Abelian extension ,Lie superalgebra ,Mathematics - Rings and Algebras ,010103 numerical & computational mathematics ,Extension (predicate logic) ,01 natural sciences ,Cohomology ,Rings and Algebras (math.RA) ,17B40, 17B56, 18G40 ,FOS: Mathematics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
Let $ 0\rightarrow \mathfrak{a} \rightarrow \mathfrak{e} \rightarrow \mathfrak{g} \rightarrow 0$ be an abelian extension of the Lie superalgebra $\mathfrak{g}$. In this article we consider the problems of extending endomorphisms of $\mathfrak{a}$ and lifting endomorphisms of $\mathfrak{g}$ to certain endomorphisms of $\mathfrak{e}$. We connect these problems to the cohomology of $\mathfrak{g}$ with coefficients in $\mathfrak{a}$ through construction of two exact sequences, which is our main result, involving various endomorphism groups and the second cohomology. The first exact sequence is obtained using the Hochschild-Serre spectral sequence corresponding to the above extension while to prove the second we rather take a direct approach. As an application of our results we obtain descriptions of certain automorphism groups of semidirect product Lie superalgebras., Comment: 17 pages
- Published
- 2021
19. Outer Automorphism Groups of Contragredient Lie Superalgebras
- Author
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Mingjing Zhang and Meng-Kiat Chuah
- Subjects
Automorphism group ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,Outer automorphism group ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,021107 urban & regional planning ,Lie superalgebra ,02 engineering and technology ,01 natural sciences ,Combinatorics ,Mathematics::Group Theory ,Dynkin diagram ,Identity component ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Abstract
Let $\mathfrak {g}$ be a complex contragredient Lie superalgebra, and let D be its distinguished extended Dynkin diagram. Let $\text {Aut}(\mathfrak {g})$ be the automorphism group of $\mathfrak {g}$ , and let $\text {Int}(\mathfrak {g})$ be its identity component. We prove that $\text {Aut}(\mathfrak {g})/\text {Int}(\mathfrak {g}) \cong \text {Aut}(\mathrm {D})$ .
- Published
- 2021
20. Cohomology of the vector fields Lie algebras on R acting on trilinear differential operators, vanishing on sl(2)
- Author
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Jean Lerbet, Maha Boujelben, and Jabeur Amina
- Subjects
Pure mathematics ,Algebra and Number Theory ,Group (mathematics) ,Lie algebra ,Vector field ,Lie superalgebra ,Space (mathematics) ,Differential operator ,Cohomology ,Mathematics - Abstract
The main topic of this paper is to compute the first relative cohomology group of the Lie algebra of smooth vector fields Vect(R), with coefficients in the space of trilinear differential operators...
- Published
- 2021
21. Generalized P(N)-graded Lie superalgebras
- Author
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Jin Cheng and Yun Gao
- Subjects
Pure mathematics ,Generalization ,Mathematics::Rings and Algebras ,010102 general mathematics ,Lie superalgebra ,010103 numerical & computational mathematics ,Type (model theory) ,01 natural sciences ,Superalgebra ,Matrix (mathematics) ,Mathematics (miscellaneous) ,Mathematics::Quantum Algebra ,0101 mathematics ,Algebra over a field ,Mathematics::Representation Theory ,Commutative property ,Associative property ,Mathematics - Abstract
We generalize the P(N)-graded Lie superalgebras of Martinez-Zelmanov. This generalization is not so restrictive but sufficient enough so that we are able to have a classification for this generalized P(N)-graded Lie superalgebras. Our result is that the generalized P(N)-graded Lie superalgebra L is centrally isogenous to a matrix Lie superalgebra coordinated by an associative superalgebra with a super-involution. Moreover, L is P(N)-graded if and only if the coordinate algebra R is commutative and the super-involution is trivial. This recovers Martinez-Zelmanov’s theorem for type P(N). We also obtain a generalization of Kac’s coordinatization via Tits-Kantor-Koecher construction. Actually, the motivation of this generalization comes from the Fermionic-Bosonic module construction.
- Published
- 2021
22. Representations of principal W-algebra for the superalgebra Q(n) and the super Yangian YQ(1)
- Author
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Elena Poletaeva and Vera Serganova
- Subjects
Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,Principal (computer security) ,W-algebra ,Lie superalgebra ,01 natural sciences ,Superalgebra ,17B35, 17B20 ,Nilpotent ,Mathematics::Quantum Algebra ,Irreducible representation ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Yangian ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
We classify irreducible representations of finite W-algebra for the queer Lie superalgebra Q ( n ) associated with the principal nilpotent coadjoint orbits. We use this classification and our previous results to obtain a classification of irreducible finite-dimensional representations of the super Yangian Y Q ( 1 ) .
- Published
- 2021
23. Classification of simple Harish-Chandra modules for map (super)algebras related to the Virasoro algebra
- Author
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Rencai Lu, Yan-an Cai, and Yan Wang
- Subjects
Noetherian ,Pure mathematics ,Algebra and Number Theory ,Jet (mathematics) ,Mathematics::Rings and Algebras ,010102 general mathematics ,Witt algebra ,Lie superalgebra ,01 natural sciences ,Superalgebra ,Simple (abstract algebra) ,Mathematics::Quantum Algebra ,0103 physical sciences ,Virasoro algebra ,010307 mathematical physics ,0101 mathematics ,Mathematics::Representation Theory ,Associative property ,Mathematics - Abstract
We classify Jet modules for the Lie (super)algebras L = W ⋉ ( g ⊗ C [ t , t − 1 ] ) , where W is the Witt algebra and g is a Lie superalgebra with an even diagonlizable derivation. Then we give a conceptional method to classify all simple Harish-Chandra modules for L and the map superalgebras, which are of the form L ⊗ R , where R is a Noetherian unital supercommutative associative superalgebra.
- Published
- 2021
24. Affine Lie superalgebras
- Author
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Fateme Shirnejad, Malihe Yousofzadeh, and Abbas Darehgazani
- Subjects
Pure mathematics ,Algebra and Number Theory ,Mathematics::Rings and Algebras ,010102 general mathematics ,Lie superalgebra ,01 natural sciences ,Number theory ,Mathematics::Quantum Algebra ,0103 physical sciences ,Cartan matrix ,010307 mathematical physics ,Affine transformation ,0101 mathematics ,Algebra over a field ,Mathematics::Representation Theory ,Quotient ,Mathematics - Abstract
In 1986, Van de Leur introduced and classified affine Lie superalgebras. An affine Lie superalgebra is defined as the quotient of certain Lie superalgebra G defined by generators and relations, corresponding to a symmetrizable generalized Cartan matrix, over the so-called radical of G . Because of the interesting applications of affine Lie (super)algebras in combinatorics, number theory and physics, it is very important to recognize how far a Lie (super)algebra is to be an affine Lie (super)algebra. In this regard, we determine affine Lie superalgebras axiomatically.
- Published
- 2021
25. Characters for projective modules in the BGG category O for the orthosymplectic Lie superalgebra osp(3|4)
- Author
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Honglin Zhu and Arun S. Kannan
- Subjects
Pure mathematics ,Algebra and Number Theory ,Functor ,Verma module ,010102 general mathematics ,0103 physical sciences ,Lie superalgebra ,010307 mathematical physics ,Category O ,0101 mathematics ,Projective test ,01 natural sciences ,Mathematics - Abstract
We determine the Verma multiplicities of standard filtrations of projective modules in integral atypical blocks in the BGG category O for the orthosymplectic Lie superalgebras osp ( 3 | 4 ) by way of translation functors. We then explicitly determine the composition factor multiplicities of Verma modules using BGG reciprocity.
- Published
- 2021
26. On Kostant root systems of Lie superalgebras
- Author
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Ivan Dimitrov, Rita Fioresi, Dimitrov I., and Fioresi R.
- Subjects
Pure mathematics ,Algebra and Number Theory ,Parabolic subalgebra ,010102 general mathematics ,Subalgebra ,Lie superalgebra ,Root system ,Eigenfunction ,Killing form ,Eigenspace decomposition ,01 natural sciences ,Action (physics) ,Simple roots ,Mathematics::Quantum Algebra ,0103 physical sciences ,Kostant root system ,Hermitian symmetric pair ,010307 mathematical physics ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
We study the eigenspace decomposition of a basic classical Lie superalgebra under the adjoint action of a toral subalgebra, thus extending results of Kostant. In recognition of Kostant's contribution we refer to the eigenfunctions appearing in the decomposition as Kostant roots. We then prove that Kostant root systems inherit the main properties of classical root systems. Our approach is combinatorial in nature and utilizes certain graphs naturally associated with Kostant root systems. In particular, we reprove Kostant's results without making use of the Killing form.
- Published
- 2021
27. Two boundary centralizer algebras for q(n)
- Author
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Jieru Zhu
- Subjects
Algebra and Number Theory ,010102 general mathematics ,Degenerate energy levels ,Parameterized complexity ,Lie superalgebra ,01 natural sciences ,Representation theory ,Centralizer and normalizer ,Combinatorics ,0103 physical sciences ,Partition (number theory) ,010307 mathematical physics ,Affine transformation ,0101 mathematics ,Mathematics::Representation Theory ,Quotient ,Mathematics - Abstract
We define the degenerate two boundary affine Hecke-Clifford algebra H d , and show it admits a well-defined q ( n ) -linear action on the tensor space M ⊗ N ⊗ V ⊗ d , where V is the natural module for q ( n ) , and M , N are arbitrary modules for q ( n ) , the Lie superalgebra of Type Q. When M and N are irreducible highest weight modules parameterized by a staircase partition and a single row, respectively, this action factors through a quotient of H d . We then construct explicit modules for this quotient, H p , d , using combinatorial tools such as shifted tableaux and the Bratteli graph. These modules belong to a family of modules which we call calibrated. Using the relations in H p , d , we also classify a specific class of calibrated modules. The irreducible summands of M ⊗ N ⊗ V ⊗ d coincide with the combinatorial construction, and provide a weak version of the Schur-Weyl type duality.
- Published
- 2021
28. Denominator identities for the periplectic Lie superalgebra
- Author
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Crystal Hoyt, Shifra Reif, and Mee Seong Im
- Subjects
Pure mathematics ,Algebra and Number Theory ,Mathematics::Number Theory ,Mathematics::Rings and Algebras ,010102 general mathematics ,Lie superalgebra ,01 natural sciences ,Simple (abstract algebra) ,Mathematics::Quantum Algebra ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Abstract
We prove denominator identities for the periplectic Lie superalgebra p ( n ) , thereby completing the problem of finding denominator identities for all simple classical finite-dimensional Lie superalgebras.
- Published
- 2021
29. R matrix for generalized quantum group of type A
- Author
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Jeongwoo Yu and Jae-Hoon Kwon
- Subjects
Pure mathematics ,Algebra and Number Theory ,Quantum group ,010102 general mathematics ,Subalgebra ,Lie superalgebra ,Type (model theory) ,01 natural sciences ,Linear subspace ,Matrix decomposition ,Tensor product ,0103 physical sciences ,010307 mathematical physics ,Affine transformation ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Abstract
The generalized quantum group U ( ϵ ) of type A is an affine analogue of quantum group associated to a general linear Lie superalgebra gl M | N . We prove that there exists a unique R matrix on the tensor product of fundamental type representations of U ( ϵ ) for arbitrary parameter sequence ϵ corresponding to a non-conjugate Borel subalgebra of gl M | N . We give an explicit description of its spectral decomposition, and then as an application, construct a family of finite-dimensional irreducible U ( ϵ ) -modules which have subspaces isomorphic to the Kirillov-Reshetikhin modules of usual affine type A M − 1 ( 1 ) or A N − 1 ( 1 ) .
- Published
- 2021
30. 3-Lie Superalgebras Induced by Lie Superalgebras
- Author
-
Viktor Abramov
- Subjects
Lie superalgebra ,supertrace ,commutative superalgebra ,3-Lie superalgebra ,Mathematics ,QA1-939 - Abstract
We show that given a Lie superalgebra and an element of its dual space, one can construct the 3-Lie superalgebra. We apply this approach to Lie superalgebra of ( m , n ) -block matrices taking a supertrace of a matrix as the element of dual space. Then we also apply this approach to commutative superalgebra and the Lie superalgebra of its derivations to construct 3-Lie superalgebra. The graded Lie brackets are constructed by means of a derivation and involution of commutative superalgebra, and we use them to construct 3-Lie superalgebras.
- Published
- 2019
- Full Text
- View/download PDF
31. Isoclinic extensions of Lie superalgebras
- Author
-
Hesam Safa
- Subjects
Mathematics::Group Theory ,Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,Lie superalgebra ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
We study the notion of isoclinism on central extensions of Lie superalgebras and discuss some conditions under which central extensions are isoclinic as well as some results on isoclinic homomorphi...
- Published
- 2020
32. A Hennings type invariant of 3-manifolds from a topological Hopf superalgebra
- Author
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Ngoc Phu Ha, Laboratoire de Mathématiques de Bretagne Atlantique (LMBA), Université de Bretagne Sud (UBS)-Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS), Université de Bretagne Sud (UBS), Ha, Ngoc Phu, and Université de Brest (UBO)-Université de Bretagne Sud (UBS)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
[MATH.MATH-QA] Mathematics [math]/Quantum Algebra [math.QA] ,Root of unity ,Mathematics::Rings and Algebras ,Geometric Topology (math.GT) ,Lie superalgebra ,16. Peace & justice ,Topology ,Mathematics::Geometric Topology ,57M27, 17B37 ,Superalgebra ,Mathematics - Geometric Topology ,[MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT] ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,Ribbon ,FOS: Mathematics ,[MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA] ,Quantum Algebra (math.QA) ,Geometry and Topology ,Invariant (mathematics) ,Mathematics::Representation Theory ,Mathematical Physics ,[MATH.MATH-GT] Mathematics [math]/Geometric Topology [math.GT] ,Mathematics - Abstract
We prove the unrolled superalgebra $\mathcal{U}_{\xi}^{H}\mathfrak{sl}(2|1)$ has a completion which is a ribbon superalgebra in a topological sense where $\xi$ is a root of unity of odd order. Using this ribbon superalgebra we construct its universal invariant of links. We use it to construct an invariant of $3$-manifolds of Hennings type.
- Published
- 2020
33. On Polynomial Representations of Strange Lie Superalgebras P(n)
- Author
-
Cuiling Luo
- Subjects
Physics::General Physics ,Pure mathematics ,Polynomial ,Composition series ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,Lie superalgebra ,02 engineering and technology ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, both canonical and noncanonical polynomial representations of strange classical Lie superalgebra P(n) are investigated. It turns out all these representations are undecomposable, and their composition series are obtained.
- Published
- 2020
34. SIMPLE BOUNDED HIGHEST WEIGHT MODULES OF BASIC CLASSICAL LIE SUPERALGEBRAS
- Author
-
Dimitar Grantcharov and Maria Gorelik
- Subjects
Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,17B10 ,Lie superalgebra ,01 natural sciences ,Character (mathematics) ,Simple (abstract algebra) ,Bounded function ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Geometry and Topology ,Representation Theory (math.RT) ,0101 mathematics ,Algebra over a field ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
We classify all simple bounded highest weight modules of a basic classical Lie superalgebra $\mathfrak g$. In particular, our classification leads to the classification of the simple weight modules with finite weight multiplicities over all classical Lie superalgebras. We also obtain some character formulas of strongly typical bounded highest weight modules of $\mathfrak g$., Comment: 22 pages
- Published
- 2020
35. Relation between categories of representations of the super-Yangian of a special linear Lie superalgebra and quantum loop superalgebra
- Author
-
V. A. Stukopin
- Subjects
Pure mathematics ,Conjecture ,Relation (database) ,Matrix mechanics ,Statistical and Nonlinear Physics ,Lie superalgebra ,01 natural sciences ,Superalgebra ,Loop (topology) ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,0103 physical sciences ,010307 mathematical physics ,Yangian ,Mathematics::Representation Theory ,010306 general physics ,Quantum ,Mathematical Physics ,Mathematics - Abstract
Using the approach developed by Gautam and Toledano Laredo, we introduce analogues of the category $$ \mathfrak{O} $$ for representations of the Yangian $$Y_\hbar(A(m,n))$$ of a special linear Lie superalgebra and the quantum loop superalgebra $$U_q(LA(m,n))$$ . We investigate the relation between them and conjecture that these categories are equivalent.
- Published
- 2020
36. On Blocks in Restricted Representations of Lie Superalgebras of Cartan Type
- Author
-
Bin Shu, Fei Fei Duan, and Yu-Feng Yao
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Restricted representation ,010102 general mathematics ,Block (permutation group theory) ,Zero (complex analysis) ,Lie superalgebra ,Type (model theory) ,01 natural sciences ,Mathematics::Quantum Algebra ,0103 physical sciences ,Prime characteristic ,010307 mathematical physics ,0101 mathematics ,Algebraically closed field ,Mathematics::Representation Theory ,Mathematics - Abstract
Let g be a restricted Lie superalgebra of Cartan type W(n), S(n)or H(n) over an algebraically closed field k of prime characteristic p > 3, in the sense of modular version of Kac’s definition in 1977. In this note, we show that the restricted representation category over g has only one block (reckoning parities in). This phenomenon is very different from the case of characteristic zero.
- Published
- 2020
37. Lie super-bialgebra structures on the Lie superalgebra of Witt type
- Author
-
Xiaoyu Zhu and Xiaoqing Yue
- Subjects
Pure mathematics ,Algebra and Number Theory ,Yang–Baxter equation ,Mathematics::Rings and Algebras ,010102 general mathematics ,Lie superalgebra ,010103 numerical & computational mathematics ,Type (model theory) ,01 natural sciences ,Bialgebra ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,0101 mathematics ,Mathematics - Abstract
In this article, Lie super-bialgebra structures on the Lie superalgebra of Witt type W are investigated. They are shown to be triangular coboundary. As a byproduct, we obtain that the first cohomol...
- Published
- 2020
38. ℤ2 × ℤ2-Generalizations of Infinite-Dimensional Lie Superalgebra of Conformal Type with Complete Classification of Central Extensions
- Author
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Naruhiko Aizawa, Phillip S. Isaac, and J. Segar
- Subjects
Pure mathematics ,Class (set theory) ,010308 nuclear & particles physics ,Mathematics::Rings and Algebras ,Dimension (graph theory) ,Statistical and Nonlinear Physics ,Universal enveloping algebra ,Lie superalgebra ,Conformal map ,Extension (predicate logic) ,Type (model theory) ,01 natural sciences ,Mathematics::Quantum Algebra ,0103 physical sciences ,Virasoro algebra ,010307 mathematical physics ,Mathematics::Representation Theory ,Mathematical Physics ,Mathematics - Abstract
We introduce a class of novel ℤ2 × ℤ2-graded color superalgebras of infinite dimension. It is done by realizing each member of the class in the universal enveloping algebra of a Lie superalgebra which is a module extension of the Virasoro algebra. Then the complete classification of central extensions of the ℤ2 × ℤ2-graded color superalgebras is presented. It turns out that infinitely many members of the class have nontrivial extensions. We also demonstrate that the color superalgebras (with and without central extensions) have adjoint and superadjoint operations.
- Published
- 2020
39. On ID*-superderivations of Lie superalgebras
- Author
-
Wende Liu and Mengmeng Cai
- Subjects
Pure mathematics ,Algebra and Number Theory ,Mathematics::Quantum Algebra ,Mathematics::Rings and Algebras ,Field (mathematics) ,Lie superalgebra ,010103 numerical & computational mathematics ,0101 mathematics ,Mathematics::Representation Theory ,01 natural sciences ,Mathematics - Abstract
Let L be a Lie superalgebra over a field of characteristic different from 2, 3 and write ID∗(L) for the Lie superalgebra consisting of superderivations mapping L to L2 and the central elements to z...
- Published
- 2020
40. On capability and the Schur multiplier of some nilpotent Lie superalgebras
- Author
-
Rudra Narayan Padhan and Saudamini Nayak
- Subjects
Multiplier (Fourier analysis) ,Pure mathematics ,Nilpotent ,Algebra and Number Theory ,Mathematics::Quantum Algebra ,Mathematics::Rings and Algebras ,Lie superalgebra ,010103 numerical & computational mathematics ,0101 mathematics ,Mathematics::Representation Theory ,01 natural sciences ,Mathematics ,Schur multiplier - Abstract
A Lie superalgebra H satisfying H2=Z(H) is called generalized Heisenberg Lie superalgebra. If d(H):=(m∣n) is the minimum number of generators required to describe H, then in this article we intend ...
- Published
- 2020
41. Mixed cohomology of Lie superalgebras
- Author
-
Yucai Su and Ruibin Zhang
- Subjects
Pure mathematics ,Algebra and Number Theory ,Differential form ,010102 general mathematics ,Lie superalgebra ,Mathematics::Algebraic Topology ,01 natural sciences ,Cohomology ,BRST quantization ,Mathematics::K-Theory and Homology ,0103 physical sciences ,Supermanifold ,010307 mathematical physics ,0101 mathematics ,Special case ,Mathematics - Abstract
Supermanifolds are known to admit both differential forms and integral forms, thus any appropriate super analogue of the de Rham theory should take both types of forms into account. However, the cohomology of Lie superalgebras studied so far in the literature involves only differential forms when interpreted as a de Rham theory for Lie supergroups. Thus a new cohomology theory of Lie superalgebras is needed to fully incorporate differential-integral forms, and we investigate such a theory here. This new cohomology is defined by a BRST complex of Lie superalgebra modules, and includes the standard Lie superalgebra cohomology as a special case. General properties expected of a cohomology theory are established for the new cohomology, and examples of the new cohomology groups are computed.
- Published
- 2020
42. Derivations of affine Lie superalgebras
- Author
-
Malihe Yousofzadeh and Abbas Darehgazani
- Subjects
Pure mathematics ,Algebra and Number Theory ,Mathematics::Quantum Algebra ,Mathematics::Rings and Algebras ,010102 general mathematics ,Lie superalgebra ,010103 numerical & computational mathematics ,Affine transformation ,0101 mathematics ,Type (model theory) ,Mathematics::Representation Theory ,01 natural sciences ,Mathematics - Abstract
In this paper, we determine the derivations of affine Lie superalgebras. As the nature of affine Lie superalgebra of type A(l,l)(1) is slightly different from the nature of the other types, we stud...
- Published
- 2020
43. Hopf algebra structure on superspace $\text{SP}_q^{2|1}$
- Author
-
Salih Celik
- Subjects
Statistics and Probability ,Matematik ,Pure mathematics ,Algebra and Number Theory ,Algebraic structure ,Quantum symplectic superspace,super $\star$-algebra,super-Hopf algebra,quantum supergroup ,Structure (category theory) ,Lie superalgebra ,Function (mathematics) ,Superspace ,Hopf algebra ,Geometry and Topology ,Quantum ,Mathematics ,Analysis ,Symplectic geometry - Abstract
Super-Hopf algebra structure on the function algebra on the extended quantum symplectic superspace $\text{SP}_q^{2|1}$, denoted by ${\mathbb F}({\text{SP}}_q^{2|1})$, is defined. A quantum Lie superalgebra derived from ${\mathbb F}({\text{SP}}_q^{2|1})$ is explicitly obtained.
- Published
- 2020
44. The Drinfeld Yangian of the Queer Lie Superalgebra. Defining Relations
- Author
-
V. A. Stukopin
- Subjects
Pure mathematics ,Polynomial ,Mathematics::Quantum Algebra ,General Mathematics ,Quantization (signal processing) ,Queer ,Lie superalgebra ,Yangian ,Algebra over a field ,Mathematics::Representation Theory ,Mathematics - Abstract
Drinfeld Yangian of a queer Lie superalgebra is defined as the quantization of a Lie bisuperelgebra of twisted polynomial currents. An analogue of the new system of generators of Drinfeld is being constructed. It is proved for the partial case of Lie superalgebra $$sq_{1}$$ that this so defined Yangian and the Yangian, introduced earlier by M. Nazarov using the Faddeev–Reshetikhin–Takhtadzhjan approach, are isomorphic.
- Published
- 2020
45. The Roots of Exceptional Modular Lie Superalgebras with Cartan Matrix
- Author
-
Olexander Lozhechnyk, Jin Shang, Dimitry Leites, and Sofiane Bouarroudj
- Subjects
Chevalley basis ,Code (set theory) ,Pure mathematics ,business.industry ,General Mathematics ,Lie superalgebra ,Modular design ,Representation theory ,Simple (abstract algebra) ,Mathematics::Quantum Algebra ,Cartan matrix ,Mathematics::Representation Theory ,Indecomposable module ,business ,Mathematics - Abstract
For each of the exceptional (not entering infinite series) finite-dimensional modular Lie superalgebras with indecomposable Cartan matrix, we give the explicit list of its roots, and the corresponding Chevalley basis, for one of its inequivalent Cartan matrices, namely the one corresponding to the greatest number of mutually orthogonal isotropic odd simple roots (this number, called the defect of the Lie superalgebra, is important in the representation theory). Our main tools: Grozman’s Mathematica-based code SuperLie, Python, and A. Lebedev’s help.
- Published
- 2020
46. NON-STANDARD VERMA TYPE MODULES FOR 𝔮(n)(2)
- Author
-
Vyacheslav Futorny and Lucas Calixto
- Subjects
Pure mathematics ,Algebra and Number Theory ,ÁLGEBRAS DE LIE ,Mathematics::Rings and Algebras ,010102 general mathematics ,Diagonal ,Lie superalgebra ,Type (model theory) ,01 natural sciences ,Mathematics::Quantum Algebra ,0103 physical sciences ,Queer ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Algebra over a field ,Mathematics::Representation Theory ,Mathematics - Abstract
We study non-standard Verma type modules over the Kac-Moody queer Lie superalgebra 𝔮(n)(2). We give a sufficient condition under which such modules are irreducible. We also give a classification of all irreducible diagonal ℤ-graded modules over certain Heisenberg Lie superalgebras contained in 𝔮(n)(2).
- Published
- 2020
47. Cartan Invariants for Witt Lie Superalgebras with p-Characters of Height at Most One
- Author
-
Xun Xie and Feifei Duan
- Subjects
General Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,0211 other engineering and technologies ,Block (permutation group theory) ,021107 urban & regional planning ,Lie superalgebra ,02 engineering and technology ,Type (model theory) ,01 natural sciences ,Superalgebra ,Combinatorics ,Mathematics::Quantum Algebra ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Abstract
We investigate projective modules of the reduced enveloping superalgebra $ U_{\chi }(\frak {g}) $ of the Lie superalgebra $ \frak {g}$ of Witt type with p-character χ of height at most one. We give a formula for the Cartan invariants of $ U_{\chi }(\frak {g}) $ which implies that $ U_{\chi }(\frak {g}) $ has only one block.
- Published
- 2020
48. A generalized super AKNS hierarchy associated with orthosymplectic Lie superalgebra OSP(2,2) and its super bi‐Hamiltonian structures
- Author
-
Chuanzhong Li, Jing Yu, and Jingwei Han
- Subjects
symbols.namesake ,General Mathematics ,General Engineering ,symbols ,Lie superalgebra ,Hamiltonian (quantum mechanics) ,Mathematics ,Mathematical physics - Published
- 2020
49. Projective modules over classical Lie algebras of infinite rank in the parabolic category
- Author
-
Ngau Lam and Chih-Whi Chen
- Subjects
Pure mathematics ,Algebra and Number Theory ,Functor ,Flag (linear algebra) ,010102 general mathematics ,Duality (mathematics) ,Lie superalgebra ,Category O ,Rank (differential topology) ,01 natural sciences ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,0103 physical sciences ,Lie algebra ,Projective cover ,010307 mathematical physics ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Abstract
We study the truncation functors and show the existence of projective cover with a finite Verma flag of each irreducible module in parabolic BGG category O over infinite rank Lie algebra of types a , b , c , d . Moreover, O is a Koszul category. As a consequence, the corresponding parabolic BGG category O ‾ over infinite rank Lie superalgebra of types a , b , c , d through the super duality is also a Koszul category.
- Published
- 2020
50. Cohomology of $$\mathfrak {aff}(n|1)$$ acting on the spaces of linear differential operators on the superspace $$\mathbb {R}^{1|n}$$
- Author
-
T. Faidi, N. Ben Fraj, Z. Abdelwaheb, and Hafedh Khalfoun
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,Lie superalgebra ,02 engineering and technology ,Space (mathematics) ,Superspace ,Differential operator ,01 natural sciences ,Cohomology ,Mathematics::K-Theory and Homology ,Mathematics::Differential Geometry ,Affine transformation ,0101 mathematics ,Mathematics::Representation Theory ,Differential (mathematics) ,Mathematics - Abstract
We compute the first differential cohomology of the affine Lie superalgebra $$\mathfrak {aff}(n|1)$$ with coefficients in the superspace of linear differential operators acting on the space of weighted densities on the (1, n)-dimensional real superspace. We also compute the same, but $$\mathfrak {aff}(n-1|1)$$-relative cohomology. We explicitly give 1-cocycles spanning these cohomology groups.
- Published
- 2019
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