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Journal articles on the topic 'Lie superalgebras'

Author: Grafiati

Published: 4 June 2021

Last updated: 28 July 2024

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1

Pouye, M., and B. Kpamegan. "Extensions, crossed modules and pseudo quadratic Lie type superalgebras." Extracta Mathematicae 37, no. 2 (December 1, 2022): 153–84. http://dx.doi.org/10.17398/2605-5686.37.2.153.

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Extensions and crossed modules of Lie type superalgebras are introduced and studied. We construct homology and cohomology theories of Lie-type superalgebras. The notion of left super-invariance for a bilinear form is defined and we consider Lie type superalgebras endowed with nondegenerate, supersymmetric and left super-invariant bilinear form. Such Lie type superalgebras are called pseudo quadratic Lie type superalgebras. We show that any pseudo quadratic Lie type superalgebra induces a Jacobi-Jordan superalgebra. By using the method of double extension, we study pseudo quadratic Lie type superalgebras and theirs associated Jacobi-Jordan superalgebras.

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2

Abramov, Viktor. "3-Lie Superalgebras Induced by Lie Superalgebras." Axioms 8, no. 1 (February 6, 2019): 21. http://dx.doi.org/10.3390/axioms8010021.

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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.

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3

Sun, Liping, and Wende Liu. "Hom-Lie superalgebra structures on exceptional simple Lie superalgebras of vector fields." Open Mathematics 15, no. 1 (November 13, 2017): 1332–43. http://dx.doi.org/10.1515/math-2017-0112.

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Abstract According to the classification by Kac, there are eight Cartan series and five exceptional Lie superalgebras in infinite-dimensional simple linearly compact Lie superalgebras of vector fields. In this paper, the Hom-Lie superalgebra structures on the five exceptional Lie superalgebras of vector fields are studied. By making use of the ℤ-grading structures and the transitivity, we prove that there is only the trivial Hom-Lie superalgebra structures on exceptional simple Lie superalgebras. This is achieved by studying the Hom-Lie superalgebra structures only on their 0-th and (−1)-th ℤ-components.

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4

Chen, Liangyun, and Daoji Meng. "On the Intersection of Maximal Subalgebras in a Lie Superalgebra." Algebra Colloquium 16, no. 03 (September 2009): 503–16. http://dx.doi.org/10.1142/s1005386709000479.

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The maximal subalgebras and their intersection of a Lie algebra or a Lie superalgebra were studied by Racine, Scheiderer, Elduque, Melikyan, et al. The purpose of the present paper is to continue the investigation in order to obtain deeper structure theorems for Lie superalgebras. We develop the Frattini theory for Lie superalgebras, generalize Barnes's results to Lie superalgebras, and obtain some necessary and sufficient conditions for solvable Lie superalgebras and nilpotent Lie superalgebras. Moreover, some necessary and sufficient conditions for ϕ-free Lie superalgebras and elementary Lie superalgebras are given.

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5

PEI, YUFENG, and CHENGMING BAI. "BALINSKY–NOVIKOV SUPERALGEBRAS AND SOME INFINITE-DIMENSIONAL LIE SUPERALGEBRAS." Journal of Algebra and Its Applications 11, no. 06 (November 14, 2012): 1250119. http://dx.doi.org/10.1142/s0219498812501198.

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In this paper, we recall the Balinsky–Novikov (BN) superalgebras and revisit the approach of constructing an infinite-dimensional Lie superalgebra by a kind of affinization of a BN superalgebra. As an example, we give an explicit construction of Beltrami and Green–Schwarz–Witten (GSW) algebras from two isomorphic BN superalgebras, respectively, which proves that they are isomorphic as a direct consequence. Moreover, we consider the central extensions of the infinite-dimensional Lie superalgebras induced from BN superalgebras through certain bilinear forms on their corresponding BN superalgebras.

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6

Elduque, Alberto. "A note on semiprime Malcev superalgebras." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 123, no. 5 (1993): 887–91. http://dx.doi.org/10.1017/s0308210500029553.

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SynopsisPrime Malcev superalgebras over fields of characteristic not two and three have been studied by Shestakov [8]. He obtains the remarkable result that if these superalgebras have a nonzero odd part then they are Lie superalgebras. The main purpose of this note is to extend this result to fields of characteristic three. To this aim, it is enough to use adequately a result of Filippov [3]. Commutative and anticommutative superalgebras will be considered too, showing that they are prime, semiprime or simple as superalgebras if and only if they are as algebras. Finally, some conclusions for finite-dimensional semisimple Malcev superalgebras will be deduced. Any such superalgebra is the direct sum of a semisimple Lie superalgebra and a direct sum of simple non-Lie algebras.

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7

Sun, Liping, Wende Liu, Xiaocheng Gao, and Boying Wu. "Restricted Envelopes of Lie Superalgebras." Algebra Colloquium 22, no. 02 (April 15, 2015): 309–20. http://dx.doi.org/10.1142/s1005386715000279.

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Certain important results concerning p-envelopes of modular Lie algebras are generalized to the super-case. In particular, any p-envelope of the Lie algebra of a Lie superalgebra can be naturally extended to a restricted envelope of the Lie superalgebra. As an application, a theorem on the representations of Lie superalgebras is given, which is a super-version of Iwasawa's theorem in Lie algebra case. As an example, the minimal restricted envelopes are computed for three series of modular Lie superalgebras of Cartan type.

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8

Guan, Baoling, Xinxin Tian, and Lijun Tian. "Induced 3-Hom-Lie superalgebras." Electronic Research Archive 31, no. 8 (2023): 4637–51. http://dx.doi.org/10.3934/era.2023237.

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<abstract><p>We construct 3-Hom-Lie superalgebras on a commutative Hom-superalgebra by means of involution and even degree derivation. We construct a representation of induced 3-Hom-Lie superalgebras by means of supertrace.</p></abstract>

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9

Huang, Zhongxian. "Conformal Super-Biderivations on Lie Conformal Superalgebras." Journal of Mathematics 2021 (June 2, 2021): 1–9. http://dx.doi.org/10.1155/2021/6624315.

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In this paper, the conformal super-biderivations of two classes of Lie conformal superalgebras are studied. By proving some general results on conformal super-biderivations, we determine the conformal super-biderivations of the loop super-Virasoro Lie conformal superalgebra and Neveu–Schwarz Lie conformal superalgebra. Especially, any conformal super-biderivation of the Neveu–Schwarz Lie conformal superalgebra is inner.

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10

Yuan, He, and Liangyun Chen. "Lie n superderivations and generalized Lie n superderivations of superalgebras." Open Mathematics 16, no. 1 (March 13, 2018): 196–209. http://dx.doi.org/10.1515/math-2018-0018.

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AbstractIn the paper, we study Lie n superderivations and generalized Lie n superderivations of superalgebras, using the theory of functional identities in superalgebras. We prove that if A = A0 ⊕ A1 is a prime superalgebra with deg(A1) ≥ 2n + 5, n ≥ 2, then any Lie n superderivation of A is the sum of a superderivation and a linear mapping, and any generalized Lie n superderivation of A is the sum of a generalized superderivation and a linear mapping.

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11

Bouarroudj, Sofiane, and Abdenacer Makhlouf. "Hom-Lie Superalgebras in Characteristic 2." Mathematics 11, no. 24 (December 14, 2023): 4955. http://dx.doi.org/10.3390/math11244955.

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The main goal of this paper was to develop the structure theory of Hom-Lie superalgebras in characteristic 2. We discuss their representations, semidirect product, and αk-derivations and provide a classification in low dimension. We introduce another notion of restrictedness on Hom-Lie algebras in characteristic 2, different from the one given by Guan and Chen. This definition is inspired by the process of the queerification of restricted Lie algebras in characteristic 2. We also show that any restricted Hom-Lie algebra in characteristic 2 can be queerified to give rise to a Hom-Lie superalgebra. Moreover, we developed a cohomology theory of Hom-Lie superalgebras in characteristic 2, which provides a cohomology of ordinary Lie superalgebras. Furthermore, we established a deformation theory of Hom-Lie superalgebras in characteristic 2 based on this cohomology.

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12

Gavarini, F. "Chevalley Supergroups of Type D(2, 1; a)." Proceedings of the Edinburgh Mathematical Society 57, no. 2 (August 21, 2013): 465–91. http://dx.doi.org/10.1017/s0013091513000503.

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AbstractWe present a construction ‘à la Chevalley’ of connected affine supergroups associated with Lie superalgebras of type D(2, 1; a), for any possible value of the parameter a. This extends the results by Fioresi and Gavarini, in which all other simple Lie superalgebras of classical type were considered. The case of simple Lie superalgebras of Cartan type is dealt with in a previous paper by the author, so this work completes the programme of constructing connected affine supergroups associated with any simple Lie superalgebra.

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13

Tian, Lijun, Baoling Guan, and Yao Ma. "On the Cohomology and Extensions of n-ary Multiplicative Hom-Nambu-Lie Superalgebras." Advances in Mathematical Physics 2020 (August 1, 2020): 1–17. http://dx.doi.org/10.1155/2020/1961836.

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In this paper, we discuss the representations of n-ary multiplicative Hom-Nambu-Lie superalgebras as a generalization of the notion of representations for n-ary multiplicative Hom-Nambu-Lie algebras. We also give the cohomology of an n-ary multiplicative Hom-Nambu-Lie superalgebra and obtain a relation between extensions of an n-ary multiplicative Hom-Nambu-Lie superalgebra b by an abelian one a and Z1b,a0¯. We also introduce the notion of T∗-extensions of n-ary multiplicative Hom-Nambu-Lie superalgebras and prove that every finite-dimensional nilpotent metric n-ary multiplicative Hom-Nambu-Lie superalgebra over an algebraically closed field of characteristic not 2 in the case α is a surjection is isometric to a suitable T∗-extension.

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14

CANTARINI, NICOLETTA. "A CLASSIFICATION OF ℤ-GRADED LIE SUPERALGEBRAS OF INFINITE DEPTH." Journal of Algebra and Its Applications 01, no. 04 (December 2002): 425–49. http://dx.doi.org/10.1142/s0219498802000276.

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In 1998 Victor Kac classified infinite-dimensional, transitive, irreducible ℤ-graded Lie superalgebras of finite depth. Here we classify bitransitive, irreducible ℤ-graded Lie superalgebras of infinite depth and finite growth which are not contragredient. In particular we show that the growth of every such superalgebra is equal to one.

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15

Zhou, Jia, Liangyun Chen, Yao Ma, and Bing Sun. "On ω-Lie superalgebras." Journal of Algebra and Its Applications 17, no. 11 (November 2018): 1850212. http://dx.doi.org/10.1142/s0219498818502122.

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Let [Formula: see text] be a finite-dimensional vector space over a field [Formula: see text] of characteristic zero, [Formula: see text] an anti-commutative product on [Formula: see text] and [Formula: see text] a bilinear form on [Formula: see text]. The triple [Formula: see text] is called an [Formula: see text]-Lie algebra if [Formula: see text] (graded [Formula: see text]-Jacobi identity) for all [Formula: see text] In this paper, we introduce the notion of an [Formula: see text]-Lie superalgebra. We study elementary properties and representations of [Formula: see text]-Lie superalgebras. We classify all 3- and 4-dimensional [Formula: see text]-Lie superalgebras over the field of complex numbers.

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16

Calixto, Lucas, Adriano Moura, and Alistair Savage. "Equivariant Map Queer Lie Superalgebras." Canadian Journal of Mathematics 68, no. 2 (April 1, 2016): 258–79. http://dx.doi.org/10.4153/cjm-2015-033-6.

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AbstractAn equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme) X to a queer Lie superalgebra q that are equivariant with respect to the action of a finite group Γ acting on X and q. In this paper, we classify all irreducible finite-dimensional representations of the equivariant map queer Lie superalgebras under the assumption that Γ is abelian and acts freely on X. We show that such representations are parameterized by a certain set of Γ-equivariant finitely supported maps from X to the set of isomorphism classes of irreducible finite-dimensional representations of q. In the special case where X is the torus, we obtain a classification of the irreducible finite-dimensional representations of the twisted loop queer superalgebra.

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17

Benayadi, Saïd, and Fahmi Mhamdi. "Odd-quadratic Leibniz superalgebras." Advances in Pure and Applied Mathematics 10, no. 4 (October 1, 2019): 287–98. http://dx.doi.org/10.1515/apam-2018-0167.

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AbstractAn odd-quadratic Leibniz superalgebra is a (left or right) Leibniz superalgebra with an odd, supersymmetric, non-degenerate and invariant bilinear form. In this paper, we prove that a left (resp. right) Leibniz superalgebra that carries this structure is symmetric (meaning that it is simultaneously a left and a right Leibniz superalgebra). Moreover, we show that any non-abelian (left or right) Leibniz superalgebra does not possess simultaneously a quadratic and an odd-quadratic structure. Further, we obtain an inductive description of odd-quadratic Leibniz superalgebras using the procedure of generalized odd double extension and we reduce the study of this class of Leibniz superalgebras to that of odd-quadratic Lie superalgebras. Finally, several non-trivial examples of odd-quadratic Leibniz superalgebras are included.

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18

Roger, Claude. "The even, the odd, the superalgebras and their derivations." Advances in Pure and Applied Mathematics 9, no. 4 (October 1, 2018): 287–94. http://dx.doi.org/10.1515/apam-2018-0083.

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Abstract We give an introduction to superalgebra, founded on the difference between even (commuting) and odd (anti-commuting) variables. We give a sketch of Graßmann’s work, and show how derivations of those structures induce various superalgebra structures, Lie superalgebras of Cartan type being obtained with even derivations, while odd derivations induce Jordan-type superalgebras.

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19

Ren, Li, Qiang Mu, and Yongzheng Zhang. "A Class of Finite-dimensional Lie Superalgebras of Hamiltonian Type." Algebra Colloquium 18, no. 02 (June 2011): 347–60. http://dx.doi.org/10.1142/s1005386711000241.

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A class of finite-dimensional Cartan-type Lie superalgebras H(n,m) over a field of prime characteristic is studied in this paper. We first determine the derivation superalgebra of H(n,m). Then we obtain that H(n,m) is restrictable and it is an extension of the Lie superalgebra [Formula: see text]. Finally, we prove that H(n,m) is isomorphic to a subalgebra of the restricted Hamiltonian Lie superalgebra [Formula: see text].

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20

Padhan, Rudra Narayan, and K. C. Pati. "Some studies on central derivation of nilpotent Lie superalgebra." Asian-European Journal of Mathematics 13, no. 04 (December 7, 2018): 2050068. http://dx.doi.org/10.1142/s1793557120500680.

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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. Theory 20 (2017) 1143–1150; On [Formula: see text]-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.

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21

Bezushchak, D. I., and O. O. Bezushchak. "Derivations of infinite-dimensional Lie superalgebras." Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, no. 1 (2023): 21–25. http://dx.doi.org/10.17721/1812-5409.2023/1.2.

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We study infinite-dimensional analogs of classical Lie superalgebras over an algebraically closed field F of zero characteristic. Let I be an infinite set. For an algebra M_∞ (I) of infinite I × I matrices over a ground field F having finitely many nonzero entries, we consider the related Lie superalgebra gl_∞ (I1, I2) and its commutator sl_∞ (I1, I2) for a disjoint union of nonempty subsets I1 and I2 of the set I; and we describe derivations of the Lie superalgebra sl_∞ (I1, I2).

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22

Yang, Ruhui, and Keli Zheng. "The Matrix Representation of Quasi Centroids of Heisenberg Superalgebra." Journal of Advances in Mathematics and Computer Science 38, no. 11 (December 11, 2023): 87–94. http://dx.doi.org/10.9734/jamcs/2023/v38i111847.

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This article studies the matrix representation of the quasi-centroids of Heisenberg superalgebras. Based on the definition of Heisenberg superalgebras and quasi-centroids, a matrix representation of quasi- centroids of Heisenberg superalgebras with even center is studied by using the method of solving system of linear equations and supersymmetry operation of Lie superalgebras. Finally, a matrix representation of the quasi centroids of a Heisenberg superalgebra with an odd center is obtained through similar calculations.

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23

GOULD, M. D., R. B. ZHANG, and A. J. BRACKEN. "LIE BI-SUPERALGEBRAS AND THE GRADED CLASSICAL YANG-BAXTER EQUATION." Reviews in Mathematical Physics 03, no. 02 (June 1991): 223–40. http://dx.doi.org/10.1142/s0129055x91000084.

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The theory of Lie bi-superalgebras and its connection with the graded classical Yang-Baxter equation are studied. The classical double construction is developed in some detail in the graded case; this allows the embedding of any given finite dimensional Lie bi-superalgebra in a quasitriangular Lie bi-superalgebra. A universal formula for the classical r-matrix is obtained in an explicit form.

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24

Xu, Xiaoning, Yongzheng Zhang, and Liangyun Chen. "The Finite-dimensional Modular Lie Superalgebra Γ." Algebra Colloquium 17, no. 03 (September 2010): 525–40. http://dx.doi.org/10.1142/s1005386710000507.

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A new family of finite-dimensional modular Lie superalgebras Γ is defined. The simplicity and generators of Γ are studied and an explicit description of the derivation superalgebra of Γ is given. Moreover, it is proved that Γ is not isomorphic to any known Lie superalgebra of Cartan type.

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Assiry, Abdullah, Sabeur Mansour, and Amir Baklouti. "S-Embedding of Lie Superalgebras and Its Implications for Fuzzy Lie Algebras." Axioms 13, no. 1 (December 19, 2023): 2. http://dx.doi.org/10.3390/axioms13010002.

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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.

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26

BRINGMANN, KATHRIN, and KARL MAHLBURG. "Asymptotic formulas for coefficients of Kac–Wakimoto Characters." Mathematical Proceedings of the Cambridge Philosophical Society 155, no. 1 (February 22, 2013): 51–72. http://dx.doi.org/10.1017/s0305004112000680.

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AbstractWe study the coefficients of Kac and Wakimoto's character formulas for the affine Lie superalgebrassℓ(n+1|1)∧. The coefficients of these characters are the weight multiplicities of irreducible modules of the Lie superalgebras, and their asymptotic study begins with Kac and Peterson's earlier use of modular forms to understand the coefficients of characters for affine Lie algebras. In the affine Lie superalgebra setting, the characters are products of weakly holomorphic modular forms and Appell-type sums, which have recently been studied using developments in the theory of mock modular forms and harmonic Maass forms. Using our previously developed extension of the Circle Method for products of mock modular forms along with the Saddle Point Method, we find asymptotic series expansions for the coefficients of the characters with polynomial error.

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27

Zhang, Tao, and Zhangju Liu. "Omni-Lie superalgebras and Lie 2-superalgebras." Frontiers of Mathematics in China 9, no. 5 (January 20, 2014): 1195–210. http://dx.doi.org/10.1007/s11464-014-0347-9.

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PENKOV, IVAN, and VERA SERGANOVA. "GENERIC IRREDUCIBLE REPRESENTATIONS OF FINITE-DIMENSIONAL LIE SUPERALGEBRAS." International Journal of Mathematics 05, no. 03 (June 1994): 389–419. http://dx.doi.org/10.1142/s0129167x9400022x.

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A theory of highest weight modules over an arbitrary finite-dimensional Lie superalgebra is constructed. A necessary and sufficient condition for the finite-dimensionality of such modules is proved. Generic finite-dimensional irreducible representations are defined and an explicit character formula for such representations is written down. It is conjectured that this formula applies to any generic finite-dimensional irreducible module over any finite-dimensional Lie superalgebra. The conjecture is proved for several classes of Lie superalgebras, in particular for all solvable ones, for all simple ones, and for certain semi-simple ones.

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29

Fan, Yusi, and Liangyun Chen. "On (σ,τ)-derivations of lie superalgebras." Filomat 37, no. 1 (2023): 179–92. http://dx.doi.org/10.2298/fil2301179f.

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This paper is primarily devoted to studying (?, ?)-derivations of finite-dimensional Lie superalgebras over an algebraically closed field F. We research some properties of (?, ?)-derivations and the relationship between the (?, ?)-derivations and other generalized derivations. Under certain conditions, a left-multiplication structure concerned with (?, ?)-derivations can induces a left-symmetric superalgebra structure. Let L be a Lie superalgebra, we give a subgroup G of Aut(L), exploiting fundamental properties, we introduce and analyze their interiors, especially focusing on the rationality of the corresponding Hilbert series when G is a cyclic group.

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AYADI, IMEN, HEDI BENAMOR, and SAÏD BENAYADI. "LIE SUPERALGEBRAS WITH SOME HOMOGENEOUS STRUCTURES." Journal of Algebra and Its Applications 11, no. 05 (September 26, 2012): 1250095. http://dx.doi.org/10.1142/s0219498812500958.

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We generalize to the case of Lie superalgebras the classical symplectic double extension of symplectic Lie algebras introduced in [A. Aubert, Structures affines et pseudo-métriques invariantes à gauche sur des groupes de Lie, Thèse, Université Montpellier II (1996)]. We use this concept to give an inductive description of nilpotent homogeneous-symplectic Lie superalgebras. Several examples are included to show the existence of homogeneous quadratic symplectic Lie superalgebras other than even-quadratic even-symplectic considered in [E. Barreiro and S. Benayadi, Quadratic symplectic Lie superalgebras and Lie bi-superalgebras, J. Algebra 321(2) (2009) 582–608]. We study the structures of even (respectively, odd)-quadratic odd (respectively, even)-symplectic Lie superalgebras and odd-quadratic odd-symplectic Lie superalgebras and we give its inductive descriptions in terms of quadratic generalized double extensions and odd quadratic generalized double extensions. This study complete the inductive descriptions of homogeneous quadratic symplectic Lie superalgebras started in [E. Barreiro and S. Benayadi, Quadratic symplectic Lie superalgebras and Lie bi-superalgebras, J. Algebra 321(2) (2009) 582–608]. Finally, we generalize to the case of homogeneous quadratic symplectic Lie superalgebras some relations between even-quadratic even-symplectic Lie superalgebras and Manin superalgebras established in [E. Barreiro and S. Benayadi, Quadratic symplectic Lie superalgebras and Lie bi-superalgebras, J. Algebra 321(2) (2009) 582–608].

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Kang, Seok-Jin, Jae-Hoon Kwon, and Young-Tak Oh. "Peterson-type dimension formulas for graded Lie superalgebras." Nagoya Mathematical Journal 163 (September 2001): 107–44. http://dx.doi.org/10.1017/s0027763000007935.

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Let be a free abelian group of finite rank, let Γ be a sub-semigroup of satisfying certain finiteness conditions, and let be a (Γ × Z2)-graded Lie superalgebra. In this paper, by applying formal differential operators and the Laplacian to the denominator identity of , we derive a new recursive formula for the dimensions of homogeneous subspaces of . When applied to generalized Kac-Moody superalgebras, our formula yields a generalization of Peterson’s root multiplicity formula. We also obtain a Freudenthal-type weight multiplicity formula for highest weight modules over generalized Kac-Moody superalgebras.

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32

Mu, Qiang. "Natural Filtrations of Infinite-Dimensional Modular Contact Superalgebras." Journal of Applied Mathematics 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/601847.

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The natural filtration of the infinite-dimensional contact superalgebra over an algebraic closed field of positive characteristic is proved to be invariant under automorphisms by characterizing ad-nilpotent elements and the subalgebras generated by certain ad-nilpotent elements. Moreover, we obtain an intrinsic characterization of contact superalgebras and a property of automorphisms of these Lie superalgebras.

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33

Armakan, A. R., and M. R. Farhangdoost. "Geometric aspects of extensions of hom-Lie superalgebras." International Journal of Geometric Methods in Modern Physics 14, no. 06 (May 4, 2017): 1750085. http://dx.doi.org/10.1142/s0219887817500852.

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In this paper, we deal with (non-abelian) extensions of a given hom-Lie superalgebra and find a cohomological obstacle to the existence of extensions of hom-Lie superalgebras. Moreover, the setting of covariant exterior derivatives, super connection, curvature and the Bianchi identity in differential geometry has been studied.

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Chen, Chih-Whi, and Kevin Coulembier. "The Primitive Spectrum and Category for the Periplectic Lie Superalgebra." Canadian Journal of Mathematics 72, no. 3 (November 16, 2018): 625–55. http://dx.doi.org/10.4153/s0008414x18000081.

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AbstractWe solve two problems in representation theory for the periplectic Lie superalgebra $\mathfrak{p}\mathfrak{e}(n)$, namely, the description of the primitive spectrum in terms of functorial realisations of the braid group and the decomposition of category ${\mathcal{O}}$ into indecomposable blocks.To solve the first problem, we establish a new type of equivalence between category ${\mathcal{O}}$ for all (not just simple or basic) classical Lie superalgebras and a category of Harish-Chandra bimodules. The latter bimodules have a left action of the Lie superalgebra but a right action of the underlying Lie algebra. To solve the second problem, we establish a BGG reciprocity result for the periplectic Lie superalgebra.

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35

Wang, Yan, Yufeng Pei, and Shaoqiang Deng. "Leibniz central extensions of Lie superalgebras." Journal of Algebra and Its Applications 13, no. 08 (June 24, 2014): 1450052. http://dx.doi.org/10.1142/s0219498814500522.

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In this paper, we develop a general theory on Leibniz central extensions of Lie superalgebras and apply it to determine the second Leibniz cohomology groups for several classes of Lie superalgebras, including classical Lie superalgebras, Neveu–Schwarz superalgebras, differentiably simple Lie superalgebras, and affine (toroidal) Kac–Moody Lie superalgebras.

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36

Guo, Shuangjian, Xiaohui Zhang, and Shengxiang Wang. "Representations and Deformations of Hom-Lie-Yamaguti Superalgebras." Advances in Mathematical Physics 2020 (June 18, 2020): 1–12. http://dx.doi.org/10.1155/2020/9876738.

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Let L,α be a Hom-Lie-Yamaguti superalgebra. We first introduce the representation and cohomology theory of Hom-Lie-Yamaguti superalgebras. Also, we introduce the notions of generalized derivations and representations of L,α and present some properties. Finally, we investigate the deformations of L,α by choosing some suitable cohomology.

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37

Guo, Shuangjian, Xiaohui Zhang, and Shengxiang Wang. "On split involutive regular BiHom-Lie superalgebras." Open Mathematics 18, no. 1 (June 4, 2020): 476–85. http://dx.doi.org/10.1515/math-2020-0144.

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Abstract The goal of this paper is to examine the structure of split involutive regular BiHom-Lie superalgebras, which can be viewed as the natural generalization of split involutive regular Hom-Lie algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split involutive regular BiHom-Lie superalgebra {\mathfrak{L}} is of the form {\mathfrak{L}}=U+{\sum }_{\alpha }{I}_{\alpha } with U a subspace of a maximal abelian subalgebra H and any I α , a well-described ideal of {\mathfrak{L}} , satisfying [I α , I β ] = 0 if [α] ≠ [β]. In the case of {\mathfrak{L}} being of maximal length, the simplicity of {\mathfrak{L}} is also characterized in terms of connections of roots.

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38

Kang, Seok-Jin, and Myung-Hwan Kim. "Borcherds superalgebras and a monstrous Lie superalgebra." Mathematische Annalen 307, no. 4 (April 2, 1997): 677–94. http://dx.doi.org/10.1007/s002080050056.

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39

Leites, D. A. "Lie superalgebras." Journal of Soviet Mathematics 30, no. 6 (September 1985): 2481–512. http://dx.doi.org/10.1007/bf02249121.

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40

Bai, Wei, and Wende Liu. "Extensions of Lie superalgebras by Heisenberg Lie superalgebras." Colloquium Mathematicum 153, no. 2 (2018): 209–18. http://dx.doi.org/10.4064/cm7243-8-2017.

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41

Barreiro, Elisabete, and Saïd Benayadi. "Quadratic symplectic Lie superalgebras and Lie bi-superalgebras." Journal of Algebra 321, no. 2 (January 2009): 582–608. http://dx.doi.org/10.1016/j.jalgebra.2008.09.026.

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42

Ammar, Faouzi, and Abdenacer Makhlouf. "Hom-Lie superalgebras and Hom-Lie admissible superalgebras." Journal of Algebra 324, no. 7 (October 2010): 1513–28. http://dx.doi.org/10.1016/j.jalgebra.2010.06.014.

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43

Liu, Wende, and Jixia Yuan. "Automorphism groups of modular graded Lie superalgebras of Cartan-type." Journal of Algebra and Its Applications 16, no. 03 (March 2017): 1750050. http://dx.doi.org/10.1142/s0219498817500505.

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Suppose the underlying field is of characteristic [Formula: see text]. In this paper, we prove that the automorphisms of the finite-dimensional graded (non-restircited) Lie superalgebras of Cartan-type [Formula: see text] [Formula: see text] [Formula: see text] and [Formula: see text] can uniquely extend to the ones of the infinite-dimensional Lie superalgebra of Cartan-type [Formula: see text]. Then a concrete group embedding from [Formula: see text] into [Formula: see text] is established, where [Formula: see text] is any finite-dimensional Lie superalgebra of Cartan-type [Formula: see text] or [Formula: see text] and [Formula: see text] is the underlying (associative) superalgebra of [Formula: see text]. The normal series of the automorphism groups of [Formula: see text] are also considered.

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44

PELIZZOLA, ALESSANDRO, and CORRADO TOPI. "GENERALIZED COHERENT STATES FOR DYNAMICAL SUPERALGEBRAS." International Journal of Modern Physics B 05, no. 19 (November 20, 1991): 3073–108. http://dx.doi.org/10.1142/s0217979291001218.

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Coherent states for a general Lie superalgebra are defined following the method originally proposed by Perelomov. Algebraic and geometrical properties of the systems of states thus obtained are examined, with particular attention to the possibility of defining a Kähler structure over the states supermanifold and to the connection between this supermanifold and the coadjoint orbits of the dynamical supergroup. The theory is then applied to some compact forms of contragradient Lie superalgebras.

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45

Kwon, Namhee. "Bosonic–fermionic realizations of root spaces and bilinear forms for Lie superalgebras." Journal of Algebra and Its Applications 19, no. 11 (October 16, 2019): 2050203. http://dx.doi.org/10.1142/s0219498820502035.

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We study bosonic–fermionic constructions of the root spaces of the orthosymplectic Lie superalgebras. We also present explicitly bosonic–fermionic realizations of the supertrace forms for the orthosymplectic Lie superalgebras. As by-products, we obtain bosonic–fermionic realizations of both the general linear Lie superalgebras and vertex representations of the affine orthosymplectic Lie superalgebras.

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46

Kang, Seok-Jin, and Jae-Hoon Kwon. "Graded Lie Superalgebras, Supertrace Formula, and Orbit Lie Superalgebras." Proceedings of the London Mathematical Society 81, no. 3 (November 2000): 675–724. http://dx.doi.org/10.1112/s0024611500012661.

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47

Rodríguez-Vallarte, M. C., G. Salgado, and O. A. Sánchez-Valenzuela. "On indecomposable solvable Lie superalgebras having a Heisenberg nilradical." Journal of Algebra and Its Applications 15, no. 10 (November 24, 2016): 1650190. http://dx.doi.org/10.1142/s0219498816501905.

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All solvable, indecomposable, finite-dimensional, complex Lie superalgebras [Formula: see text] whose first derived ideal lies in its nilradical, and whose nilradical is a Heisenberg Lie superalgebra [Formula: see text] associated to a [Formula: see text]-homogeneous supersymplectic complex vector superspace [Formula: see text], are here classified up to isomorphism. It is shown that they are all of the form [Formula: see text], where [Formula: see text] is even and consists of non-[Formula: see text]-nilpotent elements. All these Lie superalgebras depend on an element [Formula: see text] in the dual space [Formula: see text] and on a pair of linear maps defined on [Formula: see text], and taking values in the Lie algebras naturally associated to the even and odd subspaces of [Formula: see text]; namely, if the supersymplectic form is even, the pair of linear maps defined on [Formula: see text] take values in [Formula: see text], and [Formula: see text], respectively, whereas if the supersymplectic form is odd these linear maps take values on [Formula: see text]. When the supersymplectic form is even, a bilinear, skew-symmetric form defined on [Formula: see text] is further needed. Conditions on these building data are given and the isomorphism classes of the resulting Lie superalgebras are described in terms of appropriate group actions.

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48

GIÉ, PIERRE-ALEXANDRE, GEORGES PINCZON, and ROSANE USHIROBIRA. "THE AMITSUR–LEVITZKI THEOREM FOR THE ORTHOSYMPLECTIC LIE SUPERALGEBRA 𝔬𝔰𝔭(1, 2n)." Journal of Algebra and Its Applications 05, no. 03 (June 2006): 307–32. http://dx.doi.org/10.1142/s0219498806001740.

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Based on Kostant's cohomological interpretation of the Amitsur–Levitzki theorem, we prove a super version of this theorem for the Lie superalgebras 𝔬𝔰𝔭(1, 2n). We conjecture that no other classical Lie superalgebra can satisfy an Amitsur–Levitzki type super identity. We show several (super) identities for the standard super polynomials. Finally, a combinatorial conjecture on the standard skew supersymmetric polynomials is stated.

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49

Guan, Baoling, Liangyun Chen, and Yao Ma. "On the Deformations and Derivations ofn-Ary Multiplicative Hom-Nambu-Lie Superalgebras." Advances in Mathematical Physics 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/381683.

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We introduce the relevant concepts ofn-ary multiplicative Hom-Nambu-Lie superalgebras and construct three classes ofn-ary multiplicative Hom-Nambu-Lie superalgebras. As a generalization of the notion of derivations forn-ary multiplicative Hom-Nambu-Lie algebras, we discuss the derivations ofn-ary multiplicative Hom-Nambu-Lie superalgebras. In addition, the theory of one parameter formal deformation ofn-ary multiplicative Hom-Nambu-Lie superalgebras is developed by choosing a suitable cohomology.

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50

Wei, Zhu, Qingcheng Zhang, Yongzheng Zhang, and Chunyue Wang. "Simple Modules for Modular Lie SuperalgebrasW(0∣n),S(0∣n), andK(n)." Advances in Mathematical Physics 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/250570.

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This paper constructs a series of modules from modular Lie superalgebrasW(0∣n),S(0∣n), andK(n)over a field of prime characteristicp≠2. Cartan subalgebras, maximal vectors of these modular Lie superalgebras, can be solved. With certain properties of the positive root vectors, we obtain that the sufficient conditions of these modules are irreducibleL-modules, whereL=W(0∣n),S(0∣n), andK(n).

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