Non-degenerate Invariant (Super)Symmetric Bilinear Forms on Simple Lie (Super)Algebras

Bouarroudj, Sofiane; Krutov, Andrey; Leites, Dimitry; Shchepochkina, Irina

doi:10.1007/s10468-018-9802-8

Cited by 17 publications

(8 citation statements)

References 36 publications

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“…A direct corollary of the above lemma is that the deforms of wk (1) (3; α)/c and wk(4; α) corresponding to the cocyles c 0 when the deformation parameter ε is equal to α are not isomorphic to the original algebras. This fact agrees with results of [15], where it was shown that these deforms are not simple Lie algebras.…”

Section: 2supporting

confidence: 92%

“…As we know from [3] succinctly summarized in [21], the double extension is indecomposable, i.e., not isomorphic to the direct sum of two ideals a and K ⊕ K * , if the following conditions are satisfied: a) the derivation D of a must be outer for any p; if p = 2, and D ∈ (out a)1, then, moreover, the condition D 2 = 0 is a must, see [15]; b) the central extension has to be non-trivial, see [21,Section 8].…”

Section: Phenomena Indigenous To P =mentioning

confidence: 99%

“…If g is a vectorial Lie algebra, it can happen that Ker K is nontrivial, e.g., for a modular svect(3), see [15], and for Hamiltonian Lie algebras h on tori where there are invariant forms on h. Then, "Leibniz central extensions" arise, i.e., a central extension of the Lie algebra, the result of which is not a Lie algebra, but a Leibniz algebra which satisfies only the Jacobi identity, but not the anti-symmetry one.…”

Section: Derivations and Central Extensions Of Simple Lie (Super)alge...mentioning

confidence: 99%

“…For the case where all g-invariant symmetric bilinear forms (S 2 (g)) g are even, the arguments of [51] are literally applicable to Lie super algebras and help us to verify the results if p ̸ = 2. (Note that the title of [15] is misleading: there are symmetric and anti-symmetric bilinear forms on superspaces, but there are no "super symmetric" forms. For an elucidation of this, see [43,Section 2.4.2].…”

Section: Derivations and Central Extensions Of Simple Lie (Super)alge...mentioning

confidence: 99%

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Derivations and Central Extensions of Symmetric Modular Lie Algebras and Superalgebras

Bouarroudj¹,

Grozman²,

Lebedev³

et al. 2023

SIGMA

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Over algebraically closed fields of positive characteristic, for simple Lie (super)algebras, and certain Lie (super)algebras close to simple ones, with symmetric root systems (such that for each root, there is minus it of the same multiplicity) and of ranks less than or equal to 8—most needed in an approach to the classification of simple vectorial Lie superalgebras (i.e., Lie superalgebras realized by means of vector fields on a supermanifold),—we list the outer derivations and nontrivial central extensions. When the conjectural answer is clear for the infinite series, it is given for any rank. We also list the outer derivations and nontrivial central extensions of one series of non-symmetric (except when considered in characteristic 2), namely periplectic, Lie superalgebras—the one that preserves the nondegenerate symmetric odd bilinear form, and of the Lie algebras obtained from them by desuperization. We also list the outer derivations and nontrivial central extensions of an analog of the rank 2 exceptional Lie algebra discovered by Shen Guangyu. Several results indigenous to positive characteristic are of particular interest being unlike known theorems for characteristic 0, some results are, moreover, counterintuitive.

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Section: 2supporting

confidence: 92%

Section: Phenomena Indigenous To P =mentioning

confidence: 99%

Section: Derivations and Central Extensions Of Simple Lie (Super)alge...mentioning

confidence: 99%

Section: Derivations and Central Extensions Of Simple Lie (Super)alge...mentioning

confidence: 99%

See 2 more Smart Citations

Derivations and Central Extensions of Symmetric Modular Lie Algebras and Superalgebras

Bouarroudj¹,

Grozman²,

Lebedev³

et al. 2023

SIGMA

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show abstract

“…Remark 2.1. There are some authors, see [10,12], who refer to supersymmetric and skewsymmetric in Definition 2.2 as symmetric and antisymmetric, respectively, for reasons explained in the text.…”

Section: Basic Concepts and Previous Resultsmentioning

confidence: 99%

Quadratic symplectic Lie superalgebras with a filiform module as an odd part

Barreiro

Benayadi

Navarro

et al. 2023

Journal of Mathematical Physics

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The present work studies deeply quadratic symplectic Lie superalgebras, obtaining, in particular, that they are all nilpotent. Consequently, we provide classifications in low dimensions and identify the double extensions that maintain symplectic structures. By means of both elementary odd double extensions and generalized double extensions of quadratic symplectic Lie superalgebras, we obtain an inductive description of quadratic symplectic Lie superalgebras of filiform type.

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Elliptic genera from classical error-correcting codes

Kawabata,

Yahagi

2024

J. High Energ. Phys.

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We consider chiral fermionic conformal field theories constructed from classical error-correcting codes and provide a systematic way of computing their elliptic genera. We exploit the U(1) current of the $$ \mathcal{N} $$ N = 2 superconformal algebra to obtain the U(1)-graded partition function that is invariant under the modular transformation and the spectral flow. We demonstrate our method by constructing extremal $$ \mathcal{N} $$ N = 2 elliptic genera from classical codes for relatively small central charges. Also, we give near-extremal elliptic genera and decompose them into $$ \mathcal{N} $$ N = 2 superconformal characters.

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Non-degenerate Invariant (Super)Symmetric Bilinear Forms on Simple Lie (Super)Algebras

Cited by 17 publications

References 36 publications

Derivations and Central Extensions of Symmetric Modular Lie Algebras and Superalgebras

Derivations and Central Extensions of Symmetric Modular Lie Algebras and Superalgebras

Quadratic symplectic Lie superalgebras with a filiform module as an odd part

Elliptic genera from classical error-correcting codes

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