Derivations and quasi-derivations of Block Lie superalgebras S(q)

@misc{kaygorodov2026derivations,
  title = {{Derivations and quasi-derivations of Block Lie superalgebras $S(q)$}},
  author = {Ivan Kaygorodov and Abror Khudoyberdiyev and Zarina Shermatova},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23150},
  eprint = {2610.23150},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23150},
  note = {CC0},
}

TY  - UNPB
TI  - Derivations and quasi-derivations of Block Lie superalgebras S(q)
AU  - Ivan Kaygorodov
AU  - Abror Khudoyberdiyev
AU  - Zarina Shermatova
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23150
UR  - https://arxiv.org/abs/2610.23150
AB  - We compute the homogeneous derivations and the homogeneous quasi-derivations of the Block Lie superalgebras S(q), whose even part is the Block Lie algebra ℬ(q). For every q∈ℂ, the space Der_θ(S(q)) is the direct sum of the inner derivations and an explicitly given finite-dimensional space of outer derivations; in particular, for q≠0 a new odd outer derivation appears if and only if q∈2ℤ^*. For q∈ℂ^* we prove that QDer(S(q))=Der(S(q))⊕⟨ id⟩ if q∉ℤ, and that for q∈ℤ^* the superalgebra S(q) has an infinite-dimensional space of nontrivial quasi-derivations, namely all graded linear maps with values in the center. For q=0 we also describe all homogeneous quasi-derivations: modulo maps with values in the center, they are spanned by id, three further maps and infinite families of maps shifting the ℤ^2-grading; in particular, QDer(S(0))/Der(S(0)) is infinite-dimensional. As corollaries, we obtain the complete description of homogeneous δ-derivations of S(q), q≠0, and prove that, in contrast with the Block Lie algebras ℬ(q), the superalgebras S(q), q≠0, do not admit nontrivial transposed δ-Poisson superalgebra structures for any δ∉{0,1}.
ER  -