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Transposed Poisson Structures on Generalized Witt Algebras and Block Lie Algebras
We describe transposed Poisson structures on generalized Witt algebras W ( A , V , ⟨ · , · ⟩ ) and Block Lie algebras L ( A , g , f ) over a field F of characteristic zero, where ⟨ · , · ⟩ and f are non-degenerate. More specifically, if dim ( V ) > 1 , then all the transposed Poisson algebra st...
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| Published in: | Resultate der Mathematik 2023-10, Vol.78 (5), Article 186 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | We describe transposed Poisson structures on generalized Witt algebras
W
(
A
,
V
,
⟨
·
,
·
⟩
)
and Block Lie algebras
L
(
A
,
g
,
f
) over a field
F
of characteristic zero, where
⟨
·
,
·
⟩
and
f
are non-degenerate. More specifically, if
dim
(
V
)
>
1
, then all the transposed Poisson algebra structures on
W
(
A
,
V
,
⟨
·
,
·
⟩
)
are trivial; and if
dim
(
V
)
=
1
, then such structures are, up to isomorphism, mutations of the group algebra structure on
FA
. The transposed Poisson algebra structures on
L
(
A
,
g
,
f
) are in a one-to-one correspondence with commutative and associative multiplications defined on a complement of the square of
L
(
A
,
g
,
f
) with values in the center of
L
(
A
,
g
,
f
). In particular, all of them are usual Poisson structures on
L
(
A
,
g
,
f
). This generalizes earlier results about transposed Poisson structures on Block Lie algebras
B
(
q
)
. |
|---|---|
| ISSN: | 1422-6383 1420-9012 |
| DOI: | 10.1007/s00025-023-01962-y |