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Leibniz Algebras Admitting a Multiplicative Basis
In the literature, many of the descriptions of different classes of Leibniz algebras ( L , [ · , · ] ) have been made by giving the multiplication table on the elements of a basis B = { v k } k ∈ K of L , in such a way that for any i , j ∈ K we have that [ v i , v j ] = λ i , j [ v j , v i ] ∈ F v k...
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| Published in: | Bulletin of the Malaysian Mathematical Sciences Society 2017-04, Vol.40 (2), p.679-695 |
|---|---|
| Main Author: | |
| 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: | In the literature, many of the descriptions of different classes of Leibniz algebras
(
L
,
[
·
,
·
]
)
have been made by giving the multiplication table on the elements of a basis
B
=
{
v
k
}
k
∈
K
of
L
, in such a way that for any
i
,
j
∈
K
we have that
[
v
i
,
v
j
]
=
λ
i
,
j
[
v
j
,
v
i
]
∈
F
v
k
for some
k
∈
K
, where
F
denotes the base field and
λ
i
,
j
∈
F
. In order to give an unifying viewpoint of all these classes of algebras, we introduce the more general category of Leibniz algebras admitting a multiplicative basis and study its structure. We show that if a Leibniz algebra
L
admits a multiplicative basis, then it is the direct sum
L
=
⨁
α
I
α
with any
I
α
a well-described ideal of
L
admitting a multiplicative basis inherited from
B
. Also the
B
-simplicity of
L
is characterized in terms of the multiplicative basis. |
|---|---|
| ISSN: | 0126-6705 2180-4206 |
| DOI: | 10.1007/s40840-017-0446-3 |