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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 |
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| Language: | English |
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| cites | cdi_FETCH-LOGICAL-c316t-3a7108f2a927a7370e5469d646a5576e9f4e5af39a3d2f6e10616bc9d6cdded13 |
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| container_title | Bulletin of the Malaysian Mathematical Sciences Society |
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| creator | Calderón Martín, Antonio J. |
| description | 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. |
| doi_str_mv | 10.1007/s40840-017-0446-3 |
| format | article |
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(
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.</description><identifier>ISSN: 0126-6705</identifier><identifier>EISSN: 2180-4206</identifier><identifier>DOI: 10.1007/s40840-017-0446-3</identifier><language>eng</language><publisher>Singapore: Springer Singapore</publisher><subject>Algebra ; Applications of Mathematics ; Mathematics ; Mathematics and Statistics</subject><ispartof>Bulletin of the Malaysian Mathematical Sciences Society, 2017-04, Vol.40 (2), p.679-695</ispartof><rights>Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017</rights><rights>Bulletin of the Malaysian Mathematical Sciences Society is a copyright of Springer, (2017). All Rights Reserved.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-3a7108f2a927a7370e5469d646a5576e9f4e5af39a3d2f6e10616bc9d6cdded13</citedby><cites>FETCH-LOGICAL-c316t-3a7108f2a927a7370e5469d646a5576e9f4e5af39a3d2f6e10616bc9d6cdded13</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,27900,27901</link.rule.ids></links><search><creatorcontrib>Calderón Martín, Antonio J.</creatorcontrib><title>Leibniz Algebras Admitting a Multiplicative Basis</title><title>Bulletin of the Malaysian Mathematical Sciences Society</title><addtitle>Bull. Malays. Math. Sci. Soc</addtitle><description>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.</description><subject>Algebra</subject><subject>Applications of Mathematics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><issn>0126-6705</issn><issn>2180-4206</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><recordid>eNp1kD1PwzAURS0EElXpD2CLxGx4zx_PyVgqoEhFLDBbbuJUrtK02CkS_HpchYGFt9zl3Pukw9g1wi0CmLukoFTAAQ0HpYjLMzYRWAJXAuicTQAFcTKgL9kspS3k0yRI4IThyod1H76Lebfx6-hSMW92YRhCvylc8XLshnDoQu2G8OmLe5dCumIXreuSn_3mlL0_Prwtlnz1-vS8mK94LZEGLp1BKFvhKmGckQa8VlQ1pMhpbchXrfLatbJyshEteQRCWteZqJvGNyin7GbcPcT9x9GnwW73x9jnlxYrQi1KYXSmcKTquE8p-tYeYti5-GUR7EmOHeXYLMee5FiZO2LspMz2Gx__LP9b-gG8b2UZ</recordid><startdate>20170401</startdate><enddate>20170401</enddate><creator>Calderón Martín, Antonio J.</creator><general>Springer Singapore</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>8FE</scope><scope>8FG</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>BVBZV</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>HCIFZ</scope><scope>L6V</scope><scope>M7S</scope><scope>PHGZM</scope><scope>PHGZT</scope><scope>PKEHL</scope><scope>PQEST</scope><scope>PQGLB</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope></search><sort><creationdate>20170401</creationdate><title>Leibniz Algebras Admitting a Multiplicative Basis</title><author>Calderón Martín, Antonio J.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-3a7108f2a927a7370e5469d646a5576e9f4e5af39a3d2f6e10616bc9d6cdded13</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>Algebra</topic><topic>Applications of Mathematics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Calderón Martín, Antonio J.</creatorcontrib><collection>CrossRef</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni)</collection><collection>ProQuest Central UK/Ireland</collection><collection>AUTh Library subscriptions: ProQuest Central</collection><collection>Technology Collection</collection><collection>East & South Asia Database</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central</collection><collection>SciTech Premium Collection (Proquest) (PQ_SDU_P3)</collection><collection>ProQuest Engineering Collection</collection><collection>ProQuest Engineering Database</collection><collection>ProQuest Central (New)</collection><collection>ProQuest One Academic (New)</collection><collection>ProQuest One Academic Middle East (New)</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Applied & Life Sciences</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering collection</collection><jtitle>Bulletin of the Malaysian Mathematical Sciences Society</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Calderón Martín, Antonio J.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Leibniz Algebras Admitting a Multiplicative Basis</atitle><jtitle>Bulletin of the Malaysian Mathematical Sciences Society</jtitle><stitle>Bull. Malays. Math. Sci. Soc</stitle><date>2017-04-01</date><risdate>2017</risdate><volume>40</volume><issue>2</issue><spage>679</spage><epage>695</epage><pages>679-695</pages><issn>0126-6705</issn><eissn>2180-4206</eissn><abstract>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.</abstract><cop>Singapore</cop><pub>Springer Singapore</pub><doi>10.1007/s40840-017-0446-3</doi><tpages>17</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0126-6705 |
| ispartof | Bulletin of the Malaysian Mathematical Sciences Society, 2017-04, Vol.40 (2), p.679-695 |
| issn | 0126-6705 2180-4206 |
| language | eng |
| recordid | cdi_proquest_journals_1961528275 |
| source | Springer Link |
| subjects | Algebra Applications of Mathematics Mathematics Mathematics and Statistics |
| title | Leibniz Algebras Admitting a Multiplicative Basis |
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