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Characterization generalized derivations of tensor products of nonassociative algebras
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Abstract
Consider
A
{\mathcal{A}}
and
ℬ
{\mathcal{ {\mathcal B} }}
to be nonassociative unital algebras. Under the assumption that either one of them has finite dimensions or that both are finite dimensions, a generalized derivation is an additive map
ℱ
:
A
→
A
{\mathcal{ {\mathcal F} }}:{\mathcal{A}}\to {\mathcal{A}}
associated with a derivation
d
{\mathcal{d}}
of
A
{\mathcal{A}}
if
ℱ
(
u
v
)
=
ℱ
(
u
)
v
+
u
d
(
v
)
{\mathcal{ {\mathcal F} }}\left(uv)={\mathcal{ {\mathcal F} }}\left(u)v+u{\mathcal{d}}\left(v)
for all
u
,
v
∈
A
u,v\in {\mathcal{A}}
. The objective of this study is to characterize and elucidate the structure of a generalized derivation on the tensor product of nonassociative algebras. Specifically, we prove that if
ℱ
{\mathcal{ {\mathcal F} }}
is a generalized derivation of
A
⊗
ℬ
{\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }}
associated with a derivation
d
{\mathcal{d}}
of
A
⊗
ℬ
{\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }}
, then
ℱ
=
ℒ
u
+
d
{\mathcal{ {\mathcal F} }}={{\mathcal{ {\mathcal L} }}}_{u}+{\mathcal{d}}
, where
ℒ
u
{{\mathcal{ {\mathcal L} }}}_{u}
is a left multiplication by
u
u
and
u
u
belongs to the left nucleus of
A
⊗
ℬ
{\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }}
(i.e.,
ℒ
u
r
=
u
r
{{\mathcal{ {\mathcal L} }}}_{u}r=ur
for all
r
∈
A
⊗
ℬ
r\in {\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }}
and
u
∈
N
l
(
A
⊗
ℬ
)
u\in {N}_{l}\left({\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }})
). Moreover, every generalized derivation of
A
⊗
ℬ
{\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }}
can be represented as the sum of the derivations of the three categories: (i)
w
+
a
d
u
w+a{\mathcal{d}}u
, where
u
,
w
∈
N
(
A
⊗
ℬ
)
u,w\in N\left({\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }})
, (ii)
ℒ
z
⊗
f
{{\mathcal{ {\mathcal L} }}}_{z}\otimes f
, where
f
f
is a derivation of
ℬ
{\mathcal{ {\mathcal B} }}
and
z
∈
Z
(
A
)
z\in Z\left({\mathcal{A}})
(the center of
A
{\mathcal{A}}
), and (iii)
g
⊗
ℒ
w
g\otimes {{\mathcal{ {\mathcal L} }}}_{w}
, where
g
g
is a derivation of
A
{\mathcal{A}}
and
w
∈
Z
(
ℬ
)
w\in Z\left({\mathcal{ {\mathcal B} }})
.
Title: Characterization generalized derivations of tensor products of nonassociative algebras
Description:
Abstract
Consider
A
{\mathcal{A}}
and
ℬ
{\mathcal{ {\mathcal B} }}
to be nonassociative unital algebras.
Under the assumption that either one of them has finite dimensions or that both are finite dimensions, a generalized derivation is an additive map
ℱ
:
A
→
A
{\mathcal{ {\mathcal F} }}:{\mathcal{A}}\to {\mathcal{A}}
associated with a derivation
d
{\mathcal{d}}
of
A
{\mathcal{A}}
if
ℱ
(
u
v
)
=
ℱ
(
u
)
v
+
u
d
(
v
)
{\mathcal{ {\mathcal F} }}\left(uv)={\mathcal{ {\mathcal F} }}\left(u)v+u{\mathcal{d}}\left(v)
for all
u
,
v
∈
A
u,v\in {\mathcal{A}}
.
The objective of this study is to characterize and elucidate the structure of a generalized derivation on the tensor product of nonassociative algebras.
Specifically, we prove that if
ℱ
{\mathcal{ {\mathcal F} }}
is a generalized derivation of
A
⊗
ℬ
{\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }}
associated with a derivation
d
{\mathcal{d}}
of
A
⊗
ℬ
{\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }}
, then
ℱ
=
ℒ
u
+
d
{\mathcal{ {\mathcal F} }}={{\mathcal{ {\mathcal L} }}}_{u}+{\mathcal{d}}
, where
ℒ
u
{{\mathcal{ {\mathcal L} }}}_{u}
is a left multiplication by
u
u
and
u
u
belongs to the left nucleus of
A
⊗
ℬ
{\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }}
(i.
e.
,
ℒ
u
r
=
u
r
{{\mathcal{ {\mathcal L} }}}_{u}r=ur
for all
r
∈
A
⊗
ℬ
r\in {\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }}
and
u
∈
N
l
(
A
⊗
ℬ
)
u\in {N}_{l}\left({\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }})
).
Moreover, every generalized derivation of
A
⊗
ℬ
{\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }}
can be represented as the sum of the derivations of the three categories: (i)
w
+
a
d
u
w+a{\mathcal{d}}u
, where
u
,
w
∈
N
(
A
⊗
ℬ
)
u,w\in N\left({\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }})
, (ii)
ℒ
z
⊗
f
{{\mathcal{ {\mathcal L} }}}_{z}\otimes f
, where
f
f
is a derivation of
ℬ
{\mathcal{ {\mathcal B} }}
and
z
∈
Z
(
A
)
z\in Z\left({\mathcal{A}})
(the center of
A
{\mathcal{A}}
), and (iii)
g
⊗
ℒ
w
g\otimes {{\mathcal{ {\mathcal L} }}}_{w}
, where
g
g
is a derivation of
A
{\mathcal{A}}
and
w
∈
Z
(
ℬ
)
w\in Z\left({\mathcal{ {\mathcal B} }})
.
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