Университеттің 85 жылдығына арналған «Қазіргі заманғы математика:
проблемалары және қолданыстары» III халықаралық Тайманов оқуларының
материалдар жинағы, 25 қараша, 2022 жыл
11
ГРНТИ 27.17.23
WEAK LEIBNIZ ALGEBRAS AND TRANSPOSED POISSON ALGEBRAS
DZHUMADILDAEV A.S.
Kazakh-British Technical University, Almaty, Kazakhstan
A weak Leibniz
algebra is defined by the following polynomial identities
[𝑎, 𝑏]𝑐 = 2𝑎(𝑏𝑐) − 2𝑏(𝑎𝑐), 𝑎[𝑏, 𝑐] = 2(𝑎, 𝑏)𝑐 − 2(𝑎, 𝑐)𝑏.
Example.
Any two-sided Leibniz algebra is weak Leibniz. In particular, any Lie algebra
is weak Leibniz.
Example.
Let
𝜖
i
∈ 𝐾, 𝑓𝑜𝑟 𝑖 ∈ 𝐼,
and
𝐴
G
is an algebra with base
𝑒
i
, 𝑖 ∈ ℤ
,
and
multiplication
𝑒
i
𝑒
j
= (𝑖 − 𝑗)𝑒
i+j
+ ∑
s∈I
𝜖
s
𝑒
i+j+s
.
Then the algebra
𝐴
G
is non-Lie simple weak Leibniz algebra. Note that any simple Leibniz
algebra is Lie.
An algebra with two binary operations
𝐴 = (𝐴,∘,•),
is called
transposed Poisson
(see
[1]), if
(𝐴,∘)
is Lie,
(𝐴,•)
is associative commutative and associative part acts on Lie part as 1
/
2-
derivation,
2𝑎 • (𝑏 ∘ 𝑐) = (𝑎 • 𝑏) ∘ 𝑐 + 𝑏 • (𝑎 ∘ 𝑐),
∀𝑎, 𝑏, 𝑐 ∈ 𝐴.
Theorem 1.
(𝑝 ≠ 2)
If A is weak Leibniz, then the algebra
(𝐴,∘,•)
is transposed Poisson,
where
𝑎 ∘ 𝑏 = 𝑎𝑏 − 𝑏𝑎, 𝑎 • 𝑏 = 𝑎𝑏 + 𝑏𝑎
. Conversely, if
(𝐴,∘,•)
is transposed Poisson, then the
algebra A with multiplication
𝑎𝑏 = 1/2(𝑎 ∘ 𝑏 + 𝑏 ∘ 𝑎)
is weak Leibniz.
An algebra
(𝐴,∙,•)
is called
Novikov-Poisson,
if
I.
(𝐴,∙)
is (left) Novikov, for any
𝑎, 𝑏, 𝑐 ∈ 𝐴,
(𝑎 ∙ 𝑏 − 𝑏 ∙ 𝑎) ∙ 𝑐 = 𝑎 ∙ (𝑏 ∙ 𝑐) − 𝑏 ∙ (𝑎 ∙ 𝑐), (𝑎 ∙ 𝑏) ∙ 𝑐 = (𝑎 ∙ 𝑐) ∙ 𝑏,
II.
(𝐴,•)
is associative commutative, such that for any
𝑎, 𝑏, 𝑐 ∈ 𝐴,
𝑎 • (𝑏 ∙ 𝑐) = (𝑎 • 𝑏) ∙ 𝑐, 𝑎 ∙ (𝑏 • 𝑐) = (𝑎 ∙ 𝑏) • 𝑐 + 𝑏 • (𝑎 ∙ 𝑐),
Proposition.
Let
𝐴 = (𝐴,∙,•)
be Novikov-Poisson algebra. Then for any
𝑢, 𝑣 ∈ 𝐴
the
algebra
𝐴
u,v
= (𝐴,∘
u
, ,•
v
)
, where
𝑎 ∘
u
𝑣 = 𝑢 • (𝑎 ∙ 𝑏 − 𝑏 ∙ 𝑎), 𝑎 •
v
𝑏 = 𝑣 • (𝑎 • 𝑏),
is transposed Posson and the algebra
𝐴
u,v
under multiplication
𝑎𝑏 = 1/2(𝑎 ∘
u
𝑏 + 𝑎 •
v
𝑏)
is
weak Leibniz.
A weak Leibniz algebra
𝐴 = (𝐴,∙)
is called
special,
if there exists transposed Poisson
algebra
𝐵
u,v
constructed by Novikov-Poisson algebra
𝐵
for some
𝑢, 𝑣 ∈ 𝐵,
such that
𝐴
is a
subalgebra of
𝐵
u,v
.
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