Some Properties Of Tortken Superalgebras And The Classification Of Low Dimensional Tortken Superalgebras

a minimal identity of minimal degree for an algebra A that does not follow from the commutativity identity is a Jordan identity for A , Askar Dzhumadil'daev[1] proved that the Jordan identity for a Novikov-Jordan algebra is an identity with degree 4 , he named it Tortken identity. This paper emphasizes on the study of Tortken superalgebra and its properties, we give the definition of Tortken superalgebra first ,then we classfy the 2-dimensional Tortken superalgebra while A₁ ≠ 0;we give an equivalence for a 2-dimensional superalgebra to be tortken while A₁ = 0;we also discuss some properties on Tortken superalgebras and we get an identity with degree 4 on Tortken superalgebra .The main results in this paper are the following:Theorem 2.2: (A, ×) is a 2-dimensional superalgera on field R, A = A₀ A₁;A₀ =< a >, A₁ =< b > are both 1-dimensional subalgebra of A, a, b ≠ 0,then we have1) a×b = 0, b × a = 0, a × a = 0, b×b = 0, A is trivial2) a×b = b, b×a = mb, a × a = 0, b×b = 0, m∈R3) a×b = b, b × a = b, a × a = ma, b×n = 0, m ≠ 04) a×b = 0, b×a = 0, a × a = 0, b ×b = a,5) a×b = 0, b×a = 0, a× a = a, b×b = 0,6) a×b = 0, b × a = b, a × a = ma, b×b = 0. m∈RTheorem 2.3: A₁ = {0}, A₀ = , then A is a Tortken superalgebra ifand only if the following statements hold:1) (x × y) × (x × x) - (x × x) × (x × y) = (x, y, x) × x - (x, x, x) × y 2)(y × y) × (x × x) - (y × x) × (x × y) = (y,y,x) × x - (y, x, x) × y 3) (x × y) × (y × x) - (x × x) × (y × y) = (x, y, y) × x - (x, x,y) × y 4)(y ×y)×(y×x)-(y×x)× (y × y) = (y,y,y) ×x-(y,x,y) × y Proposition 3.3: A is a super-commutative Tortken superalgebra , then forx ,a ,b , c ∈ hg(A),we have((x * o) * b) * c + (-1)^{d(a)·d(c)+d(a)·d(b)}((x *b)*c)*aTheorem 4.3: Novikov superalgebras under super Jordan multiplication areTortken superalgebras.Proposition 5.4: A is (left)leibniz dual superalgebra,then A satisfies super-tortkenidentity.