• Korean Journal of Mathematics
• /
• v.26 no.2
• /
• pp.175-189
• /
• 2018

In this paper, at first we generalize the notion of algebra over a field. A ${\Gamma}$-algebra is an algebraic structure consisting of a vector space V, a groupoid ${\Gamma}$ together with a map from $V{\times}{\Gamma}{\times}V$ to V. Then, on every associative ${\Gamma}$-algebra V and for every ${\alpha}{{\in}}{\Gamma}$ we construct an ${\alpha}$-Lie algebra. Also, we discuss some properties about ${\Gamma}$-Lie algebras when V and ${\Gamma}$ are the sets of $m{\times}n$ and $n{\times}m$ matrices over a field F respectively. Finally, we define the notions of ${\alpha}$-derivation, ${\alpha}$-representation, ${\alpha}$-nilpotency and prove Engel theorem in this case.

• Journal of the Chungcheong Mathematical Society
• /
• v.23 no.1
• /
• pp.177-184
• /
• 2010

Let G be an abelian group, $\alpha$ an antisymmetric bicharacter on G and g a (G, $\alpha$)-Lie algebra. Here we give a complete proof for that the associated graded algebra determined by a natural filtration in the universal enveloping algebra U(g) is a (G, $\alpha$)-Poisson Hopf algebra.

• Communications of the Korean Mathematical Society
• /
• v.16 no.1
• /
• pp.85-94
• /
• 2001

Let g=g(A) be an affine Lie algebra of type A(sub)2ι(sup)(2)(ι>1) and W be its Weyl group. In this paper, we find $\omega$$\in$W such that $\Delta$+U{1/2($\alpha$ι+$\delta$), 1/2($\alpha$ι+3$\delta$)}⊂$\Delta$+($\omega$).

• Communications of the Korean Mathematical Society
• /
• v.31 no.1
• /
• pp.101-113
• /
• 2016

In this article, we considered the stability of the following (${\alpha}$, ${\beta}$, ${\gamma}$)-derivation $${\alpha}D[x,y]={\beta}[D(x),y]+{\gamma}[x,D(y)]$$ and homomorphisms associated to the quadratic type functional equation $$f(kx+y)+f(kx+{\sigma}(y))=2kg(x)+2g(y),\;x,y{\in}A$$, where ${\sigma}$ is an involution of the Lie $C^*$-algebra A and k is a fixed positive integer. The Hyers-Ulam stability on unbounded domains is also studied. Applications of the results for the asymptotic behavior of the generalized quadratic functional equation are provided.