Ki-Bong Nam

Lie Admissible Non-Associative Algebras

A non-associative ring which contains a well-known associative ring or Lie ring is interesting. In this paper, a method to construct a Lie admissible non-associative ring is given; a class of simple non-associative algebras is obtained; all the derivations of the non-associative simple

Derivations of a Restricted Weyl-Type Algebra Containing the Polynomial Ring

\overline {N_{0,0,1}}

Weyl Type Non-Associative Algebras Using Additive Groups I

algebra defined in this paper are determined; and finally, a solid algebra is defined.

Degree Stable Lie Algebras I

A Weyl type nonassociative algebra and its subalgebra have been defined in the articles Choi and Nam (2005a b c); Lee and Nam (2004). Several authors have found all the derivations of some given algebra (see Ahmadi et al., 2005; Choi and Nam, 2005b; Kac, 1974; Kirkman et al., 1994; Osborn, 1997; Osborn and Passman, 1995). In this article, we find all derivations of the nonassociative algebra and show that the dimension of all derivations of the algebra is (s 1 + s 2)2 + s 1 + s 2. Because of the dimension of a derivation algebra, we know that if s 1 + s 2 ≠ s 1′ + s 2′, then the algebras and are not isomorphic.

Quivers and Representations

A Weyl type algebra is defined in the book [4]. A Weyl type non-associative algebra and its restricted subalgebra are defined in various papers (see [1, 3, 11, 12]). Several authors find all the derivations of an associative (a Lie, a non-associative) algebra (see [1, 2, 4, 6, 11, 12]). We define the non-associative simple algebra and the semi-Lie algebra , where . We prove that the algebra is simple and find all its non-associative algebra derivations.

Non-associative Algebras with n-Exponential Functions

We define a degree stable Lie algebra. Since the special type Lie algebra S+(2) is degree stable, we find the automorphism group AutLie(S+(2)) of the Lie algebra S+(2) and prove the Jacobian conjecture of the Lie algebra S+(2).

Graded radical W type Lie algebras I

Publisher Summary This chapter discusses the use of the language of quivers in mathematics. Quivers arise in many areas of mathematics, including representation theory, algebraic geometry and differential geometry, Kac–Moody algebras, and quantum groups. A quiver is just a directed graph, and a representation of a quiver associates a vector space to each vertex and a linear map to each arrow. The introduction of quivers marked the starting point of the modern representation theory of finite-dimensional associative algebras. A number of remarkable connections to other mathematical fields have been discovered, in particular to Hall algebras, quantum groups, elliptic Lie algebras, and more recently cluster algebras. The representation theory of finite-dimensional algebras can be formulated using the language of quivers. In effect, the study of representations of a finite-dimensional algebra over an algebraically closed field is reduced to the study of a quotient of path algebra of a certain quiver. If the base field is not algebraically closed, “species” (or valued graphs) instead of quivers can be used to realize finite-dimensional algebras.

Combinatorial Algebra and Its Antisymmetrized Algebra I

Dernon(𝔽[x1, x2, …, xn]Mn) of the evaluation algebra 𝔽[x1, x2, …, xn]Mn and Dernon(𝔽[e± x1, e± x2, …, e± xn]Mn) of the evaluation algebra 𝔽[e± x1, e± x2, …, e± xn]Mn are found in [2] and [4], respectively, where Mn = {∂1, …, ∂n}. In this work we find of the algebra . We define a finite dimensional semi-Lie algebra which is simple. We define a simple semi-Lie ring whose dimension is finite.

New Simple Algebras Containing the Matrix Ring

We get a new ℤ-graded Witt type simple Lie algebra using a generalized polynomial ring which is the radical extension of the polynomial ring F[x] with the exponential function ex.

Automorphism Group of a Special Type Lie Algebra I

The simple non-associative algebra N(eAS, q, n, t)k and its simple subalgebras are defined in [1, 3, 5–7, 13]. In this work, we define the combinatorial algebra N(e𝔄p, n, t)k and its antisymmetrized algebra and their subalgebras. We prove that these algebras are simple. Some authors [2, 5–7, 10, 13, 14, 16, 17] found all the derivations of an associative algebra, a Lie algebra, and a non-associative algebra. We find all the derivations of the subalgebra N(e±x1x2 ⋯xn, 0, n)[1] of N(e𝔄p, n, t)k and the Lie subalgebra of .