• Communications of the Korean Mathematical Society
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• v.35 no.2
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• pp.371-399
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• 2020

Let P_s := 𝔽₂[x₁, x₂, …, x_s] = ⊕_n⩾0(P_s)_n be the polynomial algebra viewed as a graded left module over the mod 2 Steenrod algebra, ${\mathfrak{A}}$. The grading is by the degree of the homogeneous terms (P_s)_n of degree n in the variables x₁, x₂, …, x_s of grading 1. We are interested in the hit problem, set up by F. P. Peterson, of finding a minimal system of generators for ${\mathfrak{A}}$-module P_s. Equivalently, we want to find a basis for the 𝔽₂-graded vector space ${\mathbb{F}}_2{\otimes}_{\mathfrak{A}}$ P_s. In this paper, we study the hit problem in the case s = 5 and the degree n = 5(2^t - 1) + 6 · 2^t with t an arbitrary positive integer.

• Bulletin of the Korean Mathematical Society
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• v.36 no.2
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• pp.337-342
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• 1999

M. Cohen and S. Montgomery showed that a Maschke-type theorem for the smash product, which unlike the corresponding result for group actions, does not require any assumptions about the characterstic of the algebra. Our purpose in this paper is a Maschke-type theorem for the graded smash coproduct C⋊kG: let V be a right C⋊kG-comodule and W a C⋊kG-subcomoduleof V which is a C-direct summand of V. Then W is a C⋊kG-direct summand of V. Also this result is equivalent to the following : let V be a graded right C-comodule and W a graded subcomodule of V which has a complement as a C-subcomodule of V. Then W has a graded complement.

• Honam Mathematical Journal
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• v.22 no.1
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• pp.17-20
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• 2000

• Honam Mathematical Journal
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• v.28 no.2
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• pp.197-204
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• 2006

For the evaluation algebra $F[e^{{\pm}{\chi}}]_M$, if M={$\partial$}, the automorphism group $Aut_{non}$($F[e^{{\pm}{\chi}}]_M$) and $Der_{non}$($F[e^{{\pm}{\chi}}]_M$) of the evaluation algebra $F[e^{{\pm}{\chi}}]_M$ are found in the paper [12]. For M={${\partial}^n$}, we find $Aut_{non}$($F[e^{{\pm}{\chi}}]_M$) and $Der_{non}$($F[e^{{\pm}{\chi}}]_M$) of the evaluation algebra $F[e^{{\pm}{\chi}}]_M$ in this paper. We show that a derivation of some non-associative algebra is not inner.

• Communications of the Korean Mathematical Society
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• v.23 no.2
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• pp.153-159
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• 2008

For $\mathbb{F}[e^{{\pm}x}]_{\{{\partial}\}}$, all the derivations of the evaluation algebra $\mathbb{F}[e^{{\pm}x}]_{\{{\partial}\}}$ is found in the paper (see [16]). For $M=\{{\partial}_1,\;{\partial}_1^2\},\;Der_{non}(\mathbb{F}[e^{{\pm}x}]_M))$ of the evaluation algebra $\mathbb{F}[e^{{\pm}x},\;e^{{\pm}y}]_M$ is found in the paper (see [2]). For $M=({\partial}_1^2,\;{\partial}_2^2)$, we find $Der_{non}(\mathbb{F}[e^{{\pm}x},\;e^{{\pm}y}]_M))$ of the evaluation algebra $\mathbb{F}[e^{{\pm}x},\;e^{{\pm}y}]_M$ in this paper.

• Journal of the Korean Mathematical Society
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• v.60 no.2
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• pp.303-325
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• 2023

In this paper, we introduce relative Rota-Baxter systems on Leibniz algebras and give some characterizations and new constructions. Then we construct a graded Lie algebra whose Maurer-Cartan elements are relative Rota-Baxter systems. This allows us to define a cohomology theory associated with a relative Rota-Baxter system. Finally, we study formal deformations and extendibility of finite order deformations of a relative Rota-Baxter system in terms of the cohomology theory.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.517-523
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• 2007

Let R/I be an Artinian algebra of codimension 3 with Hilbert function H such that $h_{d-1}>h_d=h_{d+1}$. Ahn and Shin showed that A cannot be level if ${\beta}_{1,d+2}(Gin(I))={\beta}_{2,d+2}(Gin(I))$ where Gin(I) is a generic initial ideal of I. We prove that some certain graded Artinian algebra R/I cannot be level if either ${\beta}_{1,d}(I^{lex})={\beta}_{2,d}(I^{lex})+1\;or\;{\beta}_{1,d+1}(I^{lex})={\beta}_{2,d+1}(I^{lex})\;where\;I^{lex}$ is a lex-segment ideal associated to I.

• Journal of the Korean Mathematical Society
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• v.39 no.1
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• pp.127-135
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• 2002

Let I be an ideal in a Gorenstein local ring A with the maximal ideal m. Then we say that I is an equimultiple good ideal in A, if I contains a reduction Q = ( $a_1$, $a_2$,ㆍㆍㆍ, $a_{s}$ ) generated by s elements in A and G(I) =(equation omitted)$_{n 0}$ $I^{n}$ / $I^{n＋1}$ of I is a Gorenstein ring with a(G(I)) = 1 - s, where s = h $t_{A}$ I and a(G(I)) denotes the a-invariant of G(I). Let $X_{A}$$^{s}$ denote the set of equimultiple good ideals I in A with h $t_{A}$ I = s, R(I) = A [It] be the Rees algebra of I, and $K_{R(I)}$ denote the canonical module of R(I). Let a I such that $I^{n＋l}$ = a $I^{n}$ for some n$\geq$0 and $\mu$$_{A}$(I)$\geq$2, where $\mu$$_{A}$(I) denotes the number of elements in a minimal system of generators of I. Assume that A/I is a Cohen-Macaulay ring. We show that the following conditions are equivalent. (1) $K_{R(I)}$(equation omitted)R(I)＋as graded R(I)-modules. (2) $I^2$ = aI and aA : I$\in$ $X^1$$_{A}$._{A}$./.

• Korean Journal of Mathematics
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• v.6 no.2
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• pp.313-318
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• 1998

In this paper we prove two independent theorems concerning formality of a nilmanifold and a differential graded algebra using the well-known theorem of Deligne-Griffiths-Morgan-Sullivan. We first give a rational homotopy theoretic proof to the statement that a nilmanifold is formal if and only if it is a torus. And then we study some conditions with which formality of one dga implies formality of the other in an extension of dga's.

• Journal of applied mathematics & informatics
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• v.20 no.1_2
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• pp.541-548
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• 2006

We find a graded Artinian level O-sequence of the form $H\;:\;h_0\;h_1\;\cdots\;h_{d-1}\;h_d\cdots$ $^{(d+1-1_)-st}h_d$ < $h_{d+s}$ not having the Weak-Lefschetz property. We also introduce several algorithms for construction of some examples of non-unimodal level O-sequences using a computer program called CoCoA.