• Journal of the Korean Mathematical Society
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• 제56권5호
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• pp.1333-1354
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• 2019

In authors' paper in 2007, it was shown that the BSE-extension of $C^1_0(R)$, the algebra of continuously differentiable functions f on the real number space R such that f and df /dx vanish at infinity, is the Lipschitz algebra $Lip_1(R)$. This paper extends this result to the case of $C^n_0(R^d)$ and $C^{n-1,1}_b(R^d)$, where n and d represent arbitrary natural numbers. Here $C^n_0(R^d)$ is the space of all n-times continuously differentiable functions f on $R^d$ whose k-times derivatives are vanishing at infinity for k = 0, ${\cdots}$, n, and $C^{n-1,1}_b(R^d)$ is the space of all (n - 1)-times continuously differentiable functions on $R^d$ whose k-times derivatives are bounded for k = 0, ${\cdots}$, n - 1, and (n - 1)-times derivatives are Lipschitz. As a byproduct of our investigation we obtain an important result that $C^{n-1,1}_b(R^d)$ has a predual.

• Communications of the Korean Mathematical Society
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• 제21권1호
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• pp.45-52
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• 2006

Every non-associative algebra L corresponds to its symmetric semi-Lie algebra $L_{[,]}$ with respect to its commutator. It is an interesting problem whether the equality $Aut{non}(L)=Aut_{semi-Lie}(L)$ holds or not [2], [13]. We find the non-associative algebra automorphism groups $Aut_{non}\; \frac\;{(WN_{0,0,1}_{[0,1,r_1...,r_p])}$ and $Aut_{non-Lie}\; \frac\;{(WN_{0,0,1}_{[0,1,r_1...,r_p])}$ where every automorphism of the automorphism groups is the composition of elementary maps [3], [4], [7], [8], [9], [10], [11]. The results of the paper show that the F-algebra automorphism groups of a polynomial ring and its Laurent extension make easy to find the automorphism groups of the algebras in the paper.

• Honam Mathematical Journal
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• 제27권4호
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• pp.571-581
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• 2005

The Weyl-type non-associative algebra ${\overline{WN_{g_n,m,s_r}}$ and its subalgebra ${\overline{WN_{n,m,s_r}}$ are defined and studied in the papers [8], [9], [10], [12]. We will prove that the Weyl-type non-associative algebra ${\overline{WN_{n,0,0_{[2]}}}$ and its corresponding semi-Lie algebra are simple. We find the non-associative algebra automorphism group $Aut_{non}({\overline{WN_{1,0,0_{[2]}}})$.

• Bulletin of the Korean Mathematical Society
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• 제60권6호
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• pp.1687-1695
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• 2023

In this paper, we study the Green ring r(𝔴⁰_n) of the weak Hopf algebra 𝔴⁰_n based on Taft Hopf algebra H_n(q). Let R(𝔴⁰_n) := r(𝔴⁰_n) ⊗_ℤ ℂ be the Green algebra corresponding to the Green ring r(𝔴⁰_n). We first determine all finite dimensional simple modules of the Green algebra R(𝔴⁰_n), which is based on the observations of the roots of the generating relations associated with the Green ring r(𝔴⁰_n). Then we show that the nilpotent elements in r(𝔴⁰_n) can be written as a sum of finite dimensional indecomposable projective 𝔴⁰_n-modules. The Jacobson radical J(r(𝔴⁰_n)) of r(𝔴⁰_n) is a principal ideal, and its rank equals n - 1. Furthermore, we classify all finite dimensional non-simple indecomposable R(𝔴⁰_n)-modules. It turns out that R(𝔴⁰_n) has n² - n + 2 simple modules of dimension 1, and n non-simple indecomposable modules of dimension 2.

• Honam Mathematical Journal
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• 제36권1호
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• pp.179-186
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• 2014

In this paper, we consider the simple non-associative algebra $\overline{WN(\mathbb{F}[e^{{\pm}x^r},0,1]_{(\partial,\partial^2)})}$. There are many papers on finding the derivations of an associative algebra, a Lie algebra, and a non-associative algebra (see [2], [3], [4], [5], [6], [7], [12], [14]). We find all the derivations of the algebra $\overline{WN(\mathbb{F}[e^{{\pm}x^r},0,1]_{(\partial,\partial^2)})}$.

• Honam Mathematical Journal
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• 제27권4호
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• pp.583-593
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• 2005

The Weyl-type non-associative algebra ${\overline{WN_{g_n,m,s_r}}$ and its subalgebra ${\overline{WN_{n,m,s_r}}$ are defined and studied in the papers [2], [3], [9], [11], [12]. We find the derivation group $Der_{non}({\overline{WN_{1,0,0_{[2]}}})$ the non-associative simple algebra ${\overline{WN_{1,0,0_{[2]}}}$.

• Honam Mathematical Journal
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• 제31권3호
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• pp.407-419
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• 2009

The simple non-associative algebra $N(e^{A_S},q,n,t)_k$ and its simple sub-algebras are defined in the papers [1], [3], [4], [5], [6], [12]. We define the non-associative algebra $\overline{WN_{(g_n,\mathfrak{U}),m,s_B}}$ and its antisymmetrized algebra $\overline{WN_{(g_n,\mathfrak{U}),m,s_B}}$. We also prove that the algebras are simple in this work. There are various papers on finding all the derivations of an associative algebra, a Lie algebra, and a non-associative algebra (see [3], [5], [6], [9], [12], [14], [15]). We also find all the derivations $Der_{anti}(WN(e^{{\pm}x^r},0,2)_B^-)$ of te antisymmetrized algebra $WN(e^{{\pm}x^r}0,2)_B^-$ and every derivation of the algebra is outer in this paper.

• Journal of the Korean Mathematical Society
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• 제34권1호
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• pp.159-165
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• 1997

In this paper, k will denote an arbitrary field. If m, n are natural numbers, then $M_{m \times n}(k)$ will denote the set of all $m \times n$ matrices with entries in k. Every k-algebras will be assumed to contain a (multiplicative) identity $1 \neq 0$. A k-subspace $R_0$ of a k-algebra R will be called a k-subalgebra of R if $R_0$ is closed under multiplication from R and $R_0$ contains the identity of R. We will assume all k-algebra homomorphisms take the identity to identity.

• Honam Mathematical Journal
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• 제29권2호
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• pp.213-222
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• 2007

A Weyl type algebra is defined in the paper (see [2],[4], [6], [7]). A Weyl type non-associative algebra $\bar{WN_{m,n,s}}$ and its restricted subalgebra $\bar{WN_{m,n,s_r}}$ are defined in the papers (see [1], [14], [16]). Several authors find all the derivations of an associative (Lie or non-associative) algebra (see [3], [1], [5], [7], [10], [16]). We find Der($\bar_{WN_{0,0,1_n}}$) of the algebra $\bar_{WN_{0,0,1_n}}$ and show that the algebras $\bar_{WN_{0,0,1_n}}$ and $\bar_{WN_{0,0,s_1}}$ are not isomorphic in this work. We show that the associator of $\bar_{WN_{0,0,1_n}}$ is zero.

• Kyungpook Mathematical Journal
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• 제56권4호
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• pp.1047-1067
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• 2016

In this paper, a new class of diagram algebras which are subalgebras of signed brauer algebras, called the Walled Signed Brauer algebras denoted by ${\overrightarrow{D}}_{r,s}(x)$, where $r,s{\in}{\mathbb{N}}$ and x is an indeterminate are introduced. A presentation of walled signed Brauer algebras in terms of generators and relations is given. The cellularity of a walled signed Brauer algebra is established. Finally, ${\overrightarrow{D}}_{r,s}(x)$, is quasi- hereditary if either the characteristic of a field, say p, p = 0 or p > max(r, s) and either $x {\neq}0$ or x = 0 and $r{\neq}s$.