• 대한수학회보
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• 제52권5호
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• pp.1587-1606
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• 2015

Comonoid, bi-algebra, and Hopf algebra structures are studied within the universal-algebraic context of entropic varieties. Attention focuses on the behavior of setlike and primitive elements. It is shown that entropic $J{\acute{o}}nsson$-Tarski varieties provide a natural universal-algebraic setting for primitive elements and group quantum couples (generalizations of the group quantum double). Here, the set of primitive elements of a Hopf algebra forms a Lie algebra, and the tensor algebra on any algebra is a bi-algebra. If the tensor algebra is a Hopf algebra, then the underlying $J{\acute{o}}nsson$-Tarski monoid of the generating algebra is cancellative. The problem of determining when the $J{\acute{o}}nsson$-Tarski monoid forms a group is open.

• 호남수학학술지
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• 제7권1호
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• pp.119-127
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• 1985

A study is made of a special type of $C^{\ast}$ -dynamical systems, consisting of a class $n^{\infty}$ uniformly hyperfinite $C^{\ast}$-algebra A, the torus group $G=T^{d}$ ($$1{\leq_-}d{\leq_-}n-1$$) and a natural product action of G on A by $^{*}-automorphisms$. We give some conditions for product states on the fixed point algebra $A^{G}$ of A by G to be factorial.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제20권4호
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• pp.243-249
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• 2013

Let A be an algebra and D a derivation of A. Then D is called algebraic nil if for any $x{\in}A$ there is a positive integer n = n(x) such that $D^{n(x)}(P(x))=0$, for all $P{\in}\mathbb{C}[X]$ (by convention $D^{n(x)}({\alpha})=0$, for all ${\alpha}{\in}\mathbb{C}$). In this paper, we show that any algebraic nil derivation (possibly unbounded) on a commutative complex algebra A maps into N(A), where N(A) denotes the set of all nilpotent elements of A. As an application, we deduce that any nilpotent derivation on a commutative complex algebra A maps into N(A), Finally, we deduce two noncommutative versions of algebraic nil derivations inclusion range.

• 대한수학회지
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• 제58권3호
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• pp.643-702
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• 2021

Fix ℤ/2 is the prime field of two elements and write 𝒜₂ for the mod 2 Steenrod algebra. Denote by GL_d := GL(d, ℤ/2) the general linear group of rank d over ℤ/2 and by ${\mathfrak{P}}_d$ the polynomial algebra ℤ/2[x₁, x₂, …, x_d] as a connected unstable 𝒜₂-module on d generators of degree one. We study the Peterson "hit problem" of finding the minimal set of 𝒜₂-generators for ${\mathfrak{P}}_d$. Equivalently, we need to determine a basis for the ℤ/2-vector space $$Q{\mathfrak{P}}_d:={\mathbb{Z}}/2{\otimes}_{\mathcal{A}_2}\;{\mathfrak{P}}_d{\sim_=}{\mathfrak{P}}_d/{\mathcal{A}}^+_2{\mathfrak{P}}_d$$ in each degree n ≥ 1. Note that this space is a representation of GL_d over ℤ/2. The problem for d = 5 is not yet completely solved, and unknown in general. In this work, we give an explicit solution to the hit problem of five variables in the generic degree n = r(2^t - 1) + 2^ts with r = d = 5, s = 8 and t an arbitrary non-negative integer. An application of this study to the cases t = 0 and t = 1 shows that the Singer algebraic transfer of rank 5 is an isomorphism in the bidegrees (5, 5 + (13.2⁰ - 5)) and (5, 5 + (13.2¹ - 5)). Moreover, the result when t ≥ 2 was also discussed. Here, the Singer transfer of rank d is a ℤ/2-algebra homomorphism from GL_d-coinvariants of certain subspaces of $Q{\mathfrak{P}}_d$ to the cohomology groups of the Steenrod algebra, $Ext^{d,d+*}_{\mathcal{A}_2}$ (ℤ/2, ℤ/2). It is one of the useful tools for studying these mysterious Ext groups.

• 충청수학회지
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• 제10권1호
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• pp.127-139
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• 1997

The spherical non-commutative $\mathbb{S}_{\omega}$ were defined in [2,3]. Assume that no non-trivial matrix algebra can be factored out of the $\mathbb{S}_{\omega}$, and that the fibres are isomorphic to the tensor product of a completely irrational non-commutative torus with a matrix algebra $M_k(\mathbb{C})$. It is shown that the tensor product of the spherical non-commutative torus $\mathbb{S}_{\omega}$ with the even Cuntz algebra $\mathcal{O}_{2d}$ has a trivial bundle structure if and only if k and 2d - 1 are relatively prime, and that the tensor product of the spherical non-commutative torus $S_{\omega}$ with the generalized Cuntz algebra $\mathcal{O}_{\infty}$ has a non-trivial bundle structure when k > 1.

• 호남수학학술지
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• 제31권4호
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• pp.489-495
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• 2009

Any BCK-ideal of a d-algebra can be decomposed into the union of some sets. The notion of a complicated d-algebra is introduced and some related properties are obtained.

• 충청수학회지
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• 제27권2호
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• pp.227-236
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• 2014

In this paper, we introduce the notion of a generalized derivation in a BE-algebra, and consider the properties of generalized derivations. Also, we characterize the fixed set $Fix_d(X)$ and Kerd by generalized derivations. Moreover, we prove that if d is a generalized derivation of a BE-algebra, every filter F is a d-invariant.

• 호남수학학술지
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• 제36권1호
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• pp.167-178
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• 2014

In this paper, we introduce the notion of derivation in a BE-algebra, and consider the properties of derivations. Also, we characterize the fixed set $Fix_d(X)$ and Kerd by derivations. Moreover, we prove that if d is a derivation of BE-algebra, every filter F is a d-invariant.

• 충청수학회지
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• 제28권1호
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• pp.127-138
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• 2015

In this paper, we introduce the notion of f-derivation in a BE-algebra, and consider the properties of f-derivations. Also, we characterize the fixed set $Fix_d(X)$ and Kerd by f-derivations. Moreover, we prove that if d is a f-derivation of a BE-algebra, every f-filter F is a a d-invariant.

• 충청수학회지
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• 제34권2호
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• pp.119-129
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• 2021

In this paper, we introduce the concept of a generalized right f-derivation associated with a derivation d and a function f in 𝚪-incline algebras and give some properties of 𝚪-incline algebras. Also, the concept of d-ideal is introduced in a 𝚪-incline algebra with respect to right f-derivations.