• Journal of the Chungcheong Mathematical Society
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• v.21 no.3
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• pp.301-320
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• 2008

The smash product algebra has been generalized to general smash product algebra in [3] and we can generalize the smash coproduct coalgebra to obtain the general smash coproduct coalgebra. It is natural to replace the smash product and smash coproduct by the generalized smash product and generalized smash coproduct and consider the condition under which the generalized smash product algebra structure and the generalized smash coproduct coalgebra structure will inherit a bialgebra structure or a Hopf algebra structure. We derive necessary sufficient conditions for the problem. This generalizes the corresponding results in [7] and [4].

• Journal of the Chungcheong Mathematical Society
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• v.19 no.1
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• pp.57-60
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• 2006

Several examples of smash product are given.

• Journal of the Chungcheong Mathematical Society
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• v.8 no.1
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• pp.173-181
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• 1995

Let H and G be finite dimensional semisimple Hopf algebras and let A and B be left H and G-module algebras respectively. We use smash product algebras to show that 1) if A is right Artinian then $A^H$ is right Artinian, 2) $Soc\;V_A{\subset}Soc\;V_{A^H}$ and rad $V_A{\supset}\;radV_{A^H}$, 3) $K\;dim\;_BV_A=K\;dim\;_{B^G}V_{A^H}$.

• Bulletin of the Korean Mathematical Society
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• v.38 no.3
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• pp.527-541
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• 2001

We find the necessary and sufficient conditions for the smash product algebra structure and the crossed coproduct coalgebra structure with th dual cocycle $\alpha$ to afford a Hopf algebra (A equation,※See Full-text). If B and H are finite algebra and Hopf algebra, respectively, then the linear dual (※See Full-text) is also a Hopf algebra. We show that the weak coaction admissible mapping system characterizes the new Hopf algebras (※See Full-text).

• 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.

• Communications of the Korean Mathematical Society
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• v.11 no.1
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• pp.225-233
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• 1996

It is shown that for any space A, the cofibration X \to X \Join \sumA \to \sumA \wedge X$ decomposable when X is a co-T-space. It is also obtain necessary and sufficient conditions for the cofibration $X \to X \Join A \to A \wedge X$ is trivial, in the sense of cofibre homotopy type.

• Journal of the Chungcheong Mathematical Society
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• v.14 no.1
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• pp.85-103
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• 2001

Suppose that Hand K are paired Hopf algebras and that A is an H - K - bicomodule algebra with multiplication which is a left H-comodule map and is a right K-comodule map. We define a new twisted algebra, $A^{\tau}$ and define $M^{\tau}$ for $M{\in}M_A^K$. We find an equivalent condition for $M^{\tau}{\in}M_{A^{\tau}}^K$. We show that the above defined twisted multiplication is the special case of Beattie's twist multiplication. We show that if K is commutative, then A is an H-module algebra and show that if $H^*$ is cocommutative then the construction of smash product appears as a special case of the new twist product.

• Communications of the Korean Mathematical Society
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• v.13 no.2
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• pp.225-232
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• 1998

Let H be a finite dimensional Hopf algebra over a field k, and A be an H-module algebra over k which the H-action on A is D-continuous. We show that $Q_{max}(A)$, the maximal ring or quotients of A, is an H-module algebra. This is used to prove that if H is a finite dimensional semisimple Hopf algebra and A is a semiprime right(left) Goldie algebra than $A#H$ is a semiprime right(left) Goldie algebra. Assume that Asi a semiprime H-module algebra Then $A^H$ is left Artinian if and only if A is left Artinian.

• Journal of Microbiology and Biotechnology
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• v.33 no.6
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• pp.753-759
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• 2023

The genus Paenibacillus contains a variety of biologically active compounds that have potential applications in a range of fields, including medicine, agriculture, and livestock, playing an important role in the health and economy of society. Our study focused on the bacterium SS4^T (KCTC 43402^T = GDMCC 1.3498^T), which was characterized using a polyphasic taxonomic approach. This strain was analyzed using antiSMASH, BAGEL4, and PRISM to predict the secondary metabolites. Lassopeptide clusters were found using all three analysis methods, with the possibility of secretion. Additionally, PRISM found three biosynthetic gene clusters (BGC) and predicted the structure of the product. Genome analysis indicated that glucoamylase is present in SS4^T. 16S rRNA sequence analysis showed that strain SS4^T most closely resembled Paenibacillus marchantiophytorum DSM 29850^T (98.22%), Paenibacillus nebraskensis JJ-59^T (98.19%), and Paenibacillus aceris KCTC 13870^T (98.08%). Analysis of the 16S rRNA gene sequences and Type Strain Genome Server (TYGS) analysis revealed that SS4^T belongs to the genus Paenibacillus based on the results of the phylogenetic analysis. As a result of the matrix-assisted laser desorption/ionization-time-of-flight mass spectrometry (MALDI-TOF/MS) results, SS4^T was determined to belong to the genus Paenibacillus. Comparing P. marchantiophytorum DSM 29850^T with average nucleotide identity (ANI 78.97%) and digital DNA-DNA hybridization (dDDH 23%) revealed values that were all less than the threshold for bacterial species differentiation. The results of this study suggest that strain SS4^T can be classified as a Paenibacillus andongensis species and is a novel member of the genus Paenibacillus.