ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS AND THEIR DUALS

Atsushi NAKAJİMA

doi:https://doi.org/10.24330/ieja.325932

Makale detayları

ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS AND THEIR DUALS

Yazarlar:

Atsushi NAKAJİMA

Yayıncı:

Abdullah HARMANCI

Ana Başlıklar:

Matematik

Dergi:

International Electronic Journal of Algebra 2017 Vol.22 Issue 22, pp.103-124

Belge Tipi:

Makale

Yayın Tarihi:

7/2017

Belge Dili:

Türkçe

Document ID (Org No.):

2000718053

Sıklık:

Aylık

eISSN:

1306-6048

pISSN:

1306-6048

DOI:

https://doi.org/10.24330/ieja.325932

Özet:

 


Let AAA be an associative algebra over a commutative ring RRR,BiL(A)BiL(A)\text{BiL}(A) the set of RRR-bilinear maps from A×AA×AA \times A toAAA, and arbitrarily elements xxx, yyy in AAA. Consider thefollowing RRR-modules:Ω(A)={(f, α) | f∈HomR(A, A), α∈BiL(A)},TDer(A)={(f, f′, f′′)∈HomR(A, A)3 | f(xy)=f′(x)y+xf′′(y)}.Ω(A)={(f, α) | f∈HomR(A, A), α∈BiL(A)},TDer(A)={(f, f′, f″)∈HomR(A, A)3 | f(xy)=f′(x)y+xf″(y)}.\begin{align*} &\Omega(A) = \{(f,\ \alpha)\ \vert \ f \in \text{Hom}_R(A,\ A),\ \alpha \in \text{BiL}(A) \}, \\ &\text{TDer}(A) = \{(f,\ f',\ f'') \in \text{Hom}_R(A,\ A)^3 \ \vert \ f(xy) = f'(x)y + xf''(y)\}. \end{align*}TDer(A)TDer(A)\text{TDer}(A) is called the set of triple derivations of AAA.We define a Lie algebra structure on Ω(A)Ω(A)\Omega(A) andTDer(A)TDer(A)\text{TDer}(A) such that φA:TDer(A)→Ω(A)φA:TDer(A)→Ω(A)\varphi_A : \text{TDer}(A) \to \Omega(A) is a Lie algebra homomorphism.\parDually, for a coassociative RRR-coalgebra CCC, we define theRRR-modules Ω(C)Ω(C)\Omega(C) and TCoder(C)TCoder(C)\text{TCoder}(C) which correspond toΩ(A)Ω(A)\Omega(A) and TDer(A)TDer(A)\text{TDer}(A), and show that the similarresults to the case of algebras hold. Moreover, since C∗=HomR(C, R)C∗=HomR(C, R)C^* = \text{Hom}_R(C,\ R) is an associative RRR-algebra, we give thatthere exist anti-Lie algebra homomorphisms θ0:TCoder(C)→TDer(C∗)θ0:TCoder(C)→TDer(C∗)\theta_0 : \text{TCoder}(C) \to \text{TDer}(C^*) and θ1:Ω(C)→Ω(C∗)θ1:Ω(C)→Ω(C∗)\theta_1 : \Omega(C) \to \Omega(C^*) such that the following diagram is commutative :\begin{equation*} \begin{CD} \text{TCoder}(C) @>{\psi_C}>> \Omega(C) \\ @VV{\theta_0}V  @VV{\theta_1} V  \\ \text{TDer}(C^*) @>{\varphi_{C^*}}>>\Omega(C^*). \end{CD} \end{equation*}\begin{equation*} \begin{CD} \text{TCoder}(C) @>{\psi_C}>> \Omega(C) \\ @VV{\theta_0}V  @VV{\theta_1} V  \\ \text{TDer}(C^*) @>{\varphi_{C^*}}>>\Omega(C^*). \end{CD} \end{equation*}\begin{equation*} \begin{CD} \text{TCoder}(C) @>{\psi_C}>> \Omega(C) \\ @VV{\theta_0}V  @VV{\theta_1} V  \\ \text{TDer}(C^*) @>{\varphi_{C^*}}>>\Omega(C^*). \end{CD} \end{equation*}

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Makale detayları

ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS AND THEIR DUALS

Yazarlar: Atsushi NAKAJİMA

Yayıncı: Abdullah HARMANCI

Ana Başlıklar: Matematik

Dergi: International Electronic Journal of Algebra 2017 Vol.22 Issue 22, pp.103-124

Belge Tipi: Makale

Yayın Tarihi: 7/2017

Belge Dili: Türkçe

Document ID (Org No.): 2000718053

Sıklık: Aylık

eISSN: 1306-6048

pISSN: 1306-6048

DOI: https://doi.org/10.24330/ieja.325932

Sayı

Yazarlar:

Atsushi NAKAJİMA

Yayıncı:

Abdullah HARMANCI

Ana Başlıklar:

Matematik

Dergi:

International Electronic Journal of Algebra 2017 Vol.22 Issue 22, pp.103-124

Belge Tipi:

Makale

Yayın Tarihi:

7/2017

Belge Dili:

Türkçe

Document ID (Org No.):

2000718053

Sıklık:

Aylık

eISSN:

1306-6048

pISSN:

1306-6048

DOI:

https://doi.org/10.24330/ieja.325932