Pub Date : 2017-07-01DOI: 10.21099/TKBJM/1571968818
Toshikazu Miyashita
The compact simply connected Riemannian 4-symmetric spaces were classified by J.A. Jim{e}nez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form $G/H$, where $G$ is a connected compact simple Lie group with an automorphism $tilde{gamma}$ of order four on $G$ and $H$ is a fixed points subgroup $G^gamma$ of $G$. According to the classification by J.A. Jim{e}nez, there exist seven compact simply connected Riemannian 4-symmetric spaces $ G/H $ in the case where $ G $ is of type $ E_8 $. In the present article, %as Part II continuing from Part I, for the connected compact %exceptional Lie group $E_8$, we give the explicit form of automorphisms $tilde{w}_{{}_4} tilde{upsilon}_{{}_4}$ and $tilde{mu}_{{}_4}$ of order four on $E_8$ induced by the $C$-linear transformations $w_{{}_4}, upsilon_{{}_4}$ and $mu_{{}_4}$ of the 248-dimensional vector space ${mathfrak{e}_8}^{C}$, respectively. Further, we determine the structure of these fixed points subgroups $(E_8)^{w_{{}_4}}, (E_8)^{{}_{upsilon_{{}_4}}}$ and $(E_8)^{{} _{mu_{{}_4}}}$ of $ E_8 $. These amount to the global realizations of three spaces among seven Riemannian 4-symmetric spaces $ G/H $ above corresponding to the Lie algebras $ mathfrak{h}=ibm{R} oplus mathfrak{su}(8), ibm{R} oplus mathfrak{e}_7$ and $mathfrak{h}= mathfrak{su}(2) oplus mathfrak{su}(8)$, where $ mathfrak{h}={rm Lie}(H) $.
{"title":"Realizations of inner automorphisms of order four and fixed points subgroups by them on the connected compact exceptional Lie group $E_8$, Part II","authors":"Toshikazu Miyashita","doi":"10.21099/TKBJM/1571968818","DOIUrl":"https://doi.org/10.21099/TKBJM/1571968818","url":null,"abstract":"The compact simply connected Riemannian 4-symmetric spaces were classified by J.A. Jim{e}nez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form $G/H$, where $G$ is a connected compact simple Lie group with an automorphism $tilde{gamma}$ of order four on $G$ and $H$ is a fixed points subgroup $G^gamma$ of $G$. According to the classification by J.A. Jim{e}nez, there exist seven compact simply connected Riemannian 4-symmetric spaces $ G/H $ in the case where $ G $ is of type $ E_8 $. In the present article, %as Part II continuing from Part I, for the connected compact %exceptional Lie group $E_8$, we give the explicit form of automorphisms $tilde{w}_{{}_4} tilde{upsilon}_{{}_4}$ and $tilde{mu}_{{}_4}$ of order four on $E_8$ induced by the $C$-linear transformations $w_{{}_4}, upsilon_{{}_4}$ and $mu_{{}_4}$ of the 248-dimensional vector space ${mathfrak{e}_8}^{C}$, respectively. Further, we determine the structure of these fixed points subgroups $(E_8)^{w_{{}_4}}, (E_8)^{{}_{upsilon_{{}_4}}}$ and $(E_8)^{{} _{mu_{{}_4}}}$ of $ E_8 $. These amount to the global realizations of three spaces among seven Riemannian 4-symmetric spaces $ G/H $ above corresponding to the Lie algebras $ mathfrak{h}=ibm{R} oplus mathfrak{su}(8), ibm{R} oplus mathfrak{e}_7$ and $mathfrak{h}= mathfrak{su}(2) oplus mathfrak{su}(8)$, where $ mathfrak{h}={rm Lie}(H) $.","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2017-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45020721","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-07-01DOI: 10.21099/TKBJM/1506353558
Takuya Shimokawa, Kyoji Sugimoto
{"title":"On the groups of isometries of simple para-Hermitian symmetric spaces","authors":"Takuya Shimokawa, Kyoji Sugimoto","doi":"10.21099/TKBJM/1506353558","DOIUrl":"https://doi.org/10.21099/TKBJM/1506353558","url":null,"abstract":"","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2017-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.21099/TKBJM/1506353558","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47143371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-12-01DOI: 10.21099/TKBJM/1492104600
Balesh Kumar, M. Manickam
. In this paper, we extend the Doi-Naganuma lifting as suggested by Kudla [4], on the lines of Zagier’s work [6]. For each fundamental discriminant D associated with a real quadratic field, we prove that there exists a Hecke-equivarient map i D which maps the m th Poincare series of weight k , level M and character w D ¼ : D (cid:1) (cid:2) into a Hilbert cusp form of weight k , level M = D associated with the real quadratic field of discriminant D of class number one. Through this, we get its adjoint i (cid:1) D with respect to the Petersson inner product.
{"title":"On Doi-Naganuma lifting","authors":"Balesh Kumar, M. Manickam","doi":"10.21099/TKBJM/1492104600","DOIUrl":"https://doi.org/10.21099/TKBJM/1492104600","url":null,"abstract":". In this paper, we extend the Doi-Naganuma lifting as suggested by Kudla [4], on the lines of Zagier’s work [6]. For each fundamental discriminant D associated with a real quadratic field, we prove that there exists a Hecke-equivarient map i D which maps the m th Poincare series of weight k , level M and character w D ¼ : D (cid:1) (cid:2) into a Hilbert cusp form of weight k , level M = D associated with the real quadratic field of discriminant D of class number one. Through this, we get its adjoint i (cid:1) D with respect to the Petersson inner product.","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.21099/TKBJM/1492104600","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67831384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-12-01DOI: 10.21099/TKBJM/1492104602
R. Escobedo, V. Sánchez-Gutiérrez, J. Sánchez-Martínez
{"title":"On the hyperspace ℭ(X) of continua","authors":"R. Escobedo, V. Sánchez-Gutiérrez, J. Sánchez-Martínez","doi":"10.21099/TKBJM/1492104602","DOIUrl":"https://doi.org/10.21099/TKBJM/1492104602","url":null,"abstract":"","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.21099/TKBJM/1492104602","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67831421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-07-01DOI: 10.21099/TKBJM/1474747489
P. Danchev
. We introduce and give a comprehensive study of weakly UU rings , calling them WUU rings . This notion is a natural gen-eralization of the so-called UU rings, defined by Calugareanu (Carpath. J. Math., 2015) and investigated in details by Danchev-Lam (Publicat. Math. Debrecen, 2016). It also demarcates the strength of recent results about these kind of rings by giving a strong barrier between some of their crucial properties.
{"title":"Weakly UU rings","authors":"P. Danchev","doi":"10.21099/TKBJM/1474747489","DOIUrl":"https://doi.org/10.21099/TKBJM/1474747489","url":null,"abstract":". We introduce and give a comprehensive study of weakly UU rings , calling them WUU rings . This notion is a natural gen-eralization of the so-called UU rings, defined by Calugareanu (Carpath. J. Math., 2015) and investigated in details by Danchev-Lam (Publicat. Math. Debrecen, 2016). It also demarcates the strength of recent results about these kind of rings by giving a strong barrier between some of their crucial properties.","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2016-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.21099/TKBJM/1474747489","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67831361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-07-01DOI: 10.21099/TKBJM/1474747490
Go Yamashita
{"title":"A small remark on the filtered $varphi$-module of Fermat varieties and Stickelberger's theorem","authors":"Go Yamashita","doi":"10.21099/TKBJM/1474747490","DOIUrl":"https://doi.org/10.21099/TKBJM/1474747490","url":null,"abstract":"","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2016-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.21099/TKBJM/1474747490","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67831375","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-07-01DOI: 10.21099/TKBJM/1474747487
Y. Kasahara
. The tail probability of the first hitting time is discussed for linear di¤usions. We obtain the decay rates in terms of the spectral functions and the scale functions. The result is a general-ization of recent results of Hamana-Matsumoto for Bessel processes.
{"title":"Tails of the first hitting times of linear diffusions","authors":"Y. Kasahara","doi":"10.21099/TKBJM/1474747487","DOIUrl":"https://doi.org/10.21099/TKBJM/1474747487","url":null,"abstract":". The tail probability of the first hitting time is discussed for linear di¤usions. We obtain the decay rates in terms of the spectral functions and the scale functions. The result is a general-ization of recent results of Hamana-Matsumoto for Bessel processes.","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2016-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.21099/TKBJM/1474747487","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67831505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-04-08DOI: 10.21099/TKBJM/1541559647
Nagatoshi Sasano
We will construct standard pentads which are analogues of Cartan subalgebras, and moreover, we will study graded Lie algebras corresponding to these standard pentads. We call such pentads pentads of Cartan type and describe them by two positive integers and three matrices. The structures of their corresponding Lie algebras are related with contragredient Lie algebras.
{"title":"Contragredient Lie algebras and Lie algebras associated with a standard pentad","authors":"Nagatoshi Sasano","doi":"10.21099/TKBJM/1541559647","DOIUrl":"https://doi.org/10.21099/TKBJM/1541559647","url":null,"abstract":"We will construct standard pentads which are analogues of Cartan subalgebras, and moreover, we will study graded Lie algebras corresponding to these standard pentads. We call such pentads pentads of Cartan type and describe them by two positive integers and three matrices. The structures of their corresponding Lie algebras are related with contragredient Lie algebras.","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2016-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67631409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-03-01DOI: 10.21099/tkbjm/1461270060
Yuta Ogata
. We study the construction of spacelike constant mean curvature (CMC) surfaces with mean curvature 0 a H < 1 in 3-dimensional de Sitter space S 2 ; 1 , by using Iwasawa splitting. We also study their singularities and create some criteria for them.
。研究了三维de Sitter空间s2中平均曲率为0 a H < 1的类空间常平均曲率曲面的构造;1、采用岩泽劈法。我们还研究了它们的奇异性，并为它们建立了一些准则。
{"title":"Spacelike constant mean curvature and maximal surfaces in 3-dimensional de Sitter space via Iwasawa splitting","authors":"Yuta Ogata","doi":"10.21099/tkbjm/1461270060","DOIUrl":"https://doi.org/10.21099/tkbjm/1461270060","url":null,"abstract":". We study the construction of spacelike constant mean curvature (CMC) surfaces with mean curvature 0 a H < 1 in 3-dimensional de Sitter space S 2 ; 1 , by using Iwasawa splitting. We also study their singularities and create some criteria for them.","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2016-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.21099/tkbjm/1461270060","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67831237","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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