Universal central extensions of slm∣nsl_{m|n} over Z/2ZZ/2Z-graded algebras

We study central extensions of the Lie superalgebra $sl_{m|n}(A)$, where $A$ is a $Z/2Z$-graded superalgebra over a commutative ring $K$. The Steinberg Lie superalgebra $st_{m|n}(A)$ plays a crucial role. We show that $st_{m|n}(A)$ is a central extension of $sl_{m|n}(A)$ for $m+n\geq 3$. We use a $Z/2Z$-graded version of cyclic homology to show that the center of the extension is isomorphic to $HC_1(A)$ as $K$-modules. For $m+n\geq 5$, we prove that $st_{m|n}(A)$ is the universal central extension of $sl_{m|n}(A)$. For $m+n=3,4$, we prove that $st_{2|1}(A)$ and $st_{3|1}(A)$ are both centrally closed. The universal central extension of $st_{2|2}(A)$ is constructed explicitly.Comment: 18 pages; section 7 added; reference [KT] added; the authors thank the referee for comments on the previous versio

Complexities of Chinese Contemporary Art

Throughout the history of the People’s Republic of China (PRC), art has always been state-dominated and driven by governmental and political agendas. In comparison to fellow artists in the Western world, historically, Chinese artists have lacked the freedom to express their passion and creativity through artistic forms. The contemporary art movement in China, however, maneuvers around this challenge and provides a more positive direction—one in which artists have a stronger voice and economic benefits are combined with governmental support and encouragement of art activities that enhance social capital and one’s habitus. To some extent, this is changing, with the first significant emergence of liberalization, and the rise of artist voices in the post-Mao period. China’s art market is not only booming domestically but it has opened up to the world market over the past 20 years. This affirmative phenomenon was proven by a striking purchase of a Chinese antique vase sold at auction for $86 million. Moreover, China’s contemporary art is part of learned by China’s powerful historical arts and crafts, and previous popular in art villages. Art villages, which include Beijing’s 798 Art District, have established art studios, galleries, and local exhibitions that support modern day artists as well as expand China’s collections. In recent years, village artists create various forms of visual representation and expression through paintings in order to transform the contemporary art of China. In sum, this research paper examines the rapid transformation of Chinese art, its emergence as in an economic tool, and art as a mode to express one’s freedom of speech

Classification of Graded Left-symmetric Algebra Structures on Witt and Virasoro Algebras

We find that a compatible graded left-symmetric algebra structure on the Witt algebra induces an indecomposable module of the Witt algebra with 1-dimensional weight spaces by its left multiplication operators. From the classification of such modules of the Witt algebra, the compatible graded left-symmetric algebra structures on the Witt algebra are classified. All of them are simple and they include the examples given by Chapoton and Kupershmidt. Furthermore, we classify the central extensions of these graded left-symmetric algebras which give the compatible graded left-symmetric algebra structures on the Virasoro algebra. They coincide with the examples given by Kupershmidt.Comment: 22 page

多項式固有値問題における櫻井-杉浦法に対するバランシング及びスケーリング手法の研究

筑波大学 (University of Tsukuba)201

Algorithms for square root of semi-infinite quasi-Toeplitz MM-matrices

A quasi-Toeplitz $M$-matrix $A$ is an infinite $M$-matrix that can be written as the sum of a semi-infinite Toeplitz matrix and a correction matrix. This paper is concerned with computing the square root of invertible quasi-Toeplitz $M$-matrices which preserves the quasi-Toeplitz structure. We show that the Toeplitz part of the square root can be easily computed through evaluation/interpolation at the $m$ roots of unity. This advantage allows to propose algorithms only for the computation of correction part, whence we propose a fixed-point iteration and a structure-preserving doubling algorithm. Moreover, we show that the correction part can be approximated by solving a nonlinear matrix equation with coefficients of finite size followed by extending the solution to infinity. Numerical experiments showing the efficiency of the proposed algorithms are performed