Cohomology of solvable lie algebras and solvmanifolds

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• 作者：D. V. Millionshchikov
• 刊名：Mathematical Notes
• 出版年：2005
• 出版时间：January 2005
• 年：2005
• 卷：77
• 期：1-2
• 页码：61-71
• 全文大小：185 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Russian Library of Science
  
• 出版者：MAIK Nauka/Interperiodica distributed exclusively by Springer Science+Business Media LLC.
• ISSN：1573-8876

文摘

The cohomology H_{xlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0">}^* (G/xlarge915.gif" alt="Gamma" align="BASELINE" BORDER="0">,xlarge8450.gif" alt="Copf" align="BASELINE" BORDER="0">) of the de Rham complex xlarge923.gif" alt="Lambda" align="BASELINE" BORDER="0">*(G/xlarge915.gif" alt="Gamma" align="BASELINE" BORDER="0">) xlarge8855.gif" alt="otimes" align="MIDDLE" BORDER="0"> xlarge8450.gif" alt="Copf" align="BASELINE" BORDER="0"> of a compact solvmanifold G/xlarge915.gif" alt="Gamma" align="BASELINE" BORDER="0"> with deformed differential d_{xlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0">} = d + xlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0">, where xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0"> is a closed 1 -form, is studied. Such cohomologies naturally arise in Morse-Novikov theory. It is shown that, for any completely solvable Lie group G containing a cocompact lattice xlarge915.gif" alt="Gamma" align="BASELINE" BORDER="0"> xlarge8834.gif" alt="sub" align="MIDDLE" BORDER="0"> G, the cohomology H_{xlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0">}^*(G/xlarge915.gif" alt="Gamma" align="BASELINE" BORDER="0">, xlarge8450.gif" alt="Copf" align="BASELINE" BORDER="0">) is isomorphic to the cohomology H_{xlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0">}^*() of the tangent Lie algebra of the group G with coefficients in the one-dimensional representation _{xlarge961.gif" alt="rgr" align="MIDDLE" BORDER="0">xlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0">} : xlarge8594.gif" alt="rarr" align="BASELINE" BORDER="0"> defined by _{xlarge961.gif" alt="rgr" align="MIDDLE" BORDER="0">xlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0">}(xlarge958.gif" alt="xgr" align="MIDDLE" BORDER="0">) = xlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0">(xlarge958.gif" alt="xgr" align="MIDDLE" BORDER="0">). Moreover, the cohomology H_{xlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0">}^*(G/xlarge915.gif" alt="Gamma" align="BASELINE" BORDER="0">,xlarge8450.gif" alt="Copf" align="BASELINE" BORDER="0">) is nontrivial if and only if -xlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">[xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0">] belongs to a finite subset Omega _\mathfrak{g} $$ " align="middle" border="0"> of H¹(G/xlarge915.gif" alt="Gamma" align="BASELINE" BORDER="0">,xlarge8450.gif" alt="Copf" align="BASELINE" BORDER="0">) defined in terms of the Lie algebra .