BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Michael Heins (Julius-Maximilians University Würzburg) DTSTART:20210212T130000Z DTEND:20210212T150000Z DTSTAMP:20260423T024717Z UID:DQSeminar/3 DESCRIPTION:Title: The Universal Complexification of a Lie Group\nby Michael Heins (Jul ius-Maximilians University Würzburg) as part of Deformation Quantization Seminar\n\n\nAbstract\nIn classical Lie theory\, a\n complexification of a Lie group with Lie algebra $\\mathfrak{g}$ is a\n complex Lie group\, w hose Lie algebra is given by the\n complexification $\\mathfrak{g}_{\\mat hbb{C}}$ of $\\mathfrak{g}$ in the sense\n of vector spaces. Both from an analytical and a categorical point of\n view\, this definition turned ou t to be too naive to be truly\n useful. Historically\, this lead to the r efined concept of\n universal complexification\, which is based on an ana lytically\n desirable universal property. In this talk\, we motivate this \n definition by briefly reviewing the vector space\n situation. Afterwa rds\, we give a rather geometric construction of\n the universal complexi fication of a given Lie group\, which was\n formalized by Hochschild arou nd 1955 and refined by the Bourbaki\n group in the following decade. Alon g the way\, we review Lie's\n seminal Theorems and meet the universal cov ering group. While many\n properties of the resulting universal complexif ication align with\n what we geometrically expect\, some notable aspects turn out to\n differ\, which we discuss in detail. Finally\, we provide s ome\n examples to illustrate the power and limitations of the machinery w e\n have developed.\n LOCATION:https://researchseminars.org/talk/DQSeminar/3/ END:VEVENT END:VCALENDAR