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SUMMARY:Viviana del Barco (Universidade Estadual de Campinas)
DTSTART:20211117T160000Z
DTEND:20211117T170000Z
DTSTAMP:20260423T021230Z
UID:VSGS/36
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/VSGS/36/">Un
 iqueness of ad-invariant metrics</a>\nby Viviana del Barco (Universidade E
 stadual de Campinas) as part of Virtual seminar on geometry with symmetrie
 s\n\n\nAbstract\nAn ad-invariant metric on a Lie algebra is a nondegenerat
 e symmetric bilinear form for which inner derivations are skew-symmetric. 
 These are the algebraic counterparts of bi-invariant metrics on Lie groups
 .\n\nIt is known that a positive definite ad-invariant metric can only be 
 defined on compact semisimple Lie algebras\, direct sum with an abelian fa
 ctor. On compact simple Lie algebras\, every ad-invariant metric is a mult
 iple of the Killing form which\, in addition\, is invariant under the Lie 
 algebra automorphisms.\n\nIn the pseudo-Riemannian context ad-invariant me
 trics appear on more general Lie algebras such as semisimple (non-compact)
 \, or solvable. For non-semisimple Lie algebras\, the orbit space of ad-in
 variant metrics under the action of the automorphism group has not been sy
 stematically described yet.\n\nIn this talk\, we will discuss characterist
 ics of Lie algebras possessing a unique ad-invariant metric up to automorp
 hisms (and sign). In particular\, we will introduce the concept of "solita
 ry" metrics on Lie algebras\, which aims to encode the property of being a
  unique ad-invariant metric. As we will see\, this is actually a property 
 of a Lie algebra rather than of the metric itself.\n\nThis characterizatio
 n of uniqueness allowed us to show that Lie algebras admitting a unique ad
 -invariant metric are necessarily solvable. In addition\, we show that man
 y low dimensional Lie algebras carrying ad-invariant metrics are solitary.
 \n\nTime permitting\, generalizations of the solitary conditions will be d
 iscussed.\n\nThe talk is based on joint works with Diego Conti and Federic
 o A. Rossi (Milano Bicocca).\n
LOCATION:https://researchseminars.org/talk/VSGS/36/
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