93 research outputs found
On Lie algebra crossed modules
This article constructs a crossed module corresponding to the generator of
the third cohomology group with trivial coefficients of a complex simple Lie
algebra. This generator reads as , constructed from the Lie bracket [,]
and the Killing form . The construction is inspired by the corresponding
construction for the Lie algebra of formal vector fields in one formal variable
on R, and its subalgebra sl_2(R), where the generator is usually called
Godbillon-Vey class.Comment: 24 page
Deformations of Lie algebras of vector fields arising from families of schemes
Fialowski and Schlichenmaier constructed examples of global deformations of
Lie algebras of vector fields from deforming the underlying variety. We
formulate their approach in a conceptual way. Namely, we construct a stack of
deformations and a morphism form the moduli stack of stable marked curves. The
morphism associates to a family of marked curves the family of Lie algebras
obtained by taking the Lie algebra of vertical vector fields on the family
where one has extracted the marked points. We show that this morphism is almost
a monomorphism by Pursell-Shanks theory.Comment: 20 page
On Hopf 2-algebras
Our main goal in this paper is to translate the diagram relating groups,
Lie algebras and Hopf algebras to the corresponding 2-objects, i.e. to
categorify it. This is done interpreting 2-objects as crossed modules and
showing the compatibility of the standard functors linking groups, Lie algebras
and Hopf algebras with the concept of a crossed module. One outcome is the
construction of an enveloping algebra of the string Lie algebra of Baez-Crans,
another is the clarification of the passage from crossed modules of Hopf
algebras to Hopf 2-algebras.Comment: 26 pages, clarification of several statement
Cohomology and deformations of the infinite dimensional filiform Lie algebra m_0
Denote m_0 the infinite dimensional N-graded Lie algebra defined by basis
e_i, i>= 1 and relations [e_1,e_i] = e_(i+1) for all i>=2. We compute in this
article the bracket structure on H1(m_0,m_0), H2(m_0,m_0) and in relation to
this, we establish that there are only finitely many true deformations of m_0
in each nonpositive weight, by constructing them explicitely. It turns out that
in weight 0 one gets exactly the other two filiform Lie algebras.Comment: 25 page
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