What does “control of a deformation problem” mean?

Is the expression “control of a deformation problem’ ever defined? There are of course many examples relating a dg-Lie or L-infty algebra to a deformation problem, and the phrase is evocative. Is it ever defined? Answer AttributionSource : Link , Question Author : Jim Stasheff , Answer Author : Community

Affine Steinberg groups vs Steinberg groups over Laurent polynomials

Let R be a commutative ring and Φ be a finite (also called spherical) reduced irreducible root system of rank ≥2. I will denote by St(Φ,R) the Steinberg group of type Φ over R, i. e. the quotient of the free product ∐α∈ΦXα of root subgroups Xα=⟨xα(a)∣a∈R⟩ modulo Chevalley commutator formulae: [xα(a);xβ(b)]=∏iα+jβ∈Φxiα+jβ(Ni,jα,βaibj), i,j∈N; (here Ni,jα,β … Read more

Action of orthogonal group on the free Lie algebra

This question is somewhat related and inspired by this post of professor Montgomery. The free Lie algebra L(V) generated by an r-dimensional vector space V is, in the language of https://en.wikipedia.org/wiki/Free_Lie_algebra, the free Lie algebra generated by any choice of basis e1,…,er for the vector space V. (Work over the field R or C, whichever … Read more

Harmonicity on semisimple groups

I asked this on Math.SE and got no answer, so I’ll try my luck here. Let G be a semisimple real Lie group, U(g) its universal enveloping algebra, let Ω be the Casimir element in U(g) and let f be a smooth (or analytic) real-valued function on G. We then have the following notions 1) … Read more

Lusztig’s definition of quantum groups

In his book Introduction to quantum groups, Lusztig gives a definition (Def 3.1.1) of the rational form UQ(q)q that is rather different from the usual approach (see [1,Ch.9.1] for expample). As far as I understood, the translation goes as follows: Let g be a simple Lie algebra and I a set of simple roots in … Read more

Does the tensor algebra T(V)T(V) of VV isomorphic to the symmetric algebra of the free Lie algebra over VV?

Let V be a finite dimensional vector space. Let T(V) be the tensor algebra over V. Do we have T(V)≅S(Lie(V)) as a graded vector space? Here S(Lie(V)) is the symmetric algebra of the free Lie algebra over V. Thank you very much. Answer AttributionSource : Link , Question Author : Jianrong Li , Answer Author … Read more

DGLA related to the deformation of hopf Algebras

Recently I was considering Hopf algebras and Drinfeld’s twists. I stumbled upon a certain DGLA one can associate to a Hopf algebra (unital bialgebras actually) by copying the formulas obtained by looking at the universal enveloping algebra of a Lie algebra(oid) and the Gerstenhaber structure induced by viewing them as (left invariant) polydifferential operators. Explicitely, … Read more

Restricted universal extensions and lifting of derivations

Let L be a perfect Lie algebra. Then it is well-known that L has a universal central extension ˆL and every derivation of L can be lifted to a derivation of ˆL. (See e.g. Section 2 of https://mysite.science.uottawa.ca/neher/Papers/univ.pdf.) Now suppose that (L,[p]) is a restricted Lie algebra over a field of characteristic p>0. I remember … Read more

“Signature Changing” Generalization of Lie Algebra?

I have in mind a mathematical structure I’ve never heard of before. Does anyone know what might be? It is a manifold with vector fields whose Lie brackets have structure coefficients that are invariant. Example: zY – yZ, xZ – zX, yX – xY, wtX + x(vT – U), wtY + y(vT – U), wtZ … Read more