## 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

## Lie algebras whose derivation algebra is nilpotent

Let L be a Lie algebra and denote by Der(L) the derivation algebra of L. If L is finite-dimensional, then a theorem of Leger and Togo (Duke Math. J. 26 (1959), 623 – 628, DOI: 10.1215/S0012-7094-59-02660-2) states that Der(L) is nilpotent if and only if L is characteristically nilpotent or 1-dimensional. I recall that a … 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