Freeness of a matrix semigroup

Motivated by some questions in the dimension theory of self-affine sets, a colleague and I are interested in the freeness (or otherwise) of the subsemigroup of $SL_\pm(2,\mathbb{R})$ generated by the matrices of determinant $\pm1$ which are proportional to $$A_1:=\left(\begin{array}{cc}0.85&0.04\\-0.04&0.85\end{array}\right),\quad A_2:=\left(\begin{array}{cc}0.20&-0.26\\0.23&0.22\end{array}\right),\quad A_3:=\left(\begin{array}{cc}-0.15&0.28\\0.26&0.24\end{array}\right).$$ In fact, our question is slightly more specific: we would like to know that … Read more

The existence of zero-divisors in the universal enveloping algebra of an infinite-dimensional Lie algebra

The intuition for this problem comes from §17 Exercise 1 Humphreys’ Introduction to Lie Algebras and Representation Theory which essentially asks us to use PBW in order to prove that if a Lie algebra L is finite-dimensional, then the universal enveloping algebra U=U(L) has no zero divisors. So my question is the following: Why does … Read more

Poincaré-Birkhoff-Witt theorem for Leibniz algebras

Leibniz algebras can be seen as a non-skew-symmetric generalization of Lie algebras. I have already taken a look at some papers related to Leibniz algebras and extending main results of Lie algebras to Leibniz algebras. Here I want to ask you about the Poincaré-Birkhoff-Witt (PBW) theorem for free Leibniz algebras? Does the PBW theorem allow … Read more

An order in $\mathbb Q[G]$ which is a maximal $\mathbb Z_p$-order in $\mathbb Q_p[G]$ for finitely many primes $p$

Let $G$ be a finite group and $S$ a finite set of prime numbers. I know that every separable $\mathbb Q$-algebra $A$ contains a maximal $\mathbb Z$-order but I wonder if the following is true. Is there a $\mathbb Z$-order $\Lambda$ in $\mathbb Q[G]$ which contains $\mathbb Z[G]$ and satisfies the following two conditions 1) … Read more

Auslander-Reiten-Quivers of representation-finite algebras having different 3-dimensional forms

I am looking for references, where I can find (pictures of) connected Auslander-Reiten-Quivers of representation-finite k-algebras (k is a (preferably, but not necessarily finite) field) with one or more of the following properties: 1) the Auslander-Reiten quiver contains two cylinders (so, there are periodic modules), but also many non-periodic modules 2) the Auslander-Reiten quiver contains … Read more

Reference for classification of positive involutions

An involution on a finite dimensional algebra A over Q is an involutive anti-automorphism of A. If σ is an involution on A, we say that σ is positive if TrA/Q(xσ(x))>0 for all 0≠x∈A. A theorem of Albert [1] classifies finite dimensional division algebras over Q that admit a positive involution. I know how to … Read more

Intuition for Clifford Group

Clifford group Γ of a Clifford algebra Cℓ(V,q) is defined to be the set of elements g in Cℓ(V,q) for which there exists an inverse g−1. This group can be represented by linear transformation on the Clifford algebra by the map ρ:g↦ρg defined by, ρg:x↦gxg−1 This is a linear map and it also preserves the … Read more

Generators of the symmetric square of the group ring of an abelian group

Let A be an abelian group and R=Z[A]– its group ring. Denote by I an ideal of R given by a kernel of the map R⟶Z⊕A, sending [r] to (1,r). Next, denote by S2I the symmetric square of I as an abelian group, tensored with Q. I claim that the elements ([a]+[b]−[c]−[d])2 with ab=cd generate … Read more

Inducing surjections on GLn(−)GL_n(-)?

Suppose A,B are (possibly noncommutative) rings, and GLn(−) is the group of invertible n×n matrices over a given ring. Suppose f:A→B is surjective, does it necessarily follow that f∗:GLn(A)→GLn(B) is surjective for n>1? If not, why not? I know that this need not be true for n=1, but is true for the subgroup En(−) generated … 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