arXiv subjects
Publications and source records attributed to Mohamad Maassarani.
We show that two finite isoclinic groups of the same order have isomorphic conjugacy quandles. We also show that the converse hold under some assumptions on the groups. It happens that isoclinsim of group of the same order is equivalent to having isomorphic quandles for groups of order $n<128$. We construct two non isoclinic groups of order $128$ having isomorphic conjugacy quandles.
For many finite groups a symmetric $2$-cocycle $α$ ($α(g,h)=α(h,g)$, for all pairs $(h,g)$ of the group) with values in $\mathbb{C}^\times$ is a coboundary. We show using a theoretic arguement and GAP that there is a group of order $64$ having a symmetric $2$-cocycle with a non trivial cohomology class.
A unitary finite dimensional quandle representation is decomposable into a direct sum of irreducible represenations. Not all quandle representations satisfy this property. We prove that a finite dimensional quandle represenation $ρ:Q \to GL(V) $ of a finite quandle $Q$ over $\mathbb{C}$ is decomposable into a direct sum of irreducibles if and only if every element in the image of $ρ$ is diagonlizable. We show that an irreducible representation $ρ:Q \to GL(V)$ of a finite quandle over $\mathbb{C}$ is unitary for some inner product if and only if every element of the image of $ρ$ has determinant of modulus $1$. It follows that any irreducible representation of a finite quandle $Q$ over $\mathbb{C}$ can be twisted by a quandle character to obtain a unitary irreducible representation. We also prove that the enveloping group $G(Q)$, of a finite quandle $Q$, admit a faithfull finite dimensional unitary representation over $\mathbb{C}$ and that the irreducible representations of a finite quandle $Q$ over $\mathbb{C}$ are $1$-dimensional if and only if $G(Q)$ is abelian. Finaly, we determine the irreducible representations over $\mathbb{C}$ of a family of finite quandles.
For $G$ a finite group, one way to construct irreducible quandle representations over $\mathbb{C}$ of the conjugacy quandle $Conj(G)$ is by taking the product of an irreducible linear group representation of $G$ by what we call a quandle character of $Conj(G)$ (a quandle morphism into $\mathbb{C}^\times$ ). We show that these are all the irreducible quandle representations of $Conj(G)$ over $\mathbb{C}$ if and only if all the symmetric $2$-cocyles over $G$ ($α(g,h)=α(h,g)$ for all $g,h$) with values in $\mathbb{C}^\times$ are coboundaries. For instance, this is the case of groups with trivial Bogomolov multiplier. We apply this to study the enveloping group of $Conj(G)$. If $G$ finite satisfies the previous condition on symmetric $2$-cocycles, we obtain that the enveloping group of $Conj(G)$ injects into $G\times \mathbb{Z}^{c_G}$ where $c_G$ is the number of the conjugacy classes of $G$. If moreover $G$ is perfect the injection is an isomorphism.
We consider irreducible representations of finite quandles over $\mathbb{C}$. For $Q$ a finite quandle whose inner automorphism group $Inn(Q)$ have trivial Schur multipliers, we prove that the irreducible representations of $Q$ can be constructed out of what we call characters of $Q$ and irreducible linear represenations of the group $Inn(Q)$. For $G$ a finite groiup having trivial Schur multiplier or being a Schur cover of another group, we show that the irreducible representations of the conjugacy quandle $Conj(G)$ can be constructed out of characters of $Conj(G)$ and irreducible linear representations of the group $G$. In both cases, the finite unitary irreducible representations can be determined from the results. For instance, these results allow to solve the problem of constucting irreducible represenations of the conjugacy quandles of dihedral groups and generalised quaternion groups. In general, we relate the irreducible representations of a finite quandle $Q$ to irreducible projective representations of $Inn(Q)$ and prove that the irreducible representations of $Q$ can be in theory constructed out of characters of $Q$ and irreducible representations of a finite quotient of the enveloping group $G(Q)$. The quotient is a stem extensions of $Inn(Q)$ with nucleus a finite subgroup of the center of $G(Q)$. This allows, using a result from the litterature, to show that the irreducible quandle representations of $Conj(S_n)$ ($S_n$ the symmetric group) can be constructed out of characters of the corresponding quandle and irreducible linear group representations of the symmetric group.
We are intereseted in quandles and their enveloping groups. Various results are proven. We show that a quandle $Q$ and its image in the enveloping group $G(Q)$ have isomorphic enveloping groups. The image quandle is injective. For $Q$ a finite quandle, we show that $G(Q)$ admits a faithfull representation $ρ: G(Q)\to GL_n(\mathbb{Z})$ for some $n$; an irreducible representation of $G(Q)$ over $\mathbb{C}$ is finite dimensional an its degree divides the order of the group $Inn(Q)$ of inner automorphism of $Q$ and is bounded by $\sqrt{\vert Inn(Q) \vert }$. We determine the Malcev Lie algebra and the rational cohomology ring of $G(Q)$ for $Q$ finite. We prove that a finite injective quandle is a subquandle (for conjugacy) of a finite group. We also prove that the only finite subquandles (for conjugacy) of uniquely divisible groups are trivial quandles and that morphisms from quandles to nilpotent groups (for conjugacy) are constant on the indecomposable components. Implication of these results are considered.
Homology of non degenerate real projective quadrics was studied by Steenrod and Tucker. We Compute the rational and the $\mathbb{Z}/2\mathbb{Z}$ homology of degenerate real projective quadrics. This allows to determine the integer homology of these quadrics.
For $R_1,R_2,R_3,\dots$ a family of non isomorphic rings (or algebras) having each only 2 idempotents ($1$ and $0$), we classify up to isomorphism the rings (or algebras) obtained by taking products of powers of the different $R_i$. We show that the automorphism groups of such rings (or algebras) split naturally into the product of wreath products $Aut( R_n)\wr \mathfrak{S}_{m_n} $ for different $n$. These results are applied to algebras generated by one element over a perfect field $\mathbb{K}$. Such algebra is either $\mathbb{K}[X]$ or a quotient of $\mathbb{K}[X]$. We show that in the later case the algebra is isomorphic to a finite product of the form $A=\prod (\mathbb{L}_i[X]/(X^j))^{n_{i,j}}$, where the $\mathbb{L}_i$ are non isomomorphic finite field extensions of $\mathbb{K}$ $($not isomophic as $\mathbb{K}$-algebras$)$, with restrictions on the numbers $n_{i,j}$ if $\mathbb{K}$ is finite. We classify these algebras up to isomorphism. We have also that the $\mathbb{K}$-algebra automorphism group of $A=\prod (\mathbb{L}_i[X]/(X^j))^{n_{i,j}}$ splits naturally into the product of wreat products $Aut_\mathbb{K}(\mathbb{L}_i[X]/(X^j) )\wr \mathfrak{S}_{n_{i,j}}$ ($Aut_\mathbb{K}(-)$ is for $\mathbb{K}$-algebra automorphism group). Finally, we prove that $Aut_\mathbb{K}(\mathbb{L}_i[X]/(X^n) )$ is isomorphic to the semi-direct product $G_n(\mathbb{L}_i)\rtimes Aut_\mathbb{K}(\mathbb{L}_i)$ ($Aut_\mathbb{K}(-)$ is for $\mathbb{K}$-algebra automorphism group), where $G_n(\mathbb{L}_i)\simeq Aut_{\mathbb{L}_i}(\mathbb{L}_i[X]/(X^n) )$ ($\mathbb{L}_i$ algebra automorphism group) is an algebraic subgroup of invertible lower triangular matrices of dimension $(n-1)\times (n-1)$ with coefficients in $\mathbb{L}_i$; the conjugate of a matrix $M\in G_n(\mathbb{L}_i)$ by $σ\in Aut_\mathbb{K}(\mathbb{L}_i)$ is the matrix obtained from $M$ by applying $σ$ to its coefficients.
Given two seprable irreducible polynomials $P_1$ and $P_2$ over a filed $\mathbb{K}$. We show that the rings $\mathbb{K}[X]/(P_1^n)$ and $\mathbb{K}[X]/(P_2^n)$ are isomorphic if and only if their residue fields $\mathbb{K}[X]/(P_1)$ and $\mathbb{K}[X]/(P_2)$ are isomorphic. Partial results in this direction are obtained for the case where the polynomials are not seprable. We note that, given a seprable irreducible polynomial $P$, we prove that we have an isomorphism between $\mathbb{K}[X]/(P^n)$ and $(\mathbb{K}[X](P))[Y]/(Y^n)$.
We consider orbit configuration spaces $C_n^G(S)$, where $S$ is a surface obtained out of a closed orientable surface $\bar{S}$ by removing a finite number of points (eventually none) and $G$ is a finite group acting freely continuously on $S$. We prove that the fibration $π_{n,k} : C_{n}^G(S) \to C_k^G(S)$ obtained by projecting on the first $k$ coordinates is a rational fibration. As a consequence, the space $C_{n}^G(S)$ has a Sullivan model $A_{n,k}=ΛV_{C_k^G(S)}\otimes ΛV_{C_{n-k}^G(S_{G,k})}$ fitting in a cdga sequence: $ΛV_{C_k^G(S)}\to A_{n,k} \to ΛV_{C_{n-k}^G(S_{G,k})},$ where $ΛV_X$ denotes the minimal model of $X$, and $C_{n-k}^G(S_{G,k})$ is the fiber of $π_{n,k}$. We show that this model is minimal except for some cases when $S\simeq S^2$ and compute in all the cases the higher $ψ$-homotopy groups (related to the generators of the minimal model) of $C_n^G(S)$. We deduce from the computation that $C_n^G(S)$ having finite Betti numbers is a rational $K(π,1)$, i.e its minimal model and $1$-minimal model are the same (or equivalently the $ψ$-homotopy space vanishes in degree grater then $2$), if and only if $S$ is not homeomorphic to $S^2$. In particular, for $S$ not homeomorphic to $S^2$, the minimal model (isomorphic to the $1$-minimal model) is entirely determined by the Malcev Lie algebra of $π_1 C_n^G(S)$. When $A_{n,k}$ is minimal, we get an exact sequence of Malcev Lie algebras $0\to L_{C_{n-k}^G(S_{G,k})}\to L_{C_{n}^G(S)}\to L_{C_k^G(S)}\to 0$, where $L_X$ is the Malcev Lie algebra of $π_1X$. For $S \varsubsetneq \bar{S}=S^2$ and $G$ acting by orientation preserving homeomorphism, we prove that the cohomology ring of $C_n^G(S)$ is Koszul, and that for some of these spaces the minimal model can be obtained out of a Cartan-Chevally-Eilenberg construction applied to graded Lie algebra computed in an earlier work.
We consider orbit configuration spaces associated to finite groups acting freely by orientation preserving homeomorphisms on the $2$-sphere minus a finite number of points. Such action is equivalent to a homography action of a finite subgroup $G\subset \mathrm{PGL}(\mathbb{C}^2)$ on the complex projective line $\mathbb{P}^1$ minus a finite set $Z$ stable under $G$. We compute the cohomology ring and the Poincaré series of the orbit configuration space $C_n^G(\mathbb{P}^1 \setminus Z)$. This can be seen as a generalization of the work of Arnold for the classical configuration space $C_n(\mathbb{C})$ ($(G,Z)=(\{1\},\infty$)). It follows from the work that $C_n^G(\mathbb{P}^1\setminus Z)$ is formal in the sense of rational homotopy theory. We also prove the existence of an LCS formula relating the Poincaré series of $C_n^G(\mathbb{P}^1\setminus Z)$ to the ranks of quotients of successive terms of the lower central series of the fundamental group of $C_n^G(\mathbb{P}^1 \setminus Z)$. The successive quotients correspond to homogenous elements of graded Lie algebras introduced by the author in an earlier work. Such formula is also known for classical configuration spaces of $\mathbb{C}$, where fundamental groups are Artin braid groups and the ranks correspond to dimensions of homogenous elements of the Kohno-Drinfeld Lie algebras.
We prove that Goncharov's dihedral Lie coalgebra $D_{\bullet\bullet}:={\oplus}_{k\geq m \geq 1} D_{m,k}$ of the trivial group ($\widehat{\mathscr{D}}_{\bullet \bullet}(G)$ of (arxiv:math/0009121) for $G=\{e\}$) is the bigraded dual of Brown's linearized double shuffle Lie algebra $\mathfrak{ls}:={\oplus}_{k\geq m \geq 1}\mathfrak{ls}_m^k\subset \mathbb{Q}\langle x,z \rangle$ whose Lie bracket is the Ihara bracket initially defined over $\mathbb{Q}\langle x,z \rangle$. This by constructing an explicit isomorphism of bigraded Lie coalgebras $D_{\bullet \bullet} \to \mathfrak{ls}^\vee$, where $\mathfrak{ls}^\vee$ is the Lie coalgebra dual in the bigraded sense to $\mathfrak{ls}$. The work leads to the equivalence between the two statements: "$D_{\bullet \bullet}$ is a Lie coalgebra with respect to Goncharov's cobracket formula" and "$ \mathfrak{ls}$ is preserved by the Ihara bracket". We also prove folklore results (that apparently have no written proofs in the literature) stating that for $m \geq 2$: $D_{m,\bullet}:=\oplus_{k\geq m} D_{m,k}$ is graded isomorphic (dual) to Ihara-Kaneko-Zagier's double shuffle space $\mathrm{Dsh}_{m}:=\oplus_{k\geq m} \mathrm{Dsh}_{m}({{k}-m}) \subset \mathbb{Q}[x_1,\dots,x_m]$, and that a given linear map $f_m: \mathbb{Q}\langle x,z \rangle_m \to \mathbb{Q}[x_1,\dots,x_m]$, where $\mathbb{Q}\langle x,z \rangle_m$ is the space linearly generated by monomials of $\mathbb{Q}\langle x,z \rangle$ of degree $m$ with respect to $z$, restricts to a graded isomorphism $\bar{f}_m: \mathfrak{ls}_m:=\oplus_{k\geq m} \mathfrak{ls}_m^k \to \mathrm{Dsh}_{m}$. Here, we establish three explicit compatible isomorphisms $D_{\bullet \bullet} \to \mathfrak{ls}^\vee, D_{m\bullet}\to \mathrm{Dsh}_{m}^\vee$ and $\bar{f}_m: \mathfrak{ls}_m \to \mathrm{Dsh}_{m}$, where $\mathrm{Dsh}_{m}^\vee$ is the graded dual of $\mathrm{Dsh}_{m}$.
We study orbit configuration spaces $\mathrm{Cf}_G(n,\mathbb{P}^1_*)$ obtained from the action of a finite homography group $G$ on $\mathbb{P}^1$. We construct a flat connection on the orbit space with values in a Lie algebra $\hat{\mathfrak{p}}_n(G) $. We establish an isomorphism of filtered Lie algebras between $\hat{\mathfrak{p}}_n(G)$, the Malcev Lie algebra of the fundamental group of $\mathrm{Cf}_G(n,\mathbb{P}^1_*)$ and the degree completion of the associated graded to the latter Lie algebra. These isomorphisms are obtained using the monodromy representation of the connection and the study of the fundamental group.