Automorphisms of Some Topological Regular Parallelisms of {{\rm PG}(3, \mathbb{R})}

In a precedent article we constructed various topological regular parallelisms of the real projective 3-space \({{\rm PG}(3, \mathbb{R})}\) via hyperflock determining line sets of \({{\rm PG}(5, \mathbb{R})}\) (see Betten and Riesinger in Mh Math 161:43–58, 2010). In the present paper we discuss for some of these parallelisms their automorphism groups consisting of all automorphic collineations and all automorphic dualities, especially we compute their group dimension. Thus we are able to present: (1) topological regular 5-dimensional parallelisms of \({{\rm PG}(3, \mathbb{R})}\) of group dimension 0, (2) topological regular 4-dimensional parallelisms of \({{\rm PG}(3, \mathbb{R})}\) of group dimension 0 or 1, (3) topological regular 3-dimensional parallelisms of \({{\rm PG}(3, \mathbb{R})}\) of group dimension 1.     

A Remark on The Geometry of Spaces of Functions with Prime Frequencies

For any positive integer r, denote by \({\mathcal{P}_{r}}\) the set of all integers \({\gamma \in \mathbb{Z}}\) having at most r prime divisors. We show that \({C_{\mathcal{P}_{r}}(\mathbb{T})}\) , the space of all continuous functions on the circle \({\mathbb{T}}\) whose Fourier spectrum lies in \({\mathcal{P}_{r}}\) , contains a complemented copy of \({\ell^{1}}\) . In particular, \({C_{\mathcal{P}_{r}}(\mathbb{T})}\) is not isomorphic to \({C(\mathbb{T})}\) , nor to the disc algebra \({A(\mathbb{D})}\) . A similar result holds in the L ¹ setting.     

A Few Comments on the Clifford Algebra {C\ell_2} of the Standard Euclidean Plane

This paper is a continuation of the author’s plenary lecture given at ICCA 9 which was held in Weimar at the Bauhaus University, 15–20 July, 2011. We want to study on both the mathematical and the epistemological levels the thought of the brilliant geometer W. K. Clifford by presenting a few comments on the structure of the Clifford algebra ${C\ell_2}$ associated with the standard Euclidean plane ${\mathbb{R}^2}$ . Miquel’s theorem will be given in the algebraic context of the even Clifford algebra ${C\ell^+_2}$ isomorphic to the real algebra ${\mathbb{C}}$ . The proof of this theorem will be based on the cross ratio (the anharmonic ratio) of four complex numbers. It will lead to a group of homographies of the standard projective line ${\mathbb{C}P^1 = P(\mathbb{C}^2)}$ which appeared so attractive to W. K. Clifford in his overview of a general theory of anharmonics. In conclusion it will be shown how the classical Clifford-Hopf fibration S ¹ → S ³ → S ² leads to the space of spinors ${\mathbb{C}^2}$ of the Euclidean space ${\mathbb{R}^3}$ and to the isomorphism ${{\rm {PU}(1) = \rm {SU}(2)/\{I,-I\} \simeq SO(3)}}$ .     

Cohomological characterization of hyperquadrics of odd dimensions in characteristic two

We consider characterizations of projective varieties in terms of their tangents. Mori established the characterization of projective spaces in arbitrary characteristic by ampleness of tangent bundles. Wahl characterized projective spaces in characteristic zero by cohomological condition of tangent bundles; in addition, he remarked that a counter-example in characteristic two is constructed from odd-dimensional hyperquadrics \(Q_{2n-1}\) with \(n > 1\) . This is caused by existence of a common point in \({\mathbb {P}}^{2n}\) which every embedded tangent space to the quadric contains. In general, a projective variety in \({{\mathbb {P}}}^N\) is said to be strange if its embedded tangent spaces admit such a common point in \({{\mathbb {P}}}^N\) . A non-linear smooth projective curve is strange if and only if it is a conic in characteristic two (Lluis, Samuel). Kleiman and Piene showed that a non-linear smooth hypersurface in \({{\mathbb {P}}}^N\) is strange if and only if it is a quadric of odd-dimension in characteristic two. In this paper, we investigate complete intersections, and prove that, a non-linear smooth complete intersection in \({{\mathbb {P}}}^N\) is strange if and only if it is a quadric in \({{\mathbb {P}}}^N\) of odd dimension in characteristic two; these conditions are also equivalent to non-vanishing of \(0\) -cohomology of \((-1)\) -twist of the tangent bundle.     

Matrix Representations of the Low Order Real Clifford Algebras

In this paper we construct the matrix subalgebras ${L_{r,s}(\mathbb{R})}$ of the real matrix algebra ${M_{2^{r+s}} (\mathbb{R})}$ when 2 ≤ r + s ≤ 3 and we show that each ${L_{r,s}(\mathbb{R})}$ is isomorphic to the real Clifford algebra ${\mathcal{C} \ell_{r,s}}$ . In particular, we prove that the algebras ${L_{r,s}(\mathbb{R})}$ can be induced from ${L_{0,n}(\mathbb{R})}$ when 2 ≤ r + s = n ≤ 3 by deforming vector generators of ${L_{0,n}(\mathbb{R})}$ to multiply the specific diagonal matrices. Also, we construct two subalgebras ${T_4(\mathbb{C})}$ and ${T_2(\mathbb{H})}$ of matrix algebras ${M_4(\mathbb{C})}$ and ${M_2(\mathbb{H})}$ , respectively, which are both isomorphic to the Clifford algebra ${\mathcal{C} \ell_{0,3}}$ , and apply them to obtain the properties related to the Clifford group Γ_0,3.     

Clifford parallelism: old and new definitions, and their use

Parallelity in the real elliptic 3-space was defined by W. K. Clifford in 1873 and by F. Klein in 1890; we compare the two concepts. A Clifford parallelism consists of all regular spreads of the real projective 3-space ${{\rm PG}(3,\mathbb{R})}$ whose (complex) focal lines (=directrices) form a regulus contained in an imaginary quadric (D1 = Klein??s definition). Our new access to the topic ??Clifford parallelism?? is free of complexification and involves Klein??s correspondence ?? of line geometry together with a bijective map ?? from all regular spreads of ${{\rm PG}(3,\mathbb{R})}$ onto those lines of ${{\rm PG}(5,\mathbb{R})}$ having no common point with the Klein quadric; a regular parallelism P of ${{\rm PG}(3,\mathbb{R})}$ is Clifford, if the spreads of P are mapped by ?? onto a plane of lines (D2 = planarity definition). We prove the equivalence of (D1) and (D2). Associated with ?? is a simple dimension concept for regular parallelisms which allows us to say instead of (D2): the 2-dimensional regular parallelisms of ${{\rm PG}(3,\mathbb{R})}$ are Clifford (D3 = dimensionality definition). Submission of (D2) to ??^?1 yields a complexification free definition of a Clifford parallelism which uses only elements of ${{\rm PG}(3,\mathbb{R})}$ : A regular parallelism P is Clifford, if the union of any two distinct spreads of P is contained in a general linear complex of lines (D4 = line geometric definition). In order to see (D1) and (D2) simultaneously at work we discuss the following two examples using, at the one hand, complexification and (D1) and, at the other hand, (D2) under avoidance of complexification. Example 1. In the projectively extended real Euclidean 3-space a rotational regular spread with center o is submitted to the group of all rotations about o; we prove, that a Clifford parallelism is generated. Example 2. We determine the group ${Aut_e({\bf P}_{\bf C})}$ of all automorphic collineations and dualities of the Clifford parallelism P _C and show ${Aut_e({\bf P}_{\bf C})\hspace{1.5mm} \cong ({\rm SO}_3\mathbb{R} \times {\rm SO}_3\mathbb{R})\rtimes \mathbb{Z}_2}$ .     

Automorphisms of {{\rm M_{n}}(\mathbb{D})}, Partially Ordered by Rank Subtractivity Ordering

Let \({\mathbb{D}}\) be an arbitrary division ring and \({{\rm M_{n}}(\mathbb{D})}\) be the set of all n × n matrices over \({\mathbb{D}}\) . We define the rank subtractivity or minus partial order on \({{\rm M_{n}}(\mathbb{D})}\) as defined on \({{\rm M_{n}}(\mathbb{C})}\) , i.e., \({A \leqslant B}\) iff rank(B) = rank(A) + rank(B?A). We describe the structure of maps Φ on \({{\rm M_{n}}(\mathbb{D})}\) such that \({A\leqslant B}\) iff \({\Phi(A)\leqslant \Phi(B) (A, B\in {\rm M_{n}}(\mathbb{D}) )}\) .     

Explicit Matrix Realization of Clifford Algebras

In this paper, we construct associative subalgebras ${{L_{2}}{n}(\mathbb{R})}$ of the real ${2^{n} \times 2^{n}}$ matrix algebra ${{M_{2}}{n}(\mathbb{R})}$ , which is isomorphic to the real Clifford algebra ${C \ell_{0},n}$ for every ${n \in N}$ .     

Orbit Spaces of Free Involutions on the Product of Two Projective Spaces

Let X be a finitistic space having the mod 2 cohomology algebra of the product of two projective spaces. We study free involutions on X and determine the possible mod 2 cohomology algebra of orbit space of any free involution, using the Leray spectral sequence associated to the Borel fibration ${X \hookrightarrow X_{\mathbb{Z}_2} \longrightarrow B_{\mathbb{Z}_2}}$ . We also give an application of our result to show that if X has the mod 2 cohomology algebra of the product of two real projective spaces (respectively, complex projective spaces), then there does not exist any ${\mathbb{Z}_2}$ -equivariant map from ${\mathbb{S}^k \to X}$ for k ≥ 2 (respectively, k ≥ 3), where ${\mathbb{S}^k}$ is equipped with the antipodal involution.     

Boundedness and Compactness of Pseudodifferential Operators with Non-Regular Symbols on Weighted Lebesgue Spaces

Applying the boundedness on weighted Lebesgue spaces of the maximal singular integral operator S _* related to the Carleson?CHunt theorem on almost everywhere convergence, we study the boundedness and compactness of pseudodifferential operators a(x, D) with non-regular symbols in ${L^\infty(\mathbb{R}, V(\mathbb{R})), PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ and ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ on the weighted Lebesgue spaces ${L^p(\mathbb{R},w)}$ , with 1?< p <? ?? and ${w\in A_p(\mathbb{R})}$ . The Banach algebras ${L^\infty(\mathbb{R}, V(\mathbb{R}))}$ and ${PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ consist, respectively, of all bounded measurable or piecewise continuous ${V(\mathbb{R})}$ -valued functions on ${\mathbb{R}}$ where ${V(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded total variation, and the Banach algebra ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ consists of all Lipschitz ${V_d(\mathbb{R})}$ -valued functions of exponent ${\gamma \in (0,1]}$ on ${\mathbb{R}}$ where ${V_d(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded variation on dyadic shells. Finally, for the Banach algebra ${\mathfrak{A}_{p,w}}$ generated by all pseudodifferential operators a(x, D) with symbols ${a(x, \lambda) \in PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ on the space ${L^p(\mathbb{R}, w)}$ , we construct a non-commutative Fredholm symbol calculus and give a Fredholm criterion for the operators ${A \in \mathfrak{A}_{p,w}}$ .     

Spectra of tensor triangulated categories over category algebras

Let \({\mathcal{C}}\) be a finite EI category and k be a field. We consider the category algebra \({k\mathcal{C}}\) . Suppose \({\sf{K}(\mathcal{C})=\sf{D}^b(k \mathcal{C}-\sf{mod})}\) is the bounded derived category of finitely generated left modules. This is a tensor triangulated category, and we compute its spectrum in the sense of Balmer. When \({\mathcal{C}=G \propto \mathcal{P}}\) is a finite transporter category, the category algebra becomes Gorenstein, so we can define the stable module category \({\underline{\sf{CM}} k(G \propto \mathcal{P})}\) , of maximal Cohen–Macaulay modules, as a quotient category of \({{\sf{K}}(G \propto \mathcal{P})}\) . Since \({\underline{\sf{CM}} k(G\propto\mathcal{P})}\) is also tensor triangulated, we compute its spectrum as well. These spectra are used to classify tensor ideal thick subcategories of the corresponding tensor triangulated categories.     

A remark on the generalized Hodge conjecture

It is well known that for a smooth, projective variety \(X\) over \({\mathbb {C}}\) we have \(N^{p}H^{i}(X,{\mathbb {Q}})\subset F^{p} H^{i}(X,{\mathbb {C}})\cap H^{i}(X,{\mathbb {Q}})\) , where \(N^{\bullet }\) and \(F^{\bullet }\) are respectively the coniveau and Hodge filtrations. In general this inclusion is strict. We introduce a natural subspace \(S^{p,i}\subset F^{p}H^{i}(X,{\mathbb {C}})\) such that \(N^{p}H^{i}(X,{\mathbb {Q}})= S^{p,i}\cap H^{i}(X,{\mathbb {Q}})\) holds true for any \(i,p\) . The main technical tool is the use of semi-algebraic sets, which are available by the triangulation of complex projective varieties.     

On real local isometric immersions of {\mathbb{C}Q^2_c} into {\mathbb{C} P^{3}}

We investigate real local isometric immersions of Kähler manifolds ${\mathbb{C}Q^2_c}$ of constant holomorphic curvature 4c into complex projective 3-space. Our main result is that the standard embedding of ${\mathbb{C}P^2}$ into ${\mathbb{C}P^3}$ has strong rigidity under the class of local isometric transformations. We also prove that there are no local isometric immersions of ${\mathbb{C}Q^2_c}$ into ${\mathbb{C}P^3}$ when they have different holomorphic curvature. An important method used is a study of the relationship between the complex structure of any locally isometric immersed ${\mathbb{C}Q^2_c}$ and the complex structure of the ambient space ${\mathbb{C}P^3}$ .     

Applications of Matrix Algebra to Clifford Groups

In this paper, we will show that all of nonzero vectors and nonzero bivectors in the Clifford algebra ${\mathcal{C} \ell_{0,3}}$ are invertible and we will find some conditions for those objects to be element of the Clifford group ??_0,3 using the corresponding properties in the subalgebra L ₈ of the matrix algebra ${M_8 \mathbb{(R)}}$ .     

Dirichlet to Neumann functional and the group of disc’s biholomorphic trasformations

Studying the injectivity of the Dirichlet to Neumann functional on the unit disk in the plane, we arrive at the nonlinear differential equation $S(f)=-\frac{1}{2}([f^\prime ]^2-1)$ , where $S(f)$ is the Schwarzian derivative of $f$ (cf. (1.2)). Let $\mathcal{S}$ be the set of solutions of this equation. In this article, we establish a correspondence between the set of solutions $\mathcal{S}$ and the projective linear transformations on the real projective line. Moreover, we use this to show that the quotient group $\mathbb{H }/\mathcal{T}_{2\pi }$ it is isomorphic to the group of biholomorphic automorphisms of the unit disk in the plane. Here $\mathcal{T}_{2\pi }$ and $\mathbb{H }$ denote the subgroups of $\mathcal{S}$ consisting of translations by integral multiples of $2\pi $ and those solutions with have positive derivative, respectively.     

Partial isomorphisms over finite fields

In this paper, we construct a combinatorial algebra of partial isomorphisms that gives rise to a “projective limit” of the centers of the group algebras \(\mathbb {C}\mathrm {GL}(n,\mathbb {F}_{q})\) . It allows us to prove a \(\mathrm {GL}(n,\mathbb {F}_{q})\) -analogue of a theorem of Farahat and Higman regarding products of conjugacy classes of permutations.     

On a problem concerning the Banach algebra generated by the maps n\mapsto\lambda^{\binom{n}{k}}

In Jabbari and Namioka (Milan J. Math. 78:503?C522, 2010), the authors characterized the spectrum M(W) of the Weyl algebra W, i.e. the norm closure of the algebra generated by the family of functions $\{n\mapsto x^{n^{k}}; x\in\mathbb{T}, k\in\mathbb{N}\}$ , ( $\mathbb{T}$ the unit circle), with a closed subgroup of $E(\mathbb{T})^{\mathbb{N}}$ where $E(\mathbb{T})$ denotes the family of the endomorphisms of the multiplicative group $\mathbb{T}$ . But the size of M(W) in $E(\mathbb{T})^{\mathbb{N}}$ as well as the induced group operation were left as a problem. In this paper, we will give a solution to this problem.     

On Extending {\mathcal{A}}-Modules Through the Coefficients

In classical linear algebra, extending the ring of scalars of a free module gives rise to a new free module containing an isomorphic copy of the former and satisfying a certain universal property. Also, given two free modules on the same ring of scalars and a morphism between them, enlarging the ring of scalars results in obtaining a new morphism having the nice property that it coincides with the initial map on the isomorphic copy of the initial free module in the new one. We investigate these problems in the category of free ${\mathcal{A}}$ -modules, where ${\mathcal{A}}$ is an ${\mathbb{R}}$ -algebra sheaf. Complexification of free ${\mathcal{A}}$ -modules, which is defined to be the process of obtaining new free ${\mathcal{A}}$ -modules by enlarging the ${\mathbb{R}}$ -algebra sheaf ${\mathcal{A}}$ to a ${\mathbb{C}}$ -algebra sheaf, denoted ${\mathcal{A}_\mathbb{C}}$ , is an important particular case (see Proposition 2.1, Proposition 3.1). Attention, on the one hand, is drawn on the sub- ${_{\mathbb{R}}\mathcal{A}}$ -sheaf of almost complex structures on the sheaf ${{_\mathbb{R}}\mathcal{A}^{2n}}$ , the underlying ${\mathbb{R}}$ -algebra sheaf of a ${\mathbb{C}}$ -algebra sheaf ${\mathcal{A}}$ , and on the other hand, on the complexification of the functor ${\mathcal{H}om_\mathcal {A}}$ , with ${\mathcal{A}}$ an ${\mathbb{R}}$ -algebra sheaf.     

On the Topology of Kac–Moody groups

We study the topology of spaces related to Kac–Moody groups. Given a Kac–Moody group over $\mathbb C $ , let $\text {K}$ denote the unitary form with maximal torus ${{\mathrm{T}}}$ having normalizer ${{\mathrm{N}}}({{\mathrm{T}}})$ . In this article we study the cohomology of the flag manifold $\text {K}/{{{\mathrm{T}}}}$ as a module over the Nil-Hecke algebra, as well as the (co)homology of $\text {K}$ as a Hopf algebra. In particular, if $\mathbb F $ has positive characteristic, we show that $\text {H}_*(\text {K},\mathbb F )$ is a finitely generated algebra, and that $\text {H}^*(\text {K},\mathbb F )$ is finitely generated only if $\text {K}$ is a compact Lie group . We also study the stable homotopy type of the classifying space $\text {BK}$ and show that it is a retract of the classifying space $\text {BN(T)}$ of ${{\mathrm{N}}}({{\mathrm{T}}})$ . We illustrate our results with the example of rank two Kac–Moody groups.