On complex lie supergroups and split homogeneous supermanifolds

It is well known that the category of real Lie supergroups is equivalent to the category of the so-called (real) Harish-Chandra pairs, see [DM], [Kost], [Kosz]. That means that a Lie supergroup depends only on the underlying Lie group and its Lie superalgebra with certain compatibility conditions. More precisely, the structure sheaf of a Lie supergroup and the supergroup morphisms can be explicitly described in terms of the corresponding Lie superalgebra. In this paper we give a proof of this result in the complex-analytic case. Furthermore, if (G, $ \mathcal{O} $ _G ) is a complex Lie supergroup and H ? G is a closed Lie subgroup, i.e., it is a Lie subsupergroup of (G, $ \mathcal{O} $ _G ) and its odd dimension is zero, we show that the corresponding homogeneous supermanifold (G/H, $ \mathcal{O} $ _G/H ) is split. In particular, any complex Lie supergroup is a split supermanifold. It is well known that a complex homogeneous supermanifold may be nonsplit (see, e.g., [OS1]). We find here necessary and sufficient conditions for a complex homogeneous supermanifold to be split.     

Cocalibrated structures on Lie algebras with a codimension one Abelian ideal

Cocalibrated G₂-structures and cocalibrated ${{\rm G}_2^*}$ -structures are the natural initial values for Hitchin’s evolution equations whose solutions define (pseudo)-Riemannian manifolds with holonomy group contained in Spin(7) or Spin₀(3, 4), respectively. In this article, we classify 7-D real Lie algebras with a codimension one Abelian ideal which admit such structures. Moreover, we classify the 7-D complex Lie algebras with a codimension one Abelian ideal which admit cocalibrated ${({\rm G}_2)_{\mathbb{C}}}$ -structures.     

Complex Boosts: A Hermitian Clifford Algebra Approach

The aim of this paper is to study complex boosts in complex Minkowski space-time that preserves the Hermitian norm. Starting from the spin group Spin ${^+(2n, 2m, \mathbb{R})}$ in the real Minkowski space ${\mathbb{R}^{2n,2m}}$ we construct a Clifford realization of the pseudo-unitary group U(n,m) using the space-time Witt basis in the framework of Hermitian Clifford algebra. Restricting to the case of one complex time direction we derive a general formula for a complex boost in an arbitrary complex direction and its KAK-decomposition, generalizing the well-known formula of a real boost in an arbitrary real direction. In the end we derive the complex Einstein velocity addition law for complex relativistic velocities, by the projective model of hyperbolic n-space.     

Twisted fermionic and bosonic representations for a class of BC-graded Lie algebras

In this paper, we study the fermionic and bosonic representations for a class of BC-graded Lie algebras coordinatized by skew Laurent polynomial rings. This generalizes the fermionic and bosonic constructions for the affine Kac-Moody algebras of type A _N ⁽²⁾.     

The image of Colmez’s Montreal functor

We prove a conjecture of Colmez concerning the reduction modulo p of invariant lattices in irreducible admissible unitary p-adic Banach space representations of GL₂(Q _{?? p} ) with p≥5. This enables us to restate nicely the p-adic local Langlands correspondence for GL₂(Q _{?? p} ) and deduce a conjecture of Breuil on irreducible admissible unitary completions of locally algebraic representations.     

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 ${Aut_e({\bf P}_{\bf C})\hspace{1.5mm} \cong ({\rm SO}_3\mathbb{R} \times {\rm SO}_3\mathbb{R})\rtimes \mathbb{Z}_2}$ .     

A Clifford Algebraic Approach to Line Geometry

In this paper we combine methods from projective geometry, Klein’s model, and Clifford algebra. We develop a Clifford algebra whose Pin group is a double cover of the group of regular projective transformations. The Clifford algebra we use is constructed as homogeneous model for the five-dimensional real projective space \({\mathbb {P}^5 (\mathbb{R})}\) where Klein’s quadric \({M^4_2}\) defines the quadratic form. We discuss all entities that can be represented naturally in this homogeneous Clifford algebra model. Projective automorphisms of Klein’s quadric induce projective transformations of \({\mathbb {P}^3 (\mathbb{R})}\) and vice versa. Cayley-Klein geometries can be represented by Clifford algebras, where the group of Cayley-Klein isometries is given by the Pin group of the corresponding Clifford algebra. Therefore, we examine the versor group and study the correspondence between versors and regular projective transformations represented as 4 × 4 matrices. Furthermore, we give methods to compute a versor corresponding to a given projective transformation.     

Comparing Invariants of SK1

In this text, we compare an invariant of the reduced Whitehead group SK ₁ of a central simple algebra recently introduced by Kahn (2010) to other invariants of SK ₁. Doing so, we prove the non-triviality of Kahn’s invariant using the non-triviality of an invariant introduced by Suslin (1991) which is non-trivial for Platonov’s examples of non-trivial SK ₁ (Platonov, Math USSR Izv 10(2):211–243, 1976). We also give a formula for the value on the centre of the tensor product of two symbol algebras which generalises a formula of Merkurjev for biquaternion algebras (Merkurjev 1995).     

Graded Morita Equivalence of Clifford Superalgebras

This note uses a variation of graded Morita theory for finite dimensional superalgebras to determine explicitly the graded basic superalgebras for all real and complex Clifford superalgebras. As an application, the Grothendieck groups of the category of left ${\mathbb{Z}_2}$ -graded modules over all real and complex Clifford superalgebras are described explicitly.     

Pseudocompact Algebras and Highest Weight Categories

We develop a new approach to highest weight categories $\cal{C}$ with good (and cogood) posets of weights via pseudocompact algebras by introducing ascending (and descending) quasi-hereditary pseudocompact algebras. For $\cal{C}$ admitting a Chevalley duality, we define and investigate tilting modules and Ringel duals of the corresponding pseudocompact algebras. Finally, we illustrate all these concepts on an explicit example of the general linear supergroup GL(1|1).     

Variational Inequalities: an Elementary Approach

We will deal with the following problem: Let M be an n×n matrix with real entries. Under which conditions the family of inequalities: x∈? ⁿ ;x?0;M·x?0has non–trivial solutions? We will prove that a sufficient condition is given by m_i,j+m_j,i?0 (1?i,j?n); from this result we will derive an elementary proof of the existence theorem for Variational Inequalities in the framework of Monotone Operators.     

Independent joins of tolerance factorable varieties

Let Lat denote the variety of lattices. In 1982, the second author proved that Lat is strongly tolerance factorable, that is, the members of Lat have quotients in Lat modulo tolerances, although Lat has proper tolerances. We did not know any other nontrivial example of a strongly tolerance factorable variety. Now we prove that this property is preserved by forming independent joins (also called products) of varieties. This enables us to present infinitely many strongly tolerance factorable varieties with proper tolerances. Extending a recent result of G. Czédli and G. Grätzer, we show that if ${\mathcal{V}}$ is a strongly tolerance factorable variety, then the tolerances of ${\mathcal{V}}$ are exactly the homomorphic images of congruences of algebras in ${\mathcal{V}}$ . Our observation that (strong) tolerance factorability is not necessarily preserved when passing from a variety to an equivalent one leads to an open problem.     

Metric and Involution Scores of Clifford Algebras

The notion of a metric score is introduced as a separate tally of the total number of standard (Grassmann) basis elements spanning a Clifford algebra \({C\ell_{p,q}}\) that square to + 1 and ?1. A closed-form expression is derived for any given vector space dimension n = p+q. This is then generalized to reversion and Clifford-conjugation. A central application is that two real Clifford algebras are isomorphic if and only if their metric scores are identical.     

Positive definite superfunctions and unitary representations of lie supergroups

For a broad class of Fréchet-Lie supergroups $ \mathcal{G} $ , we prove that there exists a correspondence between positive definite smooth (resp., analytic) superfunctions on $ \mathcal{G} $ and matrix coefficients of smooth (resp., analytic) unitary representations of the Harish-Chandra pair (G, $ \mathfrak{g} $ ) associated to $ \mathcal{G} $ . As an application, we prove that a smooth positive definite superfunction on $ \mathcal{G} $ is analytic if and only if it restricts to an analytic function on the underlying manifold of $ \mathcal{G} $ . When the underlying manifold of $ \mathcal{G} $ is 1-connected we obtain a necessary and sufficient condition for a linear functional on the universal enveloping algebra U( $ {{\mathfrak{g}}_{\mathbb{C}}} $ ) to correspond to a matrix coefficient of a unitary representation of (G, $ \mathfrak{g} $ ). The class of Lie supergroups for which the aforementioned results hold is characterised by a condition on the convergence of the Trotter product formula. This condition is strictly weaker than assuming that the underlying Lie group of $ \mathcal{G} $ is a locally exponential Fréchet-Lie group. In particular, our results apply to examples of interest in representation theory such as mapping supergroups and diffeomorphism supergroups.     

De Vries Algebras and Compact Regular Frames

By the de Vries theorem, the category DeV of de Vries algebras is dually equivalent to the category KHaus of compact Hausdorff spaces. By the Isbell theorem, the category KRFrm of compact regular frames is dually equivalent to KHaus. The proofs of both theorems employ the axiom of choice. It is a consequence of the de Vries and Isbell theorems that DeV is equivalent to KRFrm. We give a direct proof of this result, which is choice-free. In the absence of the axiom of countable dependent choice (CDC), the category KCRFrm of compact completely regular frames is a proper subcategory of KRFrm. We introduce the category cDeV of completely regular de Vries algebras, which in the absence of (CDC) is a proper subcategory of DeV, and show that cDeV is equivalent to KCRFrm. Finally, we show how the restriction of the equivalence of DeV and KRFrm works in the zero-dimensional and extremally disconnected cases.     

A Short Note on Essentially Σ1 Sentences

Guaspari (J Symb Logic 48:777–789, 1983) conjectured that a modal formula is it essentially Σ₁ (i.e., it is Σ₁ under any arithmetical interpretation), if and only if it is provably equivalent to a disjunction of formulas of the form ${\square{B}}$ . This conjecture was proved first by A. Visser. Then, in (de Jongh and Pianigiani, Logic at Work: In Memory of Helena Rasiowa, Springer-Physica Verlag, Heidelberg-New York, pp. 246–255, 1999), the authors characterized essentially Σ₁ formulas of languages including witness comparisons using the interpretability logic ILM. In this note we give a similar characterization for formulas with a binary operator interpreted as interpretability in a finitely axiomatizable extension of IΔ ₀  + Supexp and we address a similar problem for IΔ ₀  + Exp.     

Hypersurfaces of Spinc Manifolds and Lawson Type Correspondence

Simply connected three-dimensional homogeneous manifolds ${\mathbb{E}(\kappa, \tau)}$ , with four-dimensional isometry group, have a canonical Spin^c structure carrying parallel or Killing spinors. The restriction to any hypersurface of these parallel or Killing spinors allows to characterize isometric immersions of surfaces into ${\mathbb{E}(\kappa, \tau)}$ . As application, we get an elementary proof of a Lawson type correspondence for constant mean curvature surfaces in ${\mathbb{E}(\kappa, \tau)}$ . Real hypersurfaces of the complex projective space and the complex hyperbolic space are also characterized via Spin^c spinors.     

Spin geometry and grand unification

Gravitation becomes unified with quantum mechanics when we recognize that the spacetime tetrads and the matter fields of Fermions are the integral and half-integral spin representations of theEinstein group, E, the global extension of the Poincaré group to a curved spacetimeM. There are8 fundamental spinor representations of theE group, interchanged byP, T, andC: the degree-one maps of spin space overM. Tensor products of2 spinor fields buildClifford vectors or 1 forms, e.g. the spacetime tetrads. It takes tensor products of all8 spinor fields to build a natural 4 form; in particular, ourE-invariant Lagrangian density . We propose a simple form for : the8-spinor factorization of theMaurer-Cartan 4-form, Ω⁴. Thespin connections Ω_α step off the conjoined left and right internalgl (2, ?) phase increments over aspacetime incremente _α. Our actionS _g measures the covering number of the spinor phases over spacetimeM∪D _J; theD _J aresingular domains or caustics, whereJ=1, 2, and 3 chiral pairs of spin waves cross. Here, the massive Dirac equations emerge to govern the mass scattering that keep the “null zig-zags” of a bispinor particle confined to a timelike worldtube. We identify the coupled envelopes of 1, 2, and 3 chiral bispinor pairs as the leptons, mesons, and hadrons, respectively. These source topologically —nontrivialgl (2,C) phase distributions in the far-field region, which appear aseffective vector potentials. Their vorticities are thespin curvatures, whose Hermitian parts —thegravitational curvatures —specify how our spacetime manifoldM must expand and curve to accommodate such anholonomic differentials. The anti-Hermitian parts reproduce the standard electroweak and strong fields, together with their actions. also contains some new cross terms between electroweak potentials and gravitational curvatures. Do these signal a failure of unification, or predict new phenomena?