Zhelobenko invariants, Bernstein-Gelfand-Gelfand operators and the analogue Kostant Clifford algebra conjecture

Let $ \mathfrak{g} $ be a complex simple Lie algebra and $ \mathfrak{h} $ a Cartan subalgebra. The Clifford algebra C( $ \mathfrak{g} $ ) of g admits a Harish-Chandra map. Kostant conjectured (as communicated to Bazlov in about 1997) that the value of this map on a (suitably chosen) fundamental invariant of degree 2?m?+?1 is just the zero weight vector of the simple (2?m?+?1)-dimensional module of the principal s-triple obtained from the Langlands dual $ {\mathfrak{g}^\vee } $ . Bazlov [1] settled this conjecture positively in type A. The hard part of the Kostant Clifford algebra conjecture is a question concerning the Harish-Chandra map for the enveloping algebra U( $ \mathfrak{g} $ ) composed with evaluation at the half sum ?? of the positive roots. The analogue Kostant conjecture is obtained by replacing the Harish-Chandra map by a ??generalized Harish-Chandra?? map. This map had been studied notably by Zhelobenko [15]. The proof given here involves a symmetric algebra version of the Kostant conjecture, the Zhelobenko invariants in the adjoint case, and, surprisingly, the Bernstein-Gelfand-Gelfand operators introduced in their study [3] of the cohomology of the flag variety.     

Some Integral Representation for Meta-Monogenic Function in Clifford Algebras Depending on Parameters

In recent years, the integral representation problems have been studied in many context and generalities. For example, for the monogenic and meta functions in some Clifford type algebras, see [10, 11]. In this paper we construct a Cauchy-Pompeiu type formula for meta-monogenic operator of order ${n, (D-\lambda)^n, \lambda \in \mathbb{R}}$ , and its conjugate ${(\overline{D} - \lambda)^n}$ in a Clifford algebra depending on parameters ${\mathcal{A}_n(2, \alpha_j, \gamma_{ij})}$ . Using these explicit representation formula of Cauchy-Pompeiu type we will show some applications.     

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.     

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.     

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 ${{\rm {PU}(1) = \rm {SU}(2)/\{I,-I\} \simeq SO(3)}}$ .     

Foulkes characters, Eulerian idempotents, and an amazing matrix

John Holte (Am. Math. Mon. 104:138?C149, 1997) introduced a family of ??amazing matrices?? which give the transition probabilities of ??carries?? when adding a list of numbers. It was subsequently shown that these same matrices arise in the combinatorics of the Veronese embedding of commutative algebra (Brenti and Welker, Adv. Appl. Math. 42:545?C556, 2009; Diaconis and Fulman, Am. Math. Mon. 116:788?C803, 2009; Adv. Appl. Math. 43:176?C196, 2009) and in the analysis of riffle shuffling (Diaconis and Fulman, Am. Math. Mon. 116:788?C803, 2009; Adv. Appl. Math. 43:176?C196, 2009). We find that the left eigenvectors of these matrices form the Foulkes character table of the symmetric group and the right eigenvectors are the Eulerian idempotents introduced by Loday (Cyclic Homology, 1992) in work on Hochschild homology. The connections give new closed formulae for Foulkes characters and allow explicit computation of natural correlation functions in the original carries problem.     

Modular Decomposition of the Orlik-Terao Algebra

Let ${\mathcal{A}}$ be a collection of n linear hyperplanes in ${\mathbb{k}^\ell}$ , where ${\mathbb{k}}$ is an algebraically closed field. The Orlik-Terao algebra of ${\mathcal{A}}$ is the subalgebra ${{\rm R}(\mathcal{A})}$ of the rational functions generated by reciprocals of linear forms vanishing on hyperplanes of ${\mathcal{A}}$ . It determines an irreducible subvariety ${Y (\mathcal{A})}$ of ${\mathbb{P}^{n-1}}$ . We show that a flat X of ${\mathcal{A}}$ is modular if and only if ${{\rm R}(\mathcal{A})}$ is a split extension of the Orlik-Terao algebra of the subarrangement ${\mathcal{A}_X}$ . This provides another refinement of Stanley’s modular factorization theorem [34] and a new characterization of modularity, similar in spirit to the fibration theorem of [27]. We deduce that if ${\mathcal{A}}$ is supersolvable, then its Orlik-Terao algebra is Koszul. In certain cases, the algebra is also a complete intersection, and we characterize when this happens.     

Endpoint Bounds for the Quartile Operator

It is a result by Lacey and Thiele (Ann. of Math. (2) 146(3):693–724, 1997; ibid. 149(2):475–496, 1999) that the bilinear Hilbert transform maps $L^{p_{1}}(\mathbb{R}) \times L^{p_{2}}(\mathbb{R}) $ into $L^{p_{3}}(\mathbb{R})$ whenever (p ₁,p ₂,p ₃) is a Hölder tuple with p ₁,p ₂>1 and $p_{3}>\frac{2}{3}$ . We study the behavior of the quartile operator, which is the Walsh model for the bilinear Hilbert transform, when $p_{3}=\frac{2}{3}$ . We show that the quartile operator maps $L^{p_{1}}(\mathbb{R}) \times L^{p_{2}}(\mathbb{R}) $ into $L^{\frac{2}{3},\infty}(\mathbb{R})$ when p ₁,p ₂>1 and one component is restricted to subindicator functions. As a corollary, we derive that the quartile operator maps $L^{p_{1}}(\mathbb{R}) \times L^{p_{2},\frac{2}{3}}(\mathbb{R}) $ into $L^{\frac{2}{3},\infty}(\mathbb{R})$ . We also provide weak type estimates and boundedness on Orlicz-Lorentz spaces near p ₁=1,p ₂=2 which improve, in the Walsh case, the results of Bilyk and Grafakos (J. Geom. Anal. 16 (4):563–584, 2006) and Carro et al. (J. Math. Anal. Appl. 357(2):479–497, 2009). Our main tool is the multi-frequency Calderón-Zygmund decomposition from (Nazarov et al. in Math. Res. Lett. 17(3):529–545, 2010).     

The Idempotents of the TL n -Module {\otimes^{n}\mathbb{C}2} in Terms of Elements of {{\rm U}_{q} \mathfrak{sl}_2}

The vector space \({\otimes^{n}\mathbb{C}^2}\) upon which the XXZ Hamiltonian with n spins acts bears the structure of a module over both the Temperley–Lieb algebra \({{\rm TL}_{n}(\beta = q + q^{-1})}\) and the quantum algebra \({{\rm U}_{q} \mathfrak{sl}_2}\) . The decomposition of \({\otimes^{n}\mathbb{C}^2}\) as a \({{\rm U}_{q} \mathfrak{sl}_2}\) -module was first described by Rosso (Commun Math Phys 117:581–593, 1988), Lusztig (Cont Math 82:58–77, 1989) and Pasquier and Saleur (Nucl Phys B 330:523–556, 1990) and that as a TL _n -module by Martin (Int J Mod Phys A 7:645–673, 1992) (see also Read and Saleur Nucl Phys B 777(3):316–351, 2007; Gainutdinov and Vasseur Nucl Phys B 868:223–270, 2013). For q generic, i.e. not a root of unity, the TL _n -module \({\otimes^{n}\mathbb{C}^2}\) is known to be a sum of irreducible modules. We construct the projectors (idempotents of the algebra of endomorphisms of \({\otimes^{n}\mathbb{C}^2}\) ) onto each of these irreducible modules as linear combinations of elements of \({{\rm U}_{q} \mathfrak{sl}_2}\) . When q = q _c is a root of unity, the TL _n -module \({\otimes^{n}\mathbb{C}^2}\) (with n large enough) can be written as a direct sum of indecomposable modules that are not all irreducible. We also give the idempotents projecting onto these indecomposable modules. Their expression now involves some new generators, whose action on \({\otimes^{n}\mathbb{C}^2}\) is that of the divided powers \({(S^{\pm})^{(r)} = \lim_{q \rightarrow q_{c}} (S^{\pm})^r/[r]!}\) .     

On the bilinearity rank of a proper cone and Lyapunov-like transformations

A real square matrix \(Q\) is a bilinear complementarity relation on a proper cone \(K\) in \(\mathbb{R }^n\) if $$\begin{aligned} x\in K, s\in K^*,\,\,\text{ and }\,\,\langle x,s\rangle =0\Rightarrow x^{T}Qs=0, \end{aligned}$$ where \(K^*\) is the dual of \(K\) . The bilinearity rank of \(K\) is the dimension of the linear space of all bilinear complementarity relations on \(K\) . In this article, we continue the study initiated by Rudolf et al. (Math Prog Ser B 129:5–31, 2011). We show that bilinear complementarity relations are related to Lyapunov-like transformations that appear in dynamical systems and in complementarity theory and further show that the bilinearity rank of \(K\) is the dimension of the Lie algebra of the automorphism group of \(K\) . In addition, we correct a result of Rudolf et al., compute the bilinearity ranks of symmetric and completely positive cones, and state some Schur-type results for Lyapunov-like transformations.     

Newform theory for Hilbert Eisenstein series

In his thesis, Weisinger (Thesis, 1977) developed a newform theory for elliptic modular Eisenstein series. This newform theory for Eisenstein series was later extended to the Hilbert modular setting by Wiles (Ann. Math. 123(3):407–456, 1986). In this paper, we extend the theory of newforms for Hilbert modular Eisenstein series. In particular, we provide a strong multiplicity-one theorem in which we prove that Hilbert Eisenstein newforms are uniquely determined by their Hecke eigenvalues for any set of primes having Dirichlet density greater than $\frac{1}{2}$ . Additionally, we provide a number of applications of this newform theory. Let denote the space of Hilbert modular Eisenstein series of parallel weight k≥3, level $\mathcal{N}$ and Hecke character Ψ over a totally real field K. For any prime $\mathfrak{q}$ dividing $\mathcal{N}$ , we define an operator $C_{\mathfrak{q}}$ generalizing the Hecke operator $T_{\mathfrak{q}}$ and prove a multiplicity-one theorem for with respect to the algebra generated by the Hecke operators $T_{\mathfrak{p}}$ ( $\mathfrak{p}\nmid\mathcal{N}$ ) and the operators $C_{\mathfrak{q}}$ ( $\mathfrak{q}\mid\mathcal{N}$ ). We conclude by examining the behavior of Hilbert Eisenstein newforms under twists by Hecke characters, proving a number of results having a flavor similar to those of Atkin and Li (Invent. Math. 48(3):221–243, 1978).     

L p -theory for second-order elliptic operators with unbounded coefficients

Second-order elliptic operators with unbounded coefficients of the form ${Au := -{\rm div}(a\nabla u) + F . \nabla u + Vu}$ in ${L^{p}(\mathbb{R}^{N}) (N \in \mathbb{N}, 1 < p < \infty)}$ are considered, which are the same as in recent papers Metafune et?al. (Z Anal Anwendungen 24:497–521, 2005), Arendt et?al. (J Operator Theory 55:185–211, 2006; J Math Anal Appl 338: 505–517, 2008) and Metafune et?al. (Forum Math 22:583–601, 2010). A new criterion for the m-accretivity and m-sectoriality of A in ${L^{p}(\mathbb{R}^{N})}$ is presented via a certain identity that behaves like a sesquilinear form over L ^p ×?L ^p'. It partially improves the results in (Metafune et?al. in Z Anal Anwendungen 24:497–521, 2005) and (Metafune et?al. in Forum Math 22:583–601, 2010) with a different approach. The result naturally extends Kato’s criterion in (Kato in Math Stud 55:253–266, 1981) for the nonnegative selfadjointness to the case of p ≠?2. The simplicity is illustrated with the typical example ${Au = -u\hspace{1pt}'' + x^{3}u\hspace{1pt}' + c |x|^{\gamma}u}$ in ${L^p(\mathbb{R})}$ which is dealt with in (Arendt et?al. in J Operator Theory 55:185–211, 2006; Arendt et?al. in J Math Anal Appl 338: 505–517, 2008).     

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.     

Some Recent Developments on Hopf��s Holomorphic Form

Hopf??s theorem on surfaces in ${\mathbb{R}^3}$ with constant mean curvature (Hopf in Math Nach 4:232?C249, 1950-51) was a turning point in the study of such surfaces. In recent years, Hopf-type theorems appeared in various ambient spaces, (Abresch and Rosenberg in Acta Math 193:141?C174, 2004 and Abresch and Rosenberg in Mat Contemp Sociedade Bras Mat 28:283-298, 2005). The simplest case is the study of surfaces with parallel mean curvature vector in ${M_k^n \times \mathbb{R}, n \ge 2}$ , where ${M_k^n}$ is a complete, simply-connected Riemannian manifold with constant sectional curvature k ?? 0. The case n?=?2 was solved in Abresch and Rosenberg 2004. Here we describe some new results for arbitrary n.     

Vector partition functions and generalized dahmen and micchelli spaces

This is the first of a series of papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory are in [4], while in [5] and [6] we shall investigate the cohomological formulas generated by this theory. Here we introduce a space of functions on a lattice which generalizes the space of quasipolynomials satisfying the difference equations associated to cocircuits of a sequence of vectors X, introduced by Dahmen and Micchelli [8]. This space $ \mathcal{F}(X) $ contains the partition function $ {\mathcal{P}_{(X)}} $ . We prove a “localization formula” for any f in $ \mathcal{F}(X) $ , inspired by Paradan's decomposition formula [12]. In particular, this implies a simple proof that the partition function $ {\mathcal{P}_{(X)}} $ is a quasi-polynomial on the Minkowski differences $ \mathfrak{c} - B(X) $ , where c is a big cell and B(X) is the zonotope generated by the vectors in X, a result due essentially to Dahmen and Micchelli.     

3-Nets realizing a group in a projective plane

In a projective plane $\mathit{PG}(2,\mathbb{K})$ defined over an algebraically closed field $\mathbb{K}$ of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky (Compos. Math. 140:1614–1624, 2004), arises from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky (Adv. Math. 219:672–688, 2008), comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family. Urzúa’s 3-nets (Adv. Geom. 10:287–310, 2010) realizing the quaternion group of order 8 are the unique sporadic examples. If p is larger than the order of the group, the above classification holds in characteristic p>0 apart from three possible exceptions $\rm{Alt}_{4}$ , $\rm{Sym}_{4}$ , and $\rm{Alt}_{5}$ . Motivation for the study of finite 3-nets in the complex plane comes from the study of complex line arrangements and from resonance theory; see (Falk and Yuzvinsky in Compos. Math. 143:1069–1088, 2007; Miguel and Buzunáriz in Graphs Comb. 25:469–488, 2009; Pereira and Yuzvinsky in Adv. Math. 219:672–688, 2008; Yuzvinsky in Compos. Math. 140:1614–1624, 2004; Yuzvinsky in Proc. Am. Math. Soc. 137:1641–1648, 2009).     

Basic Hopf Algebras of Tame Type

In continuation of the articles (Liu J Algebra 299:841–853, 2006; Huang, J Algebra 321:2650–2669, 2009) we classify all finite-dimensional basic Hopf algebras of tame type over an algebraically closed field of characteristic 0 in this paper. As consequences, we show the following statements: (1) the representation dimension of a tame basic Hopf algebra is exactly 3, (2) for a basic Hopf algebra H, if $\textrm{C}(H)\geq 3$ then it is wild. These conclusions verify a folklore conjecture and one of Rickard’s statements for the class of finite-dimensional basic Hopf algebras.