Superization of homogeneous spin manifolds and geometry of homogeneous supermanifolds

Let M ₀=G ₀/H be a (pseudo)-Riemannian homogeneous spin manifold, with reductive decomposition \(\mathfrak {g}_{0}=\mathfrak {h}+\mathfrak {m}\) and let S(M ₀) be the spin bundle defined by the spin representation \(\tilde{ \operatorname {Ad}}:H\rightarrow \mathrm {GL}_{\mathbb {R}}(S)\) of the stabilizer H. This article studies the superizations of M ₀, i.e. its extensions to a homogeneous supermanifold M=G/H whose sheaf of superfunctions is isomorphic to the sheaf of sections of Λ(S ^*(M ₀)). Here G is the Lie supergroup associated with a certain extension of the Lie algebra of symmetry \(\mathfrak {g}_{0}\) to an algebra of supersymmetry \(\mathfrak {g}=\mathfrak {g}_{\overline {0}}+\mathfrak {g}_{\overline {1}}=\mathfrak {g}_{0}+S\) via the Kostant-Koszul construction. Each algebra of supersymmetry naturally determines a flat connection \(\nabla^{\mathcal {S}}\) in the spin bundle S(M ₀). Killing vectors together with generalized Killing spinors (i.e. \(\nabla^{\mathcal {S}}\)-parallel spinors) are interpreted as the values of appropriate geometric symmetries of M, namely even and odd Killing fields. An explicit formula for the Killing representation of the algebra of supersymmetry is obtained, generalizing some results of Koszul. The generalized spin connection \(\nabla^{\mathcal {S}}\) defines a superconnection on M, via the super-version of a theorem of Wang.     

Weak module amenability of triangular Banach algebras

Let A and B be unital Banach algebras and let M be a unital Banach A,B-module. Forrest and Marcoux [6] have studied the weak amenability of triangular Banach algebra \(\mathcal{T} = \left[ {_B^{AM} } \right]\) and showed that T is weakly amenable if and only if the corner algebras A and B are weakly amenable. When \(\mathfrak{A}\) is a Banach algebra and A and B are Banach \(\mathfrak{A}\)-module with compatible actions, and M is a commutative left Banach \(\mathfrak{A}\)-A-module and right Banach \(\mathfrak{A}\)-B-module, we show that A and B are weakly \(\mathfrak{A}\)-module amenable if and only if triangular Banach algebra T is weakly \(\mathfrak{T}\)-module amenable, where \(\mathfrak{T}: = \{ [^\alpha _\alpha ]:\alpha \in \mathfrak{A}\} \).     

BOTT–BOREL–WEIL THEORY AND BERNSTEIN–GEL’FAND–GEL’FAND RECIPROCITY FOR LIE SUPERALGEBRAS

The main focus of this paper is Bott–Borel–Weil (BBW) theory for basic classical Lie superalgebras. We take a purely algebraic self-contained approach to the problem. A new element in this study is twisting functors, which we use in particular to prove that the top of the cohomology groups of BBW theory for generic weights is described by the recently introduced star action. We also study the algebra of regular functions, related to BBW theory. Then we introduce a weaker form of genericness, relative to the Borel subalgebra and show that the virtual BGG reciprocity of Gruson and Serganova becomes an actual reciprocity in the relatively generic region. We also obtain a complete solution of BBW theory for \( \mathfrak{o}\mathfrak{s}\mathfrak{p} \)(m|2), D(2, 1; α), F(4) and G(3) with distinguished Borel subalgebra. Furthermore, we derive information about the category of finite-dimensional \( \mathfrak{o}\mathfrak{s}\mathfrak{p} \)(m|2)-modules, such as BGG-type resolutions and Kostant homology of Kac modules and the structure of projective modules.     

Lie Algebras of Slow Growth and Klein-Gordon PDE

We discuss the notion of characteristic Lie algebra of a hyperbolic PDE. The integrability of a hyperbolic PDE is closely related to the properties of the corresponding characteristic Lie algebra χ. We establish two explicit isomorphisms:

1. 1)
   the first one is between the characteristic Lie algebra \(\chi (\sinh {u})\) of the sinh-Gordon equation \(u_{xy}=\sinh {u}\) and the non-negative part \({\mathcal {L}}({\mathfrak {sl}}(2,{\mathbb {C}}))^{\ge 0}\) of the loop algebra of \({\mathfrak {sl}}(2,{\mathbb {C}})\) that corresponds to the Kac-Moody algebra \(A_{1}^{(1)}\)

   $$\chi(\sinh{u})\cong {\mathcal{L}}({\mathfrak{s}\mathfrak{l}}(2,{\mathbb{C}}))^{\ge 0}={\mathfrak{s}\mathfrak{l}}(2, {\mathbb{C}}) \otimes {\mathbb{C}}[t]. $$
    
2. 2)
   the second isomorphism is for the Tzitzeica equation u_xy = e^u + e^??2u

   $$\chi(e^{u}{+}e^{-2u}) \cong {\mathcal{L}}({\mathfrak{s}\mathfrak{l}}(3,{\mathbb{C}}), \mu)^{\ge0}=\bigoplus_{j = 0}^{+\infty}{\mathfrak{g}}_{j (\text{mod} \; 2)} \otimes t^{j}, $$

   where \({\mathcal {L}}({\mathfrak {sl}}(3,{\mathbb {C}}), \mu )=\bigoplus _{j \in {\mathbb {Z}}}{\mathfrak {g}}_{j (\text {mod} \; 2)} \otimes t^{j}\) is the twisted loop algebra of the simple Lie algebra \({\mathfrak {sl}}(3,{\mathbb {C}})\) that corresponds to the Kac-Moody algebra \(A_{2}^{(2)}\).
    

Hence the Lie algebras \(\chi (\sinh {u})\) and χ(e^u + e^??2u) are slowly linearly growing Lie algebras with average growth rates \(\frac {3}{2}\) and \(\frac {4}{3}\) respectively.     

Commutativity of the centralizer of a subalgebra in a universal enveloping algebra

Let G be a reductive algebraic group over an algebraically closed field of characteristic zero, and let \(\mathfrak{h}\) be an algebraic subalgebra of the tangent Lie algebra \(\mathfrak{g}\) of G. We find all subalgebras \(\mathfrak{h}\) that have no nontrivial characters and whose centralizers \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) and \(P(\mathfrak{g})^\mathfrak{h} \) in the universal enveloping algebra \(\mathfrak{U}(\mathfrak{g})\) and in the associated graded algebra \(P(\mathfrak{g})\), respectively, are commutative. For all these subalgebras, we prove that \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} = \mathfrak{U}(\mathfrak{h})^\mathfrak{h} \otimes \mathfrak{U}(\mathfrak{g})^\mathfrak{g} \) and \(P(\mathfrak{g})^\mathfrak{h} = P(\mathfrak{h})^\mathfrak{h} \otimes P(\mathfrak{g})^\mathfrak{g} \). Furthermore, we obtain a criterion for the commutativity of \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) in terms of representation theory.     

The Schur functor on tensor powers

Let M be a left module for the Schur algebra S(n, r), and let \({s \in \mathbb{Z}^+}\) . Then \({M^{\otimes s}}\) is a \({(S(n,\,rs), F{\mathfrak{S}_{s}})}\) -bimodule, where the symmetric group \({{\mathfrak{S}_s}}\) on s letters acts on the right by place permutations. We show that the Schur functor f _rs sends \({M^{\otimes s}}\) to the \({(F{\mathfrak{S}_{rs}},F{\mathfrak{S}_s})}\) -bimodule \({F\mathfrak{S}_{rs}\otimes_{F(\mathfrak{S}_{r}\wr{\mathfrak{S}_s})} ((f_rM)^{\otimes s}\otimes_{F} F{\mathfrak{S}_s})}\) . As a corollary, we obtain the image under the Schur functor of the Lie power L ^s (M), exterior power \({\bigwedge^s(M)}\) of M and symmetric power S ^s (M).     

On rational isomorphisms of Lie algebras

Let \(\mathfrak{n}\) be a finite-dimensional noncommutative nilpotent Lie algebra for which the ring of polynomial invariants of the coadjoint representation is generated by linear functions. Let \(\mathfrak{g}\) be an arbitrary Lie algebra. We consider semidirect sums \(\mathfrak{n} \dashv _\rho \mathfrak{g}\) with respect to an arbitrary representation ρ: \(\mathfrak{g}\) → der \(\mathfrak{n}\) such that the center z \(\mathfrak{n}\) of \(\mathfrak{n}\) has a ρ-invariant complement.We establish that some localization \(\tilde P(\mathfrak{n} \dashv _\rho \mathfrak{g})\) of the Poisson algebra of polynomials in elements of the Lie algebra \(\mathfrak{n} \dashv _\rho \mathfrak{g}\) is isomorphic to the tensor product of the standard Poisson algebra of a nonzero symplectic space by a localization of the Poisson algebra of the Lie subalgebra \((z\mathfrak{n}) \dashv \mathfrak{g}\). If \([\mathfrak{n},\mathfrak{n}] \subseteq z\mathfrak{n}\), then a similar tensor product decomposition is established for the localized universal enveloping algebra of the Lie algebra \(\mathfrak{n} \dashv _\rho \mathfrak{g}\). For the case in which \(\mathfrak{n}\) is a Heisenberg algebra, we obtain explicit formulas for the embeddings of \(\mathfrak{g}_P \) in \(\tilde P(\mathfrak{n} \dashv _\rho \mathfrak{g})\). These formulas have applications, some related to integrability in mechanics and others to the Gelfand-Kirillov conjecture.     

A natural map in local cohomology

Let R be a Noetherian ring, \(\mathfrak{a}\) an ideal of R, M an R-module and n a non-negative integer. In this paper we first study the finiteness properties of the kernel and the cokernel of the natural map \(f\colon\operatorname{Ext}^{n}_{R}(R/\mathfrak{a},M)\to \operatorname{Hom}_{R}(R/\mathfrak{a},\mathrm{H}^{n}_{\mathfrak{a}}(M))\), under some conditions on the previous local cohomology modules. Then we get some corollaries about the associated primes and Artinianness of local cohomology modules. Finally we will study the asymptotic behavior of the kernel and the cokernel of the natural map in the graded case.     

Integral representations and the generalized Poincaré inequality on Carnot groups

We give some integral representations of the form f(x) = P(f)+K(?f) on two-step Carnot groups, where P(f) is a polynomial and K is an integral operator with a specific singularity. We then obtain the weak Poincaré inequality and coercive estimates as well as the generalized Poincaré inequality on the general Carnot groups.     

Idempotent Ideals and the Igusa-Todorov Functions

Let Λ be an artin algebra and \(\mathfrak {A}\) a two-sided idempotent ideal of Λ, that is, \(\mathfrak {A}\) is the trace of a projective Λ-module P in Λ. We consider the categories of finitely generated modules over the associated rings \({\Lambda }/\mathfrak {A}, {\Lambda }\) and Γ = End_Λ(P) ^{o p} and study the relationship between their homological properties via the Igusa-Todorov functions.     

On Irreducible Representations of the Zassenhaus Superalgebras with p-Characters of Height 0

Let n be a positive integer, and \(\mathfrak {A}(n)=\mathbb {F}[x]/(x^{p^{n}})\), the divided power algebra over an algebraically closed field \(\mathbb {F}\) of prime characteristic p >?2. Let π(n) be the tensor product of \(\mathfrak {A}(n)\) and the Grassmann superalgebra \(\bigwedge (1)\) in one variable. The Zassenhaus superalgebra \(\mathcal {Z}(n)\) is defined to be the Lie superalgebra of the special super derivations of the superalgebra π(n). In this paper we study simple modules over the Zassenhaus superalgebra \(\mathcal {Z}(n)\) with p-characters of height 0. We give a complete classification of the isomorphism classes of such simple modules and determine their dimensions. A sufficient and necessary condition for the irreducibility of Kac modules is obtained.     

A Nonlocal <Emphasis Type="Italic">C</Emphasis>*-Algebra of Singular Integral Operators with Shifts Having Periodic Points

Representations on Hilbert spaces for a nonlocal C*-algebra \({{\mathfrak {B}}}\) of singular integral operators with piecewise slowly oscillating coefficients and unitary shift operators are constructed. The group of unitary shift operators U _g of the C*-algebra \({{\mathfrak {B}}}\) is associated with an amenable discrete group of homeomorphisms \({g:{\mathbb{T}}\to{\mathbb{T}}}\) that have piecewise continuous derivatives and the same nonempty set of periodic points. An isometric C*-algebra homomorphism of the quotient C*-algebra \({{\mathfrak {B}}^\pi={\mathfrak {B}}/{\mathcal {K}}}\), where \({{\mathcal {K}}}\) is the ideal of compact operators, into an n × n matrix algebra associated to a C*-algebra \({{\mathfrak {B}}_0}\) of singular integral operators with shifts having only fixed points is established making use of a spectral measure. Based on this generalization of the Litvinchuk–Gohberg–Krupnik reduction scheme, a symbol calculus for the C*-algebra \({{\mathfrak {B}}}\) as well as a Fredholm criterion for the operators in \({{\mathfrak {B}}}\) are obtained.     

The Harish-Chandra isomorphism for reductive symmetric superpairs

We consider symmetric pairs of Lie superalgebras and introduce a graded Harish-Chandra homomorphism. Generalising results of Harish-Chandra and V. Kac, M. Gorelik, we prove that, assuming reductivity, its image is a certain explicit filtered subalgebra J( $ \mathfrak{a} $ ) of the Weyl invariants on a Cartan subspace whose associated graded gr J( $ \mathfrak{a} $ ) is the image of Chevalley’s restriction map on symmetric invariants. In contrast to the known cases, J( $ \mathfrak{a} $ ) is in general not isomorphic to gr J( $ \mathfrak{a} $ ).     

Graded Limits of Minimal Affinizations over the Quantum Loop Algebra of Type <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

The aim of this paper is to study the graded limits of minimal affinizations over the quantum loop algebra of type G ₂. We show that the graded limits are isomorphic to multiple generalizations of Demazure modules, and obtain defining relations of them. As an application, we obtain a polyhedral multiplicity formula for the decomposition of minimal affinizations of type G ₂ as a \(U_{q}(\mathfrak {g})\)-module, by showing the corresponding formula for the graded limits. As another application, we prove a character formula of the least affinizations of generic parabolic Verma modules of type G ₂ conjectured by Mukhin and Young.     

A Note on Derivations on the Algebra of Operators in Hilbert <Emphasis Type="Italic">C</Emphasis>*-Modules

Let \({\mathfrak{M}}\) be a Hilbert C*-module on a C*-algebra \({\mathfrak{A}}\) and let \({End_\mathfrak{A}(\mathfrak{M})}\) be the algebra of all operators on \({\mathfrak{M}}\). In this paper, first the continuity of \({\mathfrak{A}}\)-module homomorphism derivations on \({End_\mathfrak{A}(\mathfrak{M})}\) is investigated. We give some sufficient conditions on which every derivation on \({End_\mathfrak{A}(\mathfrak{M})}\) is inner. Next, we study approximately innerness of derivations on \({End_\mathfrak{A}(\mathfrak{M})}\) for a σ-unital C*-algebra \({\mathfrak{A}}\) and full Hilbert \({\mathfrak{A}}\)-module \({\mathfrak{M}}\). Finally, we show that every bounded linear mapping on \({End_\mathfrak{A}(\mathfrak{M})}\) which behave like a derivation when acting on pairs of elements with unit product, is a Jordan derivation.     

Some Hecke algebra products and corresponding random walks

Let i=1+q+???+q ^i?1. For certain sequences (r ₁,…,r _l ) of positive integers, we show that in the Hecke algebra ? _n (q) of the symmetric group \(\mathfrak{S}_{n}\), the product \((1+\boldsymbol{r}_{\boldsymbol{1}}T_{r_{1}})\cdots (1+\boldsymbol{r}_{\boldsymbol{l}}T_{r_{l}})\) has a simple explicit expansion in terms of the standard basis {T _w }. An interpretation is given in terms of random walks on \(\mathfrak{S}_{n}\).     

Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture

Let H be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold X and a real algebraic bundle \({\mathcal {E}}\) on X. Let \(\mathfrak {h}\) be the Lie algebra of H. Let \(\mathcal {S}(X,{\mathcal {E}})\) be the space of Schwartz sections of \({\mathcal {E}}\). We prove that \(\mathfrak {h}\mathcal {S}(X,{\mathcal {E}})\) is a closed subspace of \(\mathcal {S}(X,{\mathcal {E}})\) of finite codimension. We give an application of this result in the case when H is a real spherical subgroup of a real reductive group G. We deduce an equivalence of two old conjectures due to Casselman: the automatic continuity and the comparison conjecture for zero homology. Namely, let \(\pi \) be a Casselman–Wallach representation of G and V be the corresponding Harish–Chandra module. Then the natural morphism of coinvariants \(V_{\mathfrak {h}}\rightarrow \pi _{\mathfrak {h}}\) is an isomorphism if and only if any linear \(\mathfrak {h}\)-invariant functional on V is continuous in the topology induced from \(\pi \). The latter statement is known to hold in two important special cases: if H includes a symmetric subgroup, and if H includes the nilradical of a minimal parabolic subgroup of G.