Characterization of Lie higher derivations on triangular algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be unital rings, and $\mathcal{M}$ be an $\left( {\mathcal{A},\mathcal{B}} \right)$ -bimodule, which is faithful as a left $\mathcal{A}$ -module and also as a right $\mathcal{B}$ -module. Let $\mathcal{U} = Tri\left( {\mathcal{A},\mathcal{M},\mathcal{B}} \right)$ be the triangular algebra. In this paper, we give some different characterizations of Lie higher derivations on $\mathcal{U}$ .     

Some results on metric n-Lie algebras

We study the structure of a metric n-Lie algebra G over the complex field C. Let G = SR be the Levi decomposition, where R is the radical of G and S is a strong semisimple subalgebra of G. Denote by m(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R ⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R ⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to m(G) + 1. The centralizer of R in G is equal to the sum of all minimal ideals; it is the direct sum of R ⊥ and the center of G. Finally, G has no strong semisimple ideals if and only if R⊥■R.     

A commutant realization of Odake’s algebra

The bcβγ-system $ \mathcal{W} $ of rank 3 has an action of the affine vertex algebra $ {V_0}\left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ , and the commutant vertex algebra $ \mathcal{C}=\mathrm{Com}\left( {{V_0}\left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right),\mathcal{W}} \right) $ contains copies of V _?3/2 $ \left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ and Odake’s algebra $ \mathcal{O} $ . Odake’s algebra is an extension of the N = 2 super-conformal algebra with c = 9, and is generated by eight fields which close nonlinearly under operator product expansions. Our main result is that V _?3/2 $ \left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ and $ \mathcal{O} $ form a Howe pair (i.e., a pair of mutual commutants) inside $ \mathcal{C} $ . More generally, any finite-dimensional representation of a Lie algebra $ \mathfrak{g} $ gives rise to a similar Howe pair, and this example corresponds to the adjoint representation of $ \mathfrak{s}{{\mathfrak{l}}_2} $ .     

Height of a Superposition

We observe that, given a poset ${\left( {E,{\user1{\mathcal{R}}}} \right)}$ and a finite covering ${\user1{\mathcal{R}}} = {\user1{\mathcal{R}}}_{1} \cup \cdots \cup {\user1{\mathcal{R}}}_{n} $ of its ordering, the height of the poset does not exceed the natural product of the heights of the corresponding sub-relations: $$\mathfrak{h}{\left( {E,{\user1{\mathcal{R}}}} \right)} \leqslant \mathfrak{h}{\left( {E,{\user1{\mathcal{R}}}_{1} } \right)} \otimes \cdots \otimes \mathfrak{h}{\left( {E,{\user1{\mathcal{R}}}_{n} } \right)}.$$ Conversely for every finite sequence $(\xi_1,\cdots,\xi_n)$ of ordinals, every poset ${\left( {E,{\user1{\mathcal{R}}}} \right)}$ of height at most $\xi_1\otimes\cdots\otimes\xi_n$ admits a partition ${\left( {{\user1{\mathcal{R}}}_{1} , \cdots ,{\user1{\mathcal{R}}}_{n} } \right)}$ of its ordering ${\user1{\mathcal{R}}}$ such that each ${\left( {E,{\user1{\mathcal{R}}}_{k} } \right)}$ has height at most $\xi_k$ . In particular for every finite sequence $(\xi_1,\cdots,\xi_n)$ of ordinals, the ordinal $$\xi _{1} \underline{ \otimes } \cdots \underline{ \otimes } \xi _{n} : = \sup {\left\{ {{\left( {\xi ^{\prime }_{1} \otimes \cdots \otimes \xi ^{\prime }_{n} } \right)} + 1:\xi ^{\prime }_{1} < \xi _{1} , \cdots ,\xi ^{\prime }_{n} < \xi _{n} } \right\}}$$ is the least $\xi$ for which the following partition relation holds $$\mathfrak{H}_{\xi } \to {\left( {\mathfrak{H}_{{\xi _{1} }} , \cdots ,\mathfrak{H}_{{\xi _{n} }} } \right)}^{2} $$ meaning: for every poset ${\left( {A,{\user1{\mathcal{R}}}} \right)}$ of height at least $\xi$ and every finite covering ${\left( {{\user1{\mathcal{R}}}_{1} , \cdots ,{\user1{\mathcal{R}}}_{n} } \right)}$ of its ordering ${\user1{\mathcal{R}}}$ , there is a $k$ for which the relation ${\left( {A,{\user1{\mathcal{R}}}_{k} } \right)}$ has height at least $\xi_k$ . The proof will rely on analogue properties of vertex coverings w.r.t. the natural sum.     

{\boldsymbol(}\boldsymbol{\mathcal{Z}}_{\bf 1}, \boldsymbol{\mathcal{Z}}_{\bf 2}{\boldsymbol)}-Complete Partially Ordered Sets and Their Representations by \boldsymbol{\mathcal{Q}}-Spaces

The present paper proposes a general theory for $\left( \mathcal{Z}_{1}, \mathcal{Z}_{2}\right) $ -complete partially ordered sets (alias $\mathcal{Z} _{1}$ -join complete and $\mathcal{Z}_{2}$ -meet complete partially ordered sets) and their Stone-like representations. It is shown that for suitably chosen subset selections $\mathcal{Z}_{i}$ (i?=?1,...,4) and $\mathcal{Q} =\left( \mathcal{Z}_{1},\mathcal{Z}_{2},\mathcal{Z}_{3},\mathcal{Z} _{4}\right) $ , the category $\mathcal{Q}$ P of $\left( \mathcal{Z}_{1},\mathcal{Z}_{2}\right) $ -complete partially ordered sets and $\left( \mathcal{Z}_{3},\mathcal{Z}_{4}\right) $ -continuous (alias $\mathcal{ Z}_{3}$ -join preserving and $\mathcal{Z}_{4}$ -meet preserving) functions forms a useful categorical framework for various order-theoretical constructs, and has a close connection with the category $\mathcal{Q}$ S of $\mathcal{Q}$ -spaces which are generalizations of topological spaces involving subset selections. In particular, this connection turns into a dual equivalence between the full subcategory $ \mathcal{Q}$ P _s of $\mathcal{Q}$ P of all $\mathcal{Q}$ -spatial objects and the full subcategory $\mathcal{Q}$ S _s of $\mathcal{Q}$ S of all $\mathcal{Q}$ -sober objects. Here $\mathcal{Q}$ -spatiality and $\mathcal{Q}$ -sobriety extend usual notions of spatiality of locales and sobriety of topological spaces to the present approach, and their relations to $\mathcal{Z}$ -compact generation and $\mathcal{Z}$ -sobriety have also been pointed out in this paper.     

On the Hall algebra of semigroup representations over \mathbb F _1

Let $\mathrm{A }$ be a finitely generated semigroup with 0. An $\mathrm{A }$ -module over $\mathbb F _1$ (also called an $\mathrm{A }$ -set), is a pointed set $(M,*)$ together with an action of $\mathrm{A }$ . We define and study the Hall algebra $\mathbb H _{\mathrm{A }}$ of the category $\mathcal C _{\mathrm{A }}$ of finite $\mathrm{A }$ -modules. $\mathbb H _{\mathrm{A }}$ is shown to be the universal enveloping algebra of a Lie algebra $\mathfrak n _{\mathrm{A }}$ , called the Hall Lie algebra of $\mathcal C _{\mathrm{A }}$ . In the case of $\langle t \rangle $ —the free monoid on one generator $\langle t \rangle $ , the Hall algebra (or more precisely the Hall algebra of the subcategory of nilpotent $\langle t \rangle $ -modules) is isomorphic to Kreimer’s Hopf algebra of rooted forests. This perspective allows us to define two new commutative operations on rooted forests. We also consider the examples when $\mathrm{A }$ is a quotient of $\langle t \rangle $ by a congruence, and the monoid $G \cup \{ 0\}$ for a finite group $G$ .     

Linear recurring sequences and subfield subcodes of cyclic codes

Linear recurring sequences over finite fields play an important role in coding theory and cryptography. It is known that subfield subcodes of linear codes yield some good codes. In this paper, we study linear recurring sequences and subfield subcodes. Let Mqm(f(x)) denote the set of all linear recurring sequences over Fqm with characteristic polynomial f(x) over Fqm . Denote the restriction of Mqm(f(x)) to sequences over Fq and the set after applying trace function to each sequence in Mqm(f(x)) by Mqm(f(x)) | Fq and Tr( Mqm(f(x))), respectively. It is shown that these two sets are both complete sets of linear recurring sequences over Fq with some characteristic polynomials over Fq. In this paper, we firstly determine the characteristic polynomials for these two sets. Then, using these results, we determine the generator polynomials of subfield subcodes and trace codes of cyclic codes over Fqm .     

Irregular and Singular Loci of Commuting Varieties

Let $ {\user1{\mathcal{C}}} $ be the commuting variety of the Lie algebra $ \mathfrak{g} $ of a connected noncommutative reductive algebraic group G over an algebraically closed field of characteristic zero. Let $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ be the singular locus of $ {\user1{\mathcal{C}}} $ and let $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ be the locus of points whose G-stabilizers have dimension > rk G. We prove that: (a) $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ is a nonempty subset of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ ; (b) $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{irr}}}} = 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ where the maximum is taken over all simple ideals $ \mathfrak{a} $ of $ \mathfrak{g} $ and $ l{\left( \mathfrak{a} \right)} $ is the “lacety” of $ \mathfrak{a} $ ; and (c) if $ \mathfrak{t} $ is a Cartan subalgebra of $ \mathfrak{g} $ and $ \alpha \in \mathfrak{t}^{*} $ root of $ \mathfrak{g} $ with respect to $ \mathfrak{t} $ , then $ \overline{{G{\left( {{\text{Ker}}\,\alpha \times {\text{Ker }}\alpha } \right)}}} $ is an irreducible component of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ of codimension 4 in $ {\user1{\mathcal{C}}} $ . This yields the bound $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ and, in particular, $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 2 $ . The latter may be regarded as an evidence in favor of the known longstanding conjecture that $ {\user1{\mathcal{C}}} $ is always normal. We also prove that the algebraic variety $ {\user1{\mathcal{C}}} $ is rational.     

Analytic Toeplitz algebras and the Hilbert transform associated with a subdiagonal algebra

Let M be aσ-finite von Neumann algebra and let AM be a maximal subdiagonal algebra with respect to a faithful normal conditional expectationΦ.Based on the Haagerup’s noncommutative Lpspace Lp(M)associated with M,we consider Toeplitz operators and the Hilbert transform associated with A.We prove that the commutant of left analytic Toeplitz algebra on noncommutative Hardy space H2(M)is just the right analytic Toeplitz algebra.Furthermore,the Hilbert transform on noncommutative Lp(M)is shown to be bounded for 1p∞.As an application,we consider a noncommutative analog of the space BMO and identify the dual space of noncommutative H1(M)as a concrete space of operators.     

Some new classes of Kadison-Singer lattices in Hilbert spaces

Let N be a maximal and discrete nest on a separable Hilbert space H,E the projection from H onto the subspace[C]spanned by a particular separating vector for N′and Q the projection from K=H⊕H onto the closed subspace{(,):∈H}.Let L be the closed lattice in the strong operator topology generated by the projections(E 00 0),{(E 00 0):E∈N}and Q.We show that L is a Kadison-Singer lattice with trivial commutant,i.e.,L′=CI.Furthermore,we similarly construct some Kadison-Singer lattices in the matrix algebras M2n(C)and M2n.1(C).     

Covering functions by countably many functions from some families

Let $ \mathcal{A} $ be a nonempty family of functions from $ \mathbb{R} $ to $ \mathbb{R} $ . A function $ f:\mathbb{R}\to \mathbb{R} $ is said to be strongly countably $ \mathcal{A} $ -function if there is a sequence (f _n ) of functions from $ \mathcal{A} $ such that $ \mathrm{Gr}(f)\subset {\cup_n}\mathrm{Gr}\left( {{f_n}} \right) $ (Gr(f) denotes the graph of f). If $ \mathcal{A} $ is the family of all continuous functions, the strongly countable $ \mathcal{A} $ -functions are called strongly countably continuous and were investigated in [Z. Grande and A. Fatz-Grupka, On countably continuous functions, Tatra Mt. Math. Publ., 28:57–63, 2004], [G. Horbaczewska, On strongly countably continuous functions, Tatra Mt. Math. Publ., 42:81–86, 2009], and [T.A. Natkaniec, On additive countably continuous functions, Publ. Math., 79(1–2):1–6, 2011]. In this article, we prove that the families $ \mathcal{A}\left( \mathbb{R} \right) $ of all strongly countably $ \mathcal{A} $ -functions are closed with respect to some operations in dependence of analogous properties of the families $ \mathcal{A} $ , and, in particular, we show some properties of strongly countably differentiable functions, strongly countably approximately continuous functions, and strongly countably quasi-continuous functions.     

A zoo of diffeomorphism groups on \mathbb{R }^{n}

We consider the groups ${\mathrm{Diff }}_\mathcal{B }(\mathbb{R }^n)$ , ${\mathrm{Diff }}_{H^\infty }(\mathbb{R }^n)$ , and ${\mathrm{Diff }}_{\mathcal{S }}(\mathbb{R }^n)$ of smooth diffeomorphisms on $\mathbb{R }^n$ which differ from the identity by a function which is in either $\mathcal{B }$ (bounded in all derivatives), $H^\infty = \bigcap _{k\ge 0}H^k$ , or $\mathcal{S }$ (rapidly decreasing). We show that all these groups are smooth regular Lie groups.     

Categories of bounded \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) - and \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) -modules

Let $ \mathfrak{g} $ be a reductive Lie algebra over $ \mathbb{C} $ and $ \mathfrak{k} \subset \mathfrak{g} $ be a reductive in $ \mathfrak{g} $ subalgebra. We call a $ \mathfrak{g} $ -module M a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module whenever M is a direct sum of finite-dimensional $ \mathfrak{k} $ -modules. We call a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module M bounded if there exists $ {C_M} \in {\mathbb{Z}_{{ \geqslant 0}}} $ such that for any simple finite-dimensional $ \mathfrak{k} $ -module E the dimension of the E-isotypic component is not greater than C _M dim E. Bounded $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -modules form a subcategory of the category of $ \mathfrak{g} $ -modules. Let V be a finite-dimensional vector space. We prove that the categories of bounded $ \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ - and $ \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ -modules are isomorphic to the direct sum of countably many copies of the category of representations of some explicitly described quiver with relations under some mild assumptions on the dimension of V .     

Hypercentralizing generalized skew derivations on left ideals in prime rings

Let $\mathcal{R }$ be a prime ring of characteristic different from $2, \mathcal{Q }_r$ the right Martindale quotient ring of $\mathcal{R }, \mathcal{C }$ the extended centroid of $\mathcal{R }, \mathcal{I }$ a nonzero left ideal of $\mathcal{R }, F$ a nonzero generalized skew derivation of $\mathcal{R }$ with associated automorphism $\alpha $ , and $n,k \ge 1$ be fixed integers. If $[F(r^n),r^n]_k=0$ for all $r \in \mathcal{I }$ , then there exists $\lambda \in \mathcal{C }$ such that $F(x)=\lambda x$ , for all $x\in \mathcal{I }$ . More precisely one of the following holds: (1) $\alpha $ is an $X$ -inner automorphism of $\mathcal{R }$ and there exist $b,c \in \mathcal{Q }_r$ and $q$ invertible element of $\mathcal{Q }_r$ , such that $F(x)=bx-qxq^{-1}c$ , for all $x\in \mathcal{Q }_r$ . Moreover there exists $\gamma \in \mathcal{C }$ such that $\mathcal{I }(q^{-1}c-\gamma )=(0)$ and $b-\gamma q \in \mathcal{C }$ ; (2) $\alpha $ is an $X$ -outer automorphism of $\mathcal{R }$ and there exist $c \in \mathcal{Q }_r, \lambda \in \mathcal{C }$ , such that $F(x)=\lambda x-\alpha (x)c$ , for all $x\in \mathcal{Q }_r$ , with $\alpha (\mathcal{I })c=0$ .     

On a property of Abelian groups related to direct sums and products

Let T be an infinite set of prime numbers, $ \mathcal{M} $ be a set of groups $ \left\{ {\left. {\mathbb{Z}(p)} \right|p \in T} \right\} $ . An Abelian group A is said to be $ \mathcal{M} $ -large if $$ {\text{Hom}}\left( {A,\;\mathop { \bigoplus }\limits_{p \in T} \mathbb{Z}(p)} \right) = {\text{Hom}}\left( {A,\;\prod\limits_{p \in T} {\mathbb{Z}(p)} } \right). $$ This paper presents a characterization of $ \mathcal{M} $ -large torsion-free and mixed groups.     

Köthe’s upper nil radical for modules

Let M be a left R-module. In this paper a generalization of the notion of an s-system of rings to modules is given. Let N be a submodule of M. Define $\mathcal{S}(N):=\{ {m\in M}:\, \mbox{every } s\mbox{-system containing } m \mbox{ meets}~N \}$ . It is shown that $\mathcal{S}(N)$ is equal to the intersection of all s-prime submodules of M containing N. We define $\mathcal{N}({}_{R}M) = \mathcal{S}(0)$ . This is called (Köthe’s) upper nil radical of M. We show that if R is a commutative ring, then $\mathcal{N}({}_{R}M) = {\mathop{\mathrm{rad}}\nolimits}_{R}(M)$ where ${\mathop{\mathrm{rad}}\nolimits}_{R}(M)$ denotes the prime radical of M. We also show that if R is a left Artinian ring, then ${\mathop{\mathrm{rad}}\nolimits}_{R}(M)=\mathcal{N}({}_{R}M)= {\mathop{\mathrm{Rad}}\nolimits}\, (M)= {\mathop{\mathrm{Jac}}\nolimits}\, (R)M$ where ${\mathop{\mathrm{Rad}}\nolimits}\, (M)$ denotes the Jacobson radical of M and ${\mathop{\mathrm{Jac}}\nolimits}\, (R)$ the Jacobson radical of the ring R. Furthermore, we show that the class of all s-prime modules forms a special class of modules.     

Tilting Theory and Functor Categories II. Generalized Tilting

In this paper we continue the project of generalizing tilting theory to the category of contravariant functors $\mathrm{Mod}(\mathcal{C})$ , from a skeletally small preadditive category $\mathcal{C}$ to the category of abelian groups, initiated in [15]. We introduce the notion of a generalized tilting category $\mathcal{T}$ , and we concentrate here on extending Happel’s theorem to $\mathrm{Mod}(\mathcal{C})$ ; more specifically, we prove that there is an equivalence of triangulated categories $\mathcal{D}^{b}( \mathrm{Mod}(\mathcal{C}))\cong \mathcal{D}^{b}(\mathrm{Mod}(\mathcal{T}))$ . We then add some restrictions on our category $\mathcal{C}$ , in order to obtain a version of Happel’s theorem for the categories of finitely presented functors. We end the paper proving that some of the theorems for artin algebras, relating tilting with contravariantly finite categories proved in Auslander and Reiten (Adv Math 12(3):306–366, 1974; Adv Math 86(1):111–151, 1991), can be extended to the category of finitely presented functors $\mathrm{mod}(\mathcal{C})$ , with $\mathcal{C}$ a dualizing variety.     

On additive decompositions of the set of primitive roots modulo p

It is conjectured that the set ${\mathcal {G}}$ of the primitive roots modulo p has no decomposition (modulo p) of the form ${\mathcal {G}= \mathcal {A} +\mathcal {B}}$ with ${|\mathcal {A}|\ge 2}$ , ${|\mathcal {B} |\ge 2}$ . This conjecture seems to be beyond reach but it is shown that if such a decomposition of ${\mathcal {G}}$ exists at all, then ${|\mathcal {A} |}$ , ${|\mathcal {B} |}$ must be around p ^1/2, and then this result is applied to show that ${\mathcal {G}}$ has no decomposition of the form ${\mathcal {G} =\mathcal {A} + \mathcal {B} + \mathcal {C}}$ with ${|\mathcal {A} |\ge 2}$ , ${|\mathcal {B} |\ge 2}$ , ${|\mathcal {C} |\ge 2}$ .     

Lower bounds on Ricci curvature and quantitative behavior of singular sets

Let Y ⁿ denote the Gromov-Hausdorff limit $M^{n}_{i}\stackrel{d_{\mathrm{GH}}}{\longrightarrow} Y^{n}$ of v-noncollapsed Riemannian manifolds with ${\mathrm{Ric}}_{M^{n}_{i}}\geq-(n-1)$ . The singular set $\mathcal {S}\subset Y$ has a stratification $\mathcal {S}^{0}\subset \mathcal {S}^{1}\subset\cdots\subset \mathcal {S}$ , where $y\in \mathcal {S}^{k}$ if no tangent cone at y splits off a factor ? ^k+1 isometrically. Here, we define for all η>0, 0<r≤1, the k-th effective singular stratum $\mathcal {S}^{k}_{\eta,r}$ satisfying $\bigcup_{\eta}\bigcap_{r} \,\mathcal {S}^{k}_{\eta,r}= \mathcal {S}^{k}$ . Sharpening the known Hausdorff dimension bound $\dim\, \mathcal {S}^{k}\leq k$ , we prove that for all y, the volume of the r-tubular neighborhood of $\mathcal {S}^{k}_{\eta,r}$ satisfies ${\mathrm {Vol}}(T_{r}(\mathcal {S}^{k}_{\eta,r})\cap B_{\frac{1}{2}}(y))\leq c(n,{\mathrm {v}},\eta)r^{n-k-\eta}$ . The proof involves a quantitative differentiation argument. This result has applications to Einstein manifolds. Let $\mathcal {B}_{r}$ denote the set of points at which the C ²-harmonic radius is ≤r. If also the $M^{n}_{i}$ are Kähler-Einstein with L ₂ curvature bound, $\| Rm\|_{L_{2}}\leq C$ , then ${\mathrm {Vol}}( \mathcal {B}_{r}\cap B_{\frac{1}{2}}(y))\leq c(n,{\mathrm {v}},C)r^{4}$ for all y. In the Kähler-Einstein case, without assuming any integral curvature bound on the $M^{n}_{i}$ , we obtain a slightly weaker volume bound on $\mathcal {B}_{r}$ which yields an a priori L _p curvature bound for all p<2. The methodology developed in this paper is new and is applicable in many other contexts. These include harmonic maps, minimal hypersurfaces, mean curvature flow and critical sets of solutions to elliptic equations.     

Restrictions of generalized Verma modules to symmetric pairs

We initiate a new line of investigation on branching problems for generalized Verma modules with respect to reductive symmetric pairs $ \left( {\mathfrak{g},\mathfrak{g}'} \right) $ . In general, Verma modules may not contain any simple module when restricted to a reductive subalgebra. In this article we give a necessary and sufficient condition on the triple $ \left( {\mathfrak{g},\mathfrak{g}',\mathfrak{p}} \right) $ such that the restriction $ {\left. X \right|_{\mathfrak{g}'}} $ always contains simple $ \mathfrak{g}' $ -modules for any $ \mathfrak{g} $ -module X lying in the parabolic BGG category $ {\mathcal{O}^\mathfrak{p}} $ attached to a parabolic subalgebra $ \mathfrak{p} $ of $ \mathfrak{g} $ . Formulas are derived for the Gelfand?CKirillov dimension of any simple module occurring in a simple generalized Verma module. We then prove that the restriction $ {\left. X \right|_{\mathfrak{g}'}} $ is generically multiplicity-free for any $ \mathfrak{p} $ and any $ X \in {\mathcal{O}^\mathfrak{p}} $ if and only if $ \left( {\mathfrak{g},\mathfrak{g}'} \right) $ is isomorphic to (A _n , A _n-1), (B _n , D _n ), or (D _n+1, B _n ). Explicit branching laws are also presented.