(H,R)-Lie coalgebras and (H,R)-Lie bialgebras

Given an (H,R)-Lie coalgebra Γ, we construct (H,R _T )-Lie coalgebra Γ^T through a right cocycle T, where (H,R) is a triangular Hopf algebra, and prove that there exists a bijection between the set of (H,R)-Lie coalgebras and the set of ordinary Lie coalgebras. We also show that if (L, [, ], Δ, R) is an (H,R)-Lie bialgebra of an ordinary Lie algebra then (L ^T , [, ], Δ_T, R _T ) is an (H,R _T )-Lie bialgebra of an ordinary Lie algebra.     

Representations and cohomology for Frobenius-Lusztig kernels

Let U_ζ be the quantum group (Lusztig form) associated to the simple Lie algebra g, with parameter ζ specialized to an ?-th root of unity in a field of characteristic p>0. In this paper we study certain finite-dimensional normal Hopf subalgebras U_ζ(G_r) of U_ζ, called Frobenius-Lusztig kernels, which generalize the Frobenius kernels G_r of an algebraic group G. When r=0, the algebras studied here reduce to the small quantum group introduced by Lusztig. We classify the irreducible U_ζ(G_r)-modules and discuss their characters. We then study the cohomology rings for the Frobenius-Lusztig kernels and for certain nilpotent and Borel subalgebras corresponding to unipotent and Borel subgroups of G. We prove that the cohomology ring for the first Frobenius-Lusztig kernel is finitely-generated when g has type A or D, and that the cohomology rings for the nilpotent and Borel subalgebras are finitely-generated in general.     

Lie superalgebras, Clifford algebras, induced modules and nilpotent orbits

Let g be a classical simple Lie superalgebra. To every nilpotent orbit O in g₀ we associate a Clifford algebra over the field of rational functions on O. We find the rank, k(O) of the bilinear form defining this Clifford algebra, and deduce a lower bound on the multiplicity of a U(g)-module with O or an orbital subvariety of O as associated variety. In some cases we obtain modules where the lower bound on multiplicity is attained using parabolic induction. The invariant k(O) is in many cases, equal to the odd dimension of the orbit G⋅O, where G is a Lie supergroup with Lie superalgebra g.     

Factoriality for the reductive Zassenhaus variety and quantum enveloping algebra

Let U(g)$U(\mathfrak{g})$ be the enveloping algebra of a finite dimensional reductive Lie algebra g$\mathfrak{g}$ over an algebraically closed field of prime characteristic. Let _U?,P(s:)$U_{?,P}(\mathfrak{s}:)$ be the simply connected quantum enveloping algebra at the root of unity ?  , of a complex semi-simple finite dimensional Lie algebra s:$\mathfrak{s}:$. We show, by similar proofs, that the centers of both are factorial. While the first result was established by R. Tange [32] (by different methods), the second one confirms a conjecture in [4]. We also provide a general criterion for the factoriality of the centers of enveloping algebras in prime characteristic.     

Renormalization as a functor on bialgebras

The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T(T(B)⁺), the double tensor algebra of B, with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalization turns S(S(B)⁺), the double symmetric algebra of B, into a commutative bialgebra. The usual Hopf algebra of renormalization is recovered when the elements of S¹(B) are not renormalized, i.e., when Feynman diagrams containing one single vertex are not renormalized. When B is the Hopf algebra of a commutative group, a homomorphism is established between the bialgebra S(S(B)⁺) and the Faà di Bruno bialgebra of composition of series. The relation with the Connes-Moscovici Hopf algebra is given. Finally, the bialgebra S(S(B)⁺) is shown to give the same results as the standard renormalization procedure for the scalar field.     

The Zassenhaus variety of a reductive Lie algebra in positive characteristic

Let g be the Lie algebra of a connected reductive group G over an algebraically closed field k of characteristic p>0. Let Z be the centre of the universal enveloping algebra U=U(g) of g. Its maximal spectrum is called the Zassenhaus variety of g. We show that, under certain mild assumptions on G, the field of fractions Frac(Z) of Z is G-equivariantly isomorphic to the function field of the dual space g^∗ with twisted G-action. In particular Frac(Z) is rational. This confirms a conjecture of J. Alev. Furthermore we show that Z is a unique factorisation domain, confirming a conjecture of A. Braun and C. Hajarnavis. Recently, A. Premet used the above result about Frac(Z), a result of Colliot-Thelene, Kunyavskii, Popov and Reichstein and reduction mod p arguments to show that the Gelfand-Kirillov conjecture cannot hold for simple complex Lie algebras that are not of type A, C or G₂.     

The unitary group over the integers of a quaternion algebra

Let L be a lattice over the integers of a quaternion algebra with center K which is a $\text{B}$-adic field. Then the unitary group U(L) equals its own commutator subgroup $\text{Ω(L)}$ and is generated by the unitary transvections and quasitransvections contained in it. Let g be a tableau, U(g), U⁺(g), $\text{Ω(g)}$, T(g) be the corresponding congruence subgroups of order g. Then $\text{U(g)}\text{U}{}^{\mathrm{+}}\text{(g)}\text{? X}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{\tau }}\text{Z}{}_2$, and $\text{Ω(g) = T(g)}$ (the subgroup generated by the unitary transvections and quasitransvections with order ≤ g). Let G be a subgroup of U(L) with o(G) = g, then G is normal in U(L) if and only if U(g) ? G ? T(g).     

Nonlinear equations involving m-accretive operators

Let X be a Banach space and T an m-accretive operator defined on a subset D(T) of X and taking values in 2^x. For the class of spaces whose bounded closed and convex subsets have the fixed point property for nonexpansive self-mappings, it is shown here that two boundary conditions which imply existence of zeroes for T, appear to be equivalent. This fact is then used to prove that if there exists x₀?D(T) and a bounded open neighborhood U of x₀, such that $\text{¦T(x}{}_0\text{)¦ < r ? ¦T(x)¦}$ for all x??U ∩ D(T), then the open ball B(0; r) is contained in the range of T.     

A construction for coisotropic subalgebras of Lie bialgebras

Given a Lie bialgebra (g,g^∗), we present an explicit procedure to construct coisotropic subalgebras, i.e. Lie subalgebras of g whose annihilator is a Lie subalgebra of g^∗. We write down families of examples for the case that g is a classical complex simple Lie algebra.     

Symmetric Pairs for Quantized Enveloping Algebras

Let θ be an involution of a semisimple Lie algebra g, let g^θ denote the fixed Lie subalgebra, and assume the Cartan subalgebra of g has been chosen in a suitable way. We construct a quantum analog of U(g^θ) which can be characterized as the unique subalgebra of the quantized enveloping algebra of g which is a maximal right coideal that specializes to U(g^θ).     

On the combinatorial rank of a graded braided bialgebra

Let B be a graded braided bialgebra. Let S(B) denote the algebra obtained dividing out B by the two sided ideal generated by homogeneous primitive elements in B of degree at least two. We prove that S(B) is indeed a graded braided bialgebra quotient of B. It is then natural to compute S(S(B)), S(S(S(B))) and so on. This process yields a direct system whose direct limit comes out to be a graded braided bialgebra which is strongly N-graded as a coalgebra. Following V.K. Kharchenko, if the direct system is stationary exactly after n steps, we say that B has combinatorial rank n and we write κ(B)=n. We investigate conditions guaranteeing that κ(B) is finite. In particular, we focus on the case when B is the braided tensor algebra T(V,c) associated to a braided vector space (V,c), providing meaningful examples such that κ(T(V,c))≤1.     

Linear maps preserving the minimum and surjectivity moduli of Hilbert space operators

Let B(H) be the algebra of all bounded linear operators on a complex infinite-dimensional Hilbert space H. For every T∈B(H), let m(T) and q(T) denote the minimum modulus and surjectivity modulus of T respectively. Let ?:B(H)→B(H) be a surjective linear map. In this paper, we prove that the following assertions are equivalent:

(i)

m(T)=m(?(T)) for all T∈B(H),

(ii)

q(T)=q(?(T)) for all T∈B(H),

(iii)

there exist two unitary operators U,V∈B(H) such that ?(T)=UTV for all T∈B(H).

This generalizes the result of Mbekhta [7, Theorem 3.1] to the non-unital case.     

On the Rabinowitz Floer homology of twisted cotangent bundles

Let (M, g) be a closed connected orientable Riemannian manifold of dimension n????2. Let ??:?=??? ₀?+??? ^* ?? denote a twisted symplectic form on T ^* M, where ${\sigma\in\Omega^{2}(M)}$ is a closed 2-form and ?? ₀ is the canonical symplectic structure ${dp\wedge dq}$ on T ^* M. Suppose that ?? is weakly exact and its pullback to the universal cover ${\widetilde{M}}$ admits a bounded primitive. Let ${H:T^{*}M\rightarrow\mathbb{R}}$ be a Hamiltonian of the form ${(q,p)\mapsto\frac{1}{2}\left|p\right|^{2}+U(q)}$ for ${U\in C^{\infty}(M,\mathbb{R})}$ . Let ?? _k :?=?H ^?1(k), and suppose that k?>?c(g, ??, U), where c(g, ??, U) denotes the Ma?é critical value. In this paper we compute the Rabinowitz Floer homology of such hypersurfaces. Under the stronger condition that k?>?c ₀(g, ??, U), where c ₀(g, ??, U) denotes the strict Ma?é critical value, Abbondandolo and Schwarz (J Topol Anal 1:307?C405, 2009) recently computed the Rabinowitz Floer homology of such hypersurfaces, by means of a short exact sequence of chain complexes involving the Rabinowitz Floer chain complex and the Morse (co)chain complex associated to the free time action functional. We extend their results to the weaker case k?>?c(g, ??, U), thus covering cases where ?? is not exact. As a consequence, we deduce that the hypersurface ?? _k is never (stably) displaceable for any k?>?c(g, ??, U). This removes the hypothesis of negative curvature in Cieliebak et?al. (Geom Topol 14:1765?C1870, 2010, Theorem 1.3) and thus answers a conjecture of Cieliebak, Frauenfelder and Paternain raised in Cieliebak et?al. (2010). Moreover, following Albers and Frauenfelder (2009; J Topol Anal 2:77?C98, 2010) we prove that for k?>?c(g, ??, U), any ${\psi\in\mbox{Ham}_{c}(T^{*}M,\omega)}$ has a leaf-wise intersection point in ?? _k , and that if in addition ${\dim\, H_{*}(\Lambda M;\mathbb{Z}_{2})=\infty}$ , dim M????2, and the metric g is chosen generically, then for a generic ${\psi\in\mbox{Ham}_{c}(T^{*}M,\omega)}$ there exist infinitely many such leaf-wise intersection points.     

Algebras of holomorphic mappings in (DF)-spaces

Let U be an absolutely convex open subset of a complex barrelled (DF)-space E and let F be a commutative Banach algebra with identity. Let H_b(U, F) be the space of holomorphic mappings from U into F that are bounded on the U-bounded sets and let H_b (U, F) be the space of the holomorphic mappings from U into F that are uniformly weakly continuous on the U-bounded sets, both endowed with the topology τ_b of uniform convergence on the U-bounded sets. The spectra of (H_wu (U, F), τ_b) and (H_b(U, F), τ_b) are studied.     

Lie superalgebras whose enveloping algebras satisfy a non-matrix polynomial identity

Let L be a Lie superalgebra with its enveloping algebra U(L) over a field F. A polynomial identity is called non-matrix if it is not satisfied by the algebra of 2×2 matrices over F. We characterize L when U(L) satisfies a non-matrix polynomial identity. We also characterize L when U(L) is Lie solvable, Lie nilpotent, or Lie super-nilpotent.     

On the decomposition of the tensor algebra of the classical Lie algebras

Let $\text{g}$_n be the Lie algebra gl_n(C), let S($\text{g}$_n) be the symmetric algebra of $\text{g}$_n, and let T($\text{g}$_n) be the tensor algebra of $\text{g}$_n. In a recent paper, R. K. Gupta studied certain sequences of representations R_∞ = (R_n)^∞_{n = 1}, where R_n is a representation of $\text{g}$_n. These sequences have the property that every irreducible representation occurring in S($\text{g}$_n) is in exactly one of these sequences. Fixing f, she considers s(R_∞, f) which is the limit on n of the multiplicity of R_n in S^f($\text{g}$_n), the fth-graded piece of S($\text{g}$_n). She and R. P. Stanley independently showed that the limit s(R_∞, f) exists and is given by an amazingly elegant formula. They call s(R_∞, f) the stable multiplicity of R_n in S^f($\text{g}$_n). In this paper, an entirely different approach is used to extend the above result in several directions. Appropriately defined sequences R_∞ for all of the classical Lie algebras $\text{g}$_n are studied, and a simple formula for the stable multiplicity m(R)_∞, ψ, f, $\text{g}$_∞) of R_n in the ψ-isotypic component of T^f($\text{g}$_n), where ψ is any irreducible character of the symmetric group tS_f, is obtained. As in the work of Gupta and Stanley, the expressions for m(R)_∞, ψ, f, $\text{g}$_∞) are amazingly simple. Special cases include the stable decomposition of the tensor algebra, the symmetric algebra and the exterior algebra of $\text{g}$_n. As a byproduct of our proof, a “stable” decomposition of every isotypic component of T($\text{g}$_n) is obtained. This combinatorial decomposition is in some sense a generalization of Kostant's decomposition of S($\text{g}$_n) into direct sum of the harmonics and the ideal generated by the invariants of positive degree. To be precise, for f <n the combinatorial decomposition of T^f($\text{g}$_n) projects onto Kostant's decomposition of S^f($\text{g}$_n).     

The approximation property for spaces of holomorphic functions on infinite dimensional spaces II

Let H(U) denote the vector space of all complex-valued holomorphic functions on an open subset U of a Banach space E. Let τω and τδ respectively denote the compact-ported topology and the bornological topology on H(U). We show that if E is a Banach space with a shrinking Schauder basis, and with the property that every continuous polynomial on E is weakly continuous on bounded sets, then (H(U),τω) and (H(U),τδ) have the approximation property for every open subset U of E. The classical space c₀, the original Tsirelson space T^∗ and the Tsirelson^∗-James space are examples of Banach spaces which satisfy the hypotheses of our main result. Our results are actually valid for Riemann domains.     

Lie-Rinehart bialgebras for crossed products

Let g be a Lie algebra over a field K. Endow the polynomial ring K[t] with a g-action by derivations and consider the resulting crossed product K[t]⊙_θg in the category of (K,K[t])-Lie algebras. We explore Lie-Rinehart bialgebra structures on (K[t],K[t]⊙_θg) that arise via compatible pairs and ε-dynamical r-matrices. We give a complete classification for .