Normal Enveloping Algebras

A full characterization is given of ordinary and restricted enveloping algebras which are normal with respect to the principal involution.

Generalizations of Lie Algebras

The generalizations of Lie algebras appeared in the modern mathematics and mathematical physics. In this paper we consider recent developments and remaining open problems on the subject. Some of that developments have been influenced by lectures given by Professor Jaime Keller in his research seminar. The survey includes Lie superalgebras, color Lie algebras, Lie algebras in symmetric categories, free Lie tau-algebras, and some generalizations with non-associative enveloping algebras: tangent algebras to analytic loops, bialgebras and primitive elements, non-associative Hopf algebras.

COMMUTATIVE ALGEBRAS IN DRINFELD CATEGORIES OF ABELIAN LIE ALGEBRAS

We describe (braided-) commutative algebras with non-degenerate multiplicative form in certain braided monoidal categories, corresponding to abelian metric Lie algebras (so-called Drinfeld categories). We also describe local modules over these algebras and classify commutative algebras with a finite number of simple local modules.

FREE GROUP ALGEBRAS IN DIVISION RINGS

Let k be an algebraically closed field of characteristic zero and let L be an algebraic function field over k. Let sigma : L -> L be a k-automorphism of infinite order, and let D be the skew field of fractions of the skew polynomial ring L[t; sigma]. We show that D contains the group algebra kF of the free group F of rank 2.

ON THE CONSISTENCY OF TWISTED GENERALIZED WEYL ALGEBRAS

A twisted generalized Weyl algebra A of degree n depends on a. base algebra R, n commuting automorphisms sigma(i) of R, n central elements t(i) of R and on some additional scalar parameters. In a paper by Mazorchuk and Turowska, it is claimed that certain consistency conditions for sigma(i) and t(i) are sufficient for the algebra to be nontrivial. However, in this paper we give all example which shows that this is false. We also correct the statement by finding a new set of consistency conditions and prove that the old and new conditions together are necessary and sufficient for the base algebra R to map injectively into A. In particular they are sufficient for the algebra A to be nontrivial. We speculate that these consistency relations may play a role in other areas of mathematics, analogous to the role played by the Yang-Baxter equation in the theory of integrable systems.

On simple Lie algebras over a field of characteristic 2

We prove that the simple Lie algebras constructed by G. Jurman (2004) in 121 are isomorphic to Hamiltonian algebras. As a corollary we answer all questions formulated in G. Jurman (2004) [2] about isomorphisms of these algebras. (C) 2012 Elsevier Inc. All rights reserved.

Polynomial and Poisson dependence in free Poisson algebras and free Poisson fields

We prove that any two Poisson dependent elements in a free Poisson algebra and a free Poisson field of characteristic zero are algebraically dependent, thus answering positively a question from Makar-Limanov and Umirbaev (2007) [8]. We apply this result to give a new proof of the tameness of automorphisms for free Poisson algebras of rank two (see Makar-Limanov and Umirbaev (2011) [9], Makar-Limanov et al. (2009) [10]). (C) 2011 Elsevier Inc. All rights reserved.

Algebras determined by their supports

In this paper, we introduce and study a class of algebras which we call ada algebras. An artin algebra is ada if every indecomposable projective and every indecomposable injective module lies in the union of the left and the right parts of the module category. We describe the Auslander-Reiten components of an ada algebra which is not quasi-tilted, showing in particular that its representation theory is entirely contained in that of its left and right supports, which are both tilted algebras. Also, we prove that an ada algebra over an algebraically closed field is simply connected if and only if its first Hochschild cohomology group vanishes. (C) 2011 Elsevier B.V. All rights reserved.

Toeplitz operators on finite and infinite dimensional spaces with associated psi *-Fréchet algebras

The present thesis is a contribution to the multi-variable theory of Bergman and Hardy Toeplitz operators on spaces of holomorphic functions over finite and infinite dimensional domains. In particular, we focus on certain spectral invariant Frechet operator algebras F closely related to the local symbol behavior of Toeplitz operators in F. We summarize results due to B. Gramsch et.al. on the construction of Psi_0- and Psi^*-algebras in operator algebras and corresponding scales of generalized Sobolev spaces using commutator methods, generalized Laplacians and strongly continuous group actions. In the case of the Segal-Bargmann space H^2(C^n,m) of Gaussian square integrable entire functions on C^n we determine a class of vector-fields Y(C^n) supported in complex cones K. Further, we require that for any finite subset V of Y(C^n) the Toeplitz projection P is a smooth element in the Psi_0-algebra constructed by commutator methods with respect to V. As a result we obtain Psi_0- and Psi^*-operator algebras F localized in cones K. It is an immediate consequence that F contains all Toeplitz operators T_f with a symbol f of certain regularity in an open neighborhood of K. There is a natural unitary group action on H^2(C^n,m) which is induced by weighted shifts and unitary groups on C^n. We examine the corresponding Psi^*-algebra A of smooth elements in Toeplitz-C^*-algebras. Among other results sufficient conditions on the symbol f for T_f to belong to A are given in terms of estimates on its Berezin-transform. Local aspects of the Szegö projection P_s on the Heisenbeg group and the corresponding Toeplitz operators T_f with symbol f are studied. In this connection we apply a result due to Nagel and Stein which states that for any strictly pseudo-convex domain U the projection P_s is a pseudodifferential operator of exotic type (1/2, 1/2). The second part of this thesis is devoted to the infinite dimensional theory of Bergman and Hardy spaces and the corresponding Toeplitz operators. We give a new proof of a result observed by Boland and Waelbroeck. Namely, that the space of all holomorphic functions H(U) on an open subset U of a DFN-space (dual Frechet nuclear space) is a FN-space (Frechet nuclear space) equipped with the compact open topology. Using the nuclearity of H(U) we obtain Cauchy-Weil-type integral formulas for closed subalgebras A in H_b(U), the space of all bounded holomorphic functions on U, where A separates points. Further, we prove the existence of Hardy spaces of holomorphic functions on U corresponding to the abstract Shilov boundary S_A of A and with respect to a suitable boundary measure on S_A. Finally, for a domain U in a DFN-space or a polish spaces we consider the symmetrizations m_s of measures m on U by suitable representations of a group G in the group of homeomorphisms on U. In particular,in the case where m leads to Bergman spaces of holomorphic functions on U, the group G is compact and the representation is continuous we show that m_s defines a Bergman space of holomorphic functions on U as well. This leads to unitary group representations of G on L^p- and Bergman spaces inducing operator algebras of smooth elements related to the symmetries of U.

Pseudodifferential operators on Hilbert space riggings with associated psi *-algebras and generalized Hörmander classes

The present thesis is concerned with certain aspects of differential and pseudodifferential operators on infinite dimensional spaces. We aim to generalize classical operator theoretical concepts of pseudodifferential operators on finite dimensional spaces to the infinite dimensional case. At first we summarize some facts about the canonical Gaussian measures on infinite dimensional Hilbert space riggings. Considering the naturally unitary group actions in $L^2(H_-,gamma)$ given by weighted shifts and multiplication with $e^{iSkp{t}{cdot}_0}$ we obtain an unitary equivalence $F$ between them. In this sense $F$ can be considered as an abstract Fourier transform. We show that $F$ coincides with the Fourier-Wiener transform. Using the Fourier-Wiener transform we define pseudodifferential operators in Weyl- and Kohn-Nirenberg form on our Hilbert space rigging. In the case of this Gaussian measure $gamma$ we discuss several possible Laplacians, at first the Ornstein-Uhlenbeck operator and then pseudo-differential operators with negative definite symbol. In the second case, these operators are generators of $L^2_gamma$-sub-Markovian semi-groups and $L^2_gamma$-Dirichlet-forms. In 1992 Gramsch, Ueberberg and Wagner described a construction of generalized Hörmander classes by commutator methods. Following this concept and the classical finite dimensional description of $Psi_{ro,delta}^0$ ($0leqdeltaleqroleq 1$, $delta< 1$) in the $C^*$-algebra $L(L^2)$ by Beals and Cordes we construct in both cases generalized Hörmander classes, which are $Psi^*$-algebras. These classes act on a scale of Sobolev spaces, generated by our Laplacian. In the case of the Ornstein-Uhlenbeck operator, we prove that a large class of continuous pseudodifferential operators considered by Albeverio and Dalecky in 1998 is contained in our generalized Hörmander class. Furthermore, in the case of a Laplacian with negative definite symbol, we develop a symbolic calculus for our operators. We show some Fredholm-criteria for them and prove that these Fredholm-operators are hypoelliptic. Moreover, in the finite dimensional case, using the Gaussian-measure instead of the Lebesgue-measure the index of these Fredholm operators is still given by Fedosov's formula. Considering an infinite dimensional Heisenberg group rigging we discuss the connection of some representations of the Heisenberg group to pseudo-differential operators on infinite dimensional spaces. We use this connections to calculate the spectrum of pseudodifferential operators and to construct generalized Hörmander classes given by smooth elements which are spectrally invariant in $L^2(H_-,gamma)$. Finally, given a topological space $X$ with Borel measure $mu$, a locally compact group $G$ and a representation $B$ of $G$ in the group of all homeomorphisms of $X$, we construct a Borel measure $mu_s$ on $X$ which is invariant under $B(G)$.

Pseudodifferential analysis in Psi*-algebras on transmission spaces, infinite solving ideal chains and K-theory for conformally compact spaces

The present thesis is a contribution to the theory of algebras of pseudodifferential operators on singular settings. In particular, we focus on the $b$-calculus and the calculus on conformally compact spaces in the sense of Mazzeo and Melrose in connection with the notion of spectral invariant transmission operator algebras. We summarize results given by Gramsch et. al. on the construction of $Psi_0$-and $Psi*$-algebras and the corresponding scales of generalized Sobolev spaces using commutators of certain closed operators and derivations. In the case of a manifold with corners $Z$ we construct a $Psi*$-completion $A_b(Z,{}^bOmega^{1/2})$ of the algebra of zero order $b$-pseudodifferential operators $Psi_{b,cl}(Z, {}^bOmega^{1/2})$ in the corresponding $C*$-closure $B(Z,{}^bOmega^{12})hookrightarrow L(L^2(Z,{}^bOmega^{1/2}))$. The construction will also provide that localised to the (smooth) interior of Z the operators in the $A_b(Z, {}^bOmega^{1/2})$ can be represented as ordinary pseudodifferential operators. In connection with the notion of solvable $C*$-algebras - introduced by Dynin - we calculate the length of the $C*$-closure of $Psi_{b,cl}^0(F,{}^bOmega^{1/2},R^{E(F)})$ in $B(F,{}^bOmega^{1/2}),R^{E(F)})$ by localizing $B(Z, {}^bOmega^{1/2})$ along the boundary face $F$ using the (extended) indical familiy $I^B_{FZ}$. Moreover, we discuss how one can localise a certain solving ideal chain of $B(Z, {}^bOmega^{1/2})$ in neighbourhoods $U_p$ of arbitrary points $pin Z$. This localisation process will recover the singular structure of $U_p$; further, the induced length function $l_p$ is shown to be upper semi-continuous. We give construction methods for $Psi*$- and $C*$-algebras admitting only infinite long solving ideal chains. These algebras will first be realized as unconnected direct sums of (solvable) $C*$-algebras and then refined such that the resulting algebras have arcwise connected spaces of one dimensional representations. In addition, we recall the notion of transmission algebras on manifolds with corners $(Z_i)_{iin N}$ following an idea of Ali Mehmeti, Gramsch et. al. Thereby, we connect the underlying $C^infty$-function spaces using point evaluations in the smooth parts of the $Z_i$ and use generalized Laplacians to generate an appropriate scale of Sobolev spaces. Moreover, it is possible to associate generalized (solving) ideal chains to these algebras, such that to every $ninN$ there exists an ideal chain of length $n$ within the algebra. Finally, we discuss the $K$-theory for algebras of pseudodifferential operators on conformally compact manifolds $X$ and give an index theorem for these operators. In addition, we prove that the Dirac-operator associated to the metric of a conformally compact manifold $X$ is not a Fredholm operator.

Representations of semisimple Lie algebras

La tesi è dedicata allo studio delle rappresentazioni delle algebre di Lie semisemplici su un campo algebricamente chiuso di caratteristica zero. Mediante il teorema di Weyl sulla completa riducibilità, ogni rappresentazione di dimensione finita di una algebra di Lie semisemplice è scrivibile come somma diretta di sottorappresentazioni irriducibili. Questo permette di poter concentrare l'attenzione sullo studio delle rappresentazioni irriducibili. Inoltre, mediante il ricorso all'algebra inviluppante universale si ottiene che ogni rappresentazione irriducibile è una rappresentazione di peso più alto. Perciò è naturale chiedersi quando una rappresentazione di peso più alto sia di dimensione finita ottenendo che condizione necessaria e sufficiente perché una rappresentazione di peso più alto sia di dimensione finita è che il peso più alto sia dominante. Immediata è quindi l'applicazione della teoria delle rappresentazioni delle algebre di Lie semisemplici nello studio delle superalgebre di Lie, in quanto costituite da un'algebra di Lie e da una sua rappresentazione, dove viene utilizzata la tecnica della Z-graduazione che viene utilizzata per la prima volta da Victor Kac nello studio delle algebre di Lie di dimensione infinita nell'articolo ''Simple irreducible graded Lie algebras of finite growth'' del 1968.

Il teorema dei quattro vertici e il teorema di Jordan-Schönflies

Una curva di Jordan è una curva continua nel piano, semplice e chiusa. Lo scopo della tesi è presentare tre teoremi riguardanti le curve di Jordan. Il teorema dei quattro vertici afferma che ogni curva di Jordan regolare di classe C^2 ha almeno quattro punti in cui la curvatura orientata ha un massimo o un minimo locali. Il teorema della curva di Jordan asserisce che una curva di Jordan divide il piano esattamente in due parti, l'interno e l'esterno della curva. Secondo il teorema di Schönflies, la chiusura dell'interno di una curva di Jordan è omeomorfa a un disco chiuso.

Real forms of Lie algebras and Lie superalgebras

In questa tesi abbiamo studiato le forme reali di algebre e superalgebre di Lie. Il lavoro si suddivide in tre capitoli diversi, il primo è di introduzione alle algebre di Lie e serve per dare le prime basi di questa teoria e le notazioni. Nel secondo capitolo abbiamo introdotto le algebre compatte e le forme reali. Abbiamo visto come sono correlate tra di loro tramite strumenti potenti come l'involuzione di Cartan e relativa decomposizione ed i diagrammi di Vogan e abbiamo introdotto un algoritmo chiamato "push the button" utile per verificare se due diagrammi di Vogan sono equivalenti. Il terzo capitolo segue la struttura dei primi due, inizialmente abbiamo introdotto le superalgebre di Lie con relativi sistemi di radici e abbiamo proseguito studiando le relative forme reali, diagrammi di Vogan e abbiamo introdotto anche qua l'algoritmo "push the button".