The study of deformations of an algebra has been a topic of interest for quite some time, since it allows us to not only produce new algebras but also better understand the original algebra. Given an algebra, finding all its deformations is, if at all possible, quite a challenging problem. For this reason, several specializations of this question have been proposed. For instance, some authors concentrate their efforts in the study of deformations of an algebra arising from an action of a Hopf algebra. The purpose of this dissertation is to discuss a general construction of a deformation of a smash product algebra coming from an action of a particular Hopf algebra. This Hopf algebra is generated by skew-primitive and group-like elements, and depends on a complex parameter. The smash product algebra is defined on the quantum symmetric algebra of a nite-dimensional vector space and a group. In particular, an application of this result has enabled us to find a deformation of such a smash product algebra which is, to the best of our knowledge, the first known example of a deformation in which the new relations in the deformed algebra involve elements of the original vector space. Finally, using Hochschild cohomology, we show that these deformations are nontrivial.

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How do retail service industry employees perform appearance and identity to generate capital? This thesis is data collected through interviews with 5 female retail employees, analyzed in conjunction with relevant readings about performance at work through appearance and identity (Chapter 1 Erving Goffman), the involvement of the self at work for financial gain of the self and the employer (Chapter 2 Arlie Russell Hochschild) and the sale of self for the accumulation of forms of capital (Chapter 3 Pierre Bourdieu, Henri Lefebvre, and other theorists concerned with consumer culture). The thesis begins at the level of the individual, expands to the realm of the retail environment and its associated relationships, and finally moves to examining the circulation of forms of capital. I assert that the research participants, as embodied employees of lifestyle brands, provide a form of capital for their employers. The employees performances positively reinforce the retailers brands and images, and thereby reproduce consumer culture by inciting desire.

Doctor of Philosophy / Department of Mathematics / David Yetter / This dissertation begins the development of the deformation theorem of monoidal categories which accounts for the function that all arrow-valued operations, composition, the arrow part of the monoidal product, and structural natural transformation are deformed. The first chapter is review of algebra deformation theory. It includes the Hochschild complex of an algebra, Gerstenhaber's deformation theory of rings and algebras, Yetter's deformation theory of a monoidal category, Gerstenhaber and Schack's bialgebra deformation theory and Markl and Shnider's deformation theory for Drinfel'd algebras. The second chapter examines deformations of a small $k$-linear monoidal category. It examines deformations beginning with a naive computational approach to discover that as in Markl and Shnider's theory for Drinfel'd algebras, deformations of monoidal categories are governed by the cohomology of a multicomplex. The standard results concerning first order deformations are established. Obstructions are shown to be cocycles in the special case of strict monoidal categories when one of composition or tensor or the associator is left undeformed. At the end there is a brief conclusion with conjectures.

Although sexual abuse is a problem in all age groups, the group which is the most exposed is women between 16 and 24. Many of these youths, boys and girls, are contacting youth centres for support and help regarding sexual assault, mental illness and sexuality. Counsellors working at the youth centres treats youths with experience of sexual abuse in their daily work.This study’s purpose is to demonstrate the counsellor’s experience when meeting these young people. Issues concerned are how we talk about sexual abuse, which emotions are raised among the counsellors during these conversations and which of these emotions that the counsellors feel the need to hide in the conversation.  The study’s material is based on four different interviews with counsellors from youth centres. The questions are being analysed with help of Goffman’s sociological theory expound in The Presentation of Self in Everyday Life (1956) and Hochschild’s theory of emotional labour. The results show that the counsellors focus a lot on reducing guilt and shame among the children and youths. The result also shows that the meeting causes strong and heavy emotions among the counsellors. Opinions about which emotions that are okay to show and not okay to show when taking care of the youths are diverged.

Emotionernas roll inom myndighetsutövning och socialt arbete är både självklar och svårdefinierad. Å ena sida kräver myndighetsutövning rationell rättssäkerhet, å andra sidan anser många socialarbetare att deras känslor, i synnerhet empati, är en ytterst viktig del av deras arbete. Emotionellt arbete är ett forskningsfält som har vuxit fram sedan 1980-talet, till stor del med utgångspunkt i Arlie Hochschilds studie The managed heart – commercialization of human feeling (2012 [1983]). Genom att använda Hochschilds (2012) begrepp djup- och ytagerande, tillsammans med emotionell reglering (Gross, 2015) och uttrycksregler (Grandey & Melloy, 2017) undersöker denna uppsats hur myndighetsutövande socialarbetares emotionella arbete utförs och kan förklaras, samt studerar hur uttrycksreglerna för gruppen kan se ut och beskrivas. Det empiriska materialet består av 14 intervjuer med myndighetsutövande socialarbetare på barn- och ungdomsenheter, ekonomiskt bistånd och inom biståndshandläggning. Genom att använda mig av tematisk analys (Braun & Clarke, 2006, 2022) har jag analyserat och tematiserat materialet.  Studiens resultat visar att socialarbetare utför ett emotionellt arbete som kräver tid, mental förberedelse och erfarenhet. Emotionellt arbete används både för att skapa närhet och distans till klienterna. Emotionsregleringen sker före, under och efter möten med klienter. För att djupagera använder sig socialarbetarna av sin empati för att skapa en professionell relation med klienterna och för att sätta sig in i klienternas situation. När socialarbetarna däremot känner en frustration eller upplever sina klienter som för falska i sitt uppförande börjar de i stället ytagera och snarare använda sig av emotionell reglering för att trycka ner de genuint negativa känslorna.  De uttrycksregler socialarbetarna förhåller sig till balanserar mellan ett logiskt förhållande till myndighetsutövning och ett empatiskt och känslomässigt närvarande synsätt inför klienterna. I och med att socialarbetarna anser att känslorna spelar roll i yrkesutövningen, så försöker de förhålla sig rationellt till sina känslor och anstränger sig för att använda dem i tydliga syften. Socialarbetarna stöttar även varandra i känsloarbetet, dels i informella samtal, dels i formella forum så som i handledning. De stöttar varandra i olika typer av emotionell reglering, ofta genom kognitiv förändring som syftar till att omvärdera synen på en situation och så även känslorna för situationen.  Studiens implikationer för det sociala arbetets praktik är att det krävs organisatoriska och praktiska förutsättningar för att socialarbetare på ett effektivt sätt ska kunna utföra emotionellt arbete; det krävs tid och förberedelse för emotionellt arbete, samt även plats för socialarbetare att bearbeta de känslor som klientarbete rör upp och för de emotionella processer som krävs för emotionell reglering.

The Hochschild cohomology of an associative algebra records information about the deformations of that algebra, and hence the first step toward understanding its deformations is an examination of the Hochschild cohomology. In this dissertation, we use techniques from homological algebra, invariant theory, and combinatorics to analyze the Hochschild cohomology of skew group algebras arising from finite cyclic groups acting on polynomial rings over fields of arbitrary characteristic. These algebras are the natural semidirect product of the group ring with the polynomial ring. Many families of algebras arise as deformations of skew group algebras, such as symplectic reflection algebras and rational Cherednik algebras. We give an explicit description of the Hochschild cohomology governing graded deformations of skew group algebras for cyclic groups acting on polynomial rings. For skew group algebras, a description of the Hochschild cohomology is known in the nonmodular setting (i.e., when the characteristic of the field and the order of the group are coprime). However, in the modular setting (i.e., when the characteristic of the field divides the order of the group), much less is known, as techniques commonly used in the nonmodular setting are not available.

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La quantification par déformation et la correspondance de McKay forment les grands thèmes de l'étude qui porte sur des variétés algébriques singulières, des quotients d'algèbres de polynômes et des algèbres de polynômes invariants sous l'action d'un groupe fini. Nos principaux outils sont les cohomologies de Poisson et de Hochschild et la théorie des représentations. Certains calculs formels sont effectués avec Maple et GAP. Nous calculons les espaces d'homologie et de cohomologie de Hochschild des surfaces de Klein, en développant une généralisation du Théorème de HKR au cas de variétés non lisses et utilisons la division multivariée et les bases de Gröbner. La clôture de l'orbite nilpotente minimale d'une algèbre de Lie simple est une variété algébrique singulière sur laquelle nous construisons des star-produits invariants, grâce à la décomposition BGS de l'homologie et de la cohomologie de Hochschild, et à des résultats sur les invariants des groupes classiques. Nous explicitons les générateurs de l'idéal de Joseph associé à cette orbite et calculons les caractères infinitésimaux. Pour les algèbres de Lie simples B, C, D, nous établissons des résultats généraux sur l'espace d'homologie de Poisson en degré 0 de l'algèbre des invariants, qui vont dans le sens de la conjecture d'Alev et traitons les rangs 2 et 3. Nous calculons des séries de Poincaré à 2 variables pour des sous-groupes finis du groupe spécial linéaire en dimension 3, montrons que ce sont des fractions rationnelles, et associons aux sous-groupes une matrice de Cartan généralisée pour obtenir une correspondance de McKay algébrique en dimension 3. Toute l'étude a donné lieu à 4 articles / Deformation quantization and McKay correspondence form the main themes of the study which deals with singular algebraic varieties, quotients of polynomial algebras, and polynomial algebras invariant under the action of a finite group. Our main tools are Poisson and Hochschild cohomologies and representation theory. Certain calculations are made with Maple and GAP. We calculate Hochschild homology and cohomology spaces of Klein surfaces by developing a generalization of HKR theorem in the case of non-smooth varieties and use the multivariate division and the Groebner bases. The closure of the minimal nilpotent orbit of a simple Lie algebra is a singular algebraic variety : on this one we construct invariant star-products, with the help of the BGS decomposition of Hochschild homology and cohomology, and of results on the invariants of the classical groups. We give the generators of the Joseph ideal associated to this orbit and calculate the infinitesimal characters. For simple Lie algebras of type B, C, D, we establish general results on the Poisson homology space in degree 0 of the invariant algebra, which support Alev's conjecture, then we are interested in the ranks 2 and 3. We compute Poincaré series of 2 variables for the finite subgroups of the special linear group in dimension 3, show that they are rational fractions, and associate to the subgroups a generalized Cartan matrix in order to obtain a McKay correspondence in dimension 3. All the study comes from 4 papers

Cette thèse est consacrée à l'étude des modules de quantification par déformation ou DQ-modules. Elle explore dans quelle mesure certains théorèmes de géométrie algébrique s'étendent aux DQ-modules et plus généralement à un cadre non-commutatif. Nous établissons un théorème de type Riemann-Roch pour les algèbres différentielles graduées propres et homologiquement lisses, généralisant ainsi un résultat de Shklyarov. Nous donnons un analogue non-commutatif d'un résultat de Bondal et Van den Bergh affirmant que la catégorie dérivée des faisceaux quasi-cohérents d'une variété algébrique est engendrée par un générateur compact. Il apparaît que la notion d'objet quasi-cohérent n'est pas adaptée à la théorie des DQ-modules. Nous introduisons donc, en nous appuyant sur la notion de complétude cohomologique de Kashiwara-Schapira, la notion d'objet cohomologiquement complet à gradué quasi-cohérent. Nous montrons que ces objets forment une catégorie triangulée, engendrée par un générateur compact et nous en caractérisons les objets compacts. Nous adaptons au cas des DQ-modules une formule due à Lunts, qui calcule la trace d'un noyau cohérent agissant sur l'homologie de Hochschild d'un DQ-algébroïde. La méthode de Lunts ne semble pas s'appliquer aux DQ-modules. Nous développons donc un formalisme permettant d'obtenir un théorème similaire à celui de Lunts puis nous l'appliquons aux DQ-modules. Enfin, nous nous intéressons, dans le cadre des DQ-modules, aux transformations intégrales pour lesquelles nous donnons des résultats d'adjonction et démontrons une condition nécessaire et suffisante pour qu'une telle transformation soit une équivalence.