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From the written sources we do not have information about deportations to lumigan 3ml free shipping treatment rosacea the Galilee and Gilead 3ml lumigan otc medicine 1900s spruce cough balsam fir. Rural sites that deportees may have settled have been discovered in the Galilee order 3 ml lumigan fast delivery treatments, at Yiftachel lumigan 3 ml otc symptoms 32 weeks pregnant, and at a small site near ‘En Zippori (Gal 2009). Source: Israel changes in the material culture can be Antiquities Authority collection. The Four-Room House that was very common in the kingdom of Israel, in both urban and rural settlements, com pletely disappeared during the Assyrian period. Although they were not very common, some architectural traditions originating from Mesopotamia appeared in the area from this time on. These can be seen, for example, at Tel Megiddo, complex 1052/1369, Tel Hazor building 3002, Tel Kinrot building 737, and per haps also at Tel Dan, although some of the buildings were built only after the Neo Assyrian period (for a detailed study, see Kertai 2018). In addition, some of the new rural sites found in southwestern Samaria (and elsewhere) were composed of a courtyard complex, which seems alien to the local architectural tradition of Sa maria (Faust 2006). Imi tation Assyrian vessels, some of which are known as “Palace Ware,” have been found in many sites across the former northern kingdom (Singer-Avitz 2007; Stern 2015). Usually they can be found in very small quantities, appearing for the first time just prior to the kingdom’s final destruction. Although scholars previ ously agreed that the “Palace Ware” pottery found in the region was probably used the Kingdom of Israel in the Eighth Century 73 by Assyrian officials, it seems that the matter is much more complex and most (if not all) of the Assyrian pottery unearthed was produced locally and it is only an imitation of the Assyrian ware. It is likely that most of the pottery found was used by local people who wished to imitate Assyrian pottery and perhaps even Assyrian behavior. While the Assyrian officials in the new provinces likely used such pot tery, a set of Assyrian Palace Ware belonging to them has yet to be found (for detailed discussion see Hunt 2015). While most of the popu lation probably continue to bury in simple pit graves, the “bathtub coffin” appears for the first time. This type of burial involves placing the dead in large clay cof fins, imitating the metal coffins used in Assyria (Wicks 2015). Only a few such clay coffins were found in the territory of the former kingdom of Israel (Stern 2001, 33–34), suggesting that only a small number of inhabitants (maybe Assyr ian officials) used this form of burial. The first half of the eighth century were the glory days during which the kingdom of Israel prospered and was one of the most powerful states in the Levant. The pros perity also led to negative socioeconomic processes: namely the concentration of wealth in the hands of the elite, which harmed the more vulnerable members of society and weakened the social fabric of the kingdom. The Assyrian conquest thus dramatically changed the lives of the kingdom’s population. The area of the former kingdom of Israel was transformed into provinces in a matter of decades, and all the signs of the once independent state disappeared. While some deportees were brought in as settlers, joining the existing population, most of Israel’s land lay desolate. The Kingdom of Israel in the Eighth Century 75 Frankel, Rafi, Nimrod Getzov, Modechai Aviam, and Avi Degani. Archaeology of the City: Urban Planning in Ancient Israel and Its Social Implications. Remarks on the Material Culture of a Border Site in Northern Palestine at the Turn of an Era. The Mechanics of Empire: the Northern Frontier of Assyria as a Case Study in Imperial Dynamics. The region to the east of the Jordan River, known as the Golan Heights, is in fact a basalt plateau. The plateau declines gradually from an altitude of 1000 m at the foothills of the Hermon Mountain in the northeast, to 200 m below sea level at the Sea of Galilee. This decline extends over a relatively short distance of only about 50 km (Meiler 2011). The basalt plateau is geologically and geographically connected to the eastern region of the Bashan and the Hauran farther to the east. The terrain is rugged in the north northeast but turns into fertile, agricultural land in the south and the east. The part of the plateau that is above sea level was covered in the past with Tabor oak forests, some remnants of which still exist. Travelers and cartographers recorded the name as et-Tell, meaning the mound, since it is the most prominent mound in this area. Other names recorded for this site are Tel Amriya (the constructed mound), Tel Shafi (the mound of health), Tel el-Tala wiyah (the mound of the Bedouin tribe of Arab el Talawiyah, or the Arabs of the mounds). The identification of et-Tell with Bethsaida is based on the testimony of Jo sephus. His precise description of the location of Bethsaida, at the lower Golan near the estuary of the River Jordan, is compatible with the location of et-Tell and with the Roman remains excavated there. However, there were some confusions due to the Gospel of John 1:44, which states that Bethsaida is located in Galilee. For more than a century travelers and scholars 1 have been searching for the Galilean Bethsaida with no avail. Among the sites proposed were Capernaum, Tabgha (Byzantine site), Al Minya (An Umayyad site), Masudiya (Byzantine) and Huseniya (Byzantine) and al Araj (Byzantine, Medieval). However some years ago, I proposed identifying the place-names Zer (), or Zed, men tioned in Josh 19:35, as one of the fishermen cities located around the Sea of Galilee, with et-Tell. I also proposed interpreting the word in this verse as “the fishermen” (Arav 1995, 193–201). The translation of this verse, in this pro posal is: “The fortified cities of the fishermen were Zer, Hamat, Raqat and Kineret.
Progressive symmetrical muscle weak ness is usually the first and most important clinical manifestation in the majority of patients with der 21 buy 3ml lumigan with visa hb treatment. Autoimmune Diseases Mixed Connective Tissue Disease include a recurrent enlargement of the parotid 3ml lumigan amex medicine kit, submandibular (Fig generic lumigan 3ml amex medications zocor. Females are more com include dysphagia buy 3ml lumigan otc moroccanoil oil treatment, candidosis, cheilitis, and dental monly affected, with a mean of 35 years. Artificial saliva and sialagogues may teristically high titers of antibody to nuclear alleviate dryness of the mouth. Most frequently, it affects women in the fourth and fifth decades and is characterized by xerostomia and keratoconjunc tivitis sicca. Recent clinical, serologic, and genetic criteria have been used to distinguish two forms of the disease: primary and secondary. Autoimmune Diseases Benign Lymphoepithelial Lesion Lupoid Hepatitis the term "benign lymphoepithelial lesion" is used Lupoid hepatitis is a form of chronic active to define a localized lymphocytic infiltration of the hepatitis of autoimmune origin, which most fre salivary and lacrimal glands. It affects most frequently lung, and bowel manifestations, hemolytic middle-aged women. The only painless symmetrical enlargement that may cause difference from desquamative gingivitis is that mild xerostomia and an uncomfortable feeling. The differential diagnosis includes necrotizing Laboratory tests helpful for diagnosis include sialometaplasia and minor salivary gland tumors. Steroids and nonsteroid anti-inflam matory agents are the usual therapeutic measures. Primary Biliary Cirrhosis Primary biliary cirrhosis is a serious autoimmune disease characterized by intrahepatic cholestasis leading to hepatic cirrhosis. The cardinal clinical manifestations are jaundice, pruritus, and cutaneous xanthomas. Late manifes tations are portal hypertension and the sequelae of cirrhosis (ascites, esophageal varices, enceph alopathy, osteomalacia, etc. During the late stages of the disease, the oral mucosa is red, thin, and atrophic with telangiectasias (Fig. Laboratory tests helpful for diagnosis include serologic and immunologic tests and liver biopsy. Skin Diseases Erythema Multiforme Rarely, bullae develop on preexisting maculo papular lesions, giving rise to the bullous form of Erythema multiforme is an acute or subacute self the disease. In the oral cavity small vesicles limiting disease that mainly involves the skin and develop that rupture and leave an eroded surface mucous membranes. Lesions obscure, a plethora of different agents, such as may be seen anywhere in the mouth, but the lips drugs, infections, radiation, endocrine factors, and the anterior part of the mouth are most com neoplasia, collagen diseases, and physical factors monly involved (Fig. The occurs chiefly in young adults between 20 and 40 diagnosis is primarily based on clinical criteria. Men are more frequently affected the differential diagnosis includes stomatitis than women. The disease affects mainly the skin and has a sudden onset with the occurrence of red medicamentosa, Stevens-Johnson syndrome, toxic macules and papules in a symmetrical pattern on epidermal necrolysis, pemphigus, bullous and ero sive lichen planus, cicatricial pemphigoid, bullous the palms and soles and less commonly on the face, neck, and trunk. These lesions are small and pemphigoid, primary herpetic gingivostomatitis, may increase in size centrifugally, reaching a and recurrent aphthous ulcers. A histopathologic examina periphery remains erythematous, but the center tion of the lesions is suggestive of the disease. Stevens-Johnson Syndrome extremely painful erosions covered by grayish white or hemorrhagic pseudomembranes (Fig. The lips usually show characteristic bloody severe form of erythema multiforme that predom crusting. The dromal systemic illness (fever, cough, weakness, ocular lesions consist of conjunctivitis, but corneal malaise, sore throat, arthralgias, myalgias, ulceration, anterior uveitis, or panophthalmitis diarrhea, etc. Stevens-Johnson syndrome, widespread erosions covered by hemorrhagic crusting on the lips and tongue. They may be either pathogenesis of the disease still remains unclear, the typical maculopapular eruption of erythema and an underlying immune mechanism seems multiforme, but more commonly are bullous or most probable. The mortality rate of the conjunctivae, and erythema, which begins on untreated patients ranges from 5 to 15%. Diag the face and extremities and rapidly extends to the nosis is based mainly on clinical criteria. In the oral mucosa there is severe inflammation, vesiculation, and Laboratory findings. Histopathologic examination painful widespread erosions, primarily on the lips, is supportive of the diagnosis. Large doses of systemic steroids and Similar lesions may be seen on the eyelids, con antibiotics if considered necessary. A great variety of etiologic factors have been incriminated, but mainly drugs, such as antibiot ics, sulfonamides, sulfones, nonopiate analgesics, nonsteroidal anti-inflammatory agents, and anti epileptic drugs, are thought to be responsible for the disease. Viral, bacterial, and fungal infection, malignant diseases, and radiation have also been 22. Toxic epidermal necrolysis, characteristic detachment of epidermis, resembling scalding. Toxic epidermal necrolysis, severe erosions covered by hemorrhagic crusting on the lips. Pemphigus be involved, but the soft palate, buccal mucosa, and lower lip predominate. Lesions on other Pemphigus is a chronic autoimmune bullous dis mucosal surfaces (conjunctivae, larynx, nose, ease that affects the skin and mucous membranes pharynx, genitals, anus) may eventually develop and has a reasonable prognosis. On the skin, a high incidence in Mediterranean races (Jews, bullae that rupture easily, leaving eroded areas, Greeks, Italians) without, however, usually are seen and exhibit a tendency to enlarge as the exhibiting any familial distribution.
In patients taking risperidone 4 to discount lumigan 3ml on-line medications known to cause nightmares 6 mg/day (n = 51) for 6 weeks the following adverse events were reported: salivary hypersecretion (10%) discount lumigan 3 ml line anima sound medicine, sedation (12%) generic 3 ml lumigan with visa symptoms xanax treats, parkinsonism (28%) discount lumigan 3 ml free shipping medicine 657, tremor (10%), akathisia 199,200 (10%), dizziness (14%), dystonia (6%), and anxiety (6%). Other Second-Generation Antipsychotics Key Points • Two small, short-term studies addressed quetiapine, aripiprazole, or ziprasidone. Overview of the Literature Aripiprazole, ziprasidone, and quetiapine were used in the management of disruptive 180 188 behavior disorders in two studies (1 good and 1 poor quality for harms) meeting our criteria. The Cochrane review addressed the adverse effects of weight gain and metabolic changes as primary outcomes and provided no significant analysis of the limited harms data in the study. The review reported that quetiapine was associated with significantly less weight gain than olanzapine (moderate strength of the evidence) but with more weight gain when compared with aripiprazole (low strength of the evidence). Quetiapine was also associated with more dyslipidemia than aripiprazole (low strength of the evidence). Aripiprazole was associated with fewer prolactin-related adverse events than placebo (moderate strength of the evidence), and differences between the effects of second generation antipsychotics related to extrapyramidal symptoms, insulin resistance, and sedation were not significant (low strength of the evidence). Finally, one review and meta-analysis evaluated metabolic and neurologic adverse events associated with second-generation antipsychotic use in children with any mental health disorder. Studies included in the review did not report significant changes in blood 94 pressure, heart rate, or laboratory values, and only one short-term study addressed ziprasidone. The review suggested that risks of metabolic adverse effects are greatest for olanzapine followed by clozapine and quetiapine, while risks were lower for risperidone and aripiprazole. The risk for neurologic harms appeared greatest with risperidone, olanzapine, and aripiprazole. The mean number of parent-reported side effects and the mean severity did not differ significantly between groups nor did child-reported side effects including sedation, social withdrawal, and weight gain. Three adverse events were reported significantly more often by parents of children in the placebo arm compared with quetiapine: decreased mental alertness (n = 9 in placebo arm vs. Weight gain and prolactin levels did not differ significantly between groups, and laboratory parameters were in normal levels in both groups. Children in the quetiapine group had a higher resting pulse than did children in the placebo arm (p=0. Use of stimulant medication was allowed (8% of the aripiprazole group; 36% of the ziprasidone group). Six children in the ziprasidone arm and two in the aripiprazole arm discontinued the study due to sedation. Harms reported in studies of other second-generation antipsychotics (continued) Author, Year Groups (Final Study Design Dose), N at Final Harms in Treatment Group, n (%) Harms in Comparison Group, n (%) Quality Analysis Bastiaens G1: Aripiprazole Aripiprazole Ziprasidone 188,203 2009 (4. Adverse events referenced in the warnings/precautions section of the package insert include: neuroleptic malignant syndrome, tardive dyskinesia, hyperglycemia/diabetes mellitus, dyslipidemia, body weight gain, orthostatic hypotension, leukopenia, neutropenia, agranulocytosis, seizures, cognitive motor 205 impairment, suicide, and suicidal ideation. Pediatric patients (n = 920), aged 6 to 17 years, being treated with aripiprazole for schizophrenia, bipolar mania, or autistic disorder were included in clinical trials that assessed 205 safety. Of these patients, 117 were treated for at least 1 year and 465 were treated for at least 205 180 days. Adverse events reported in these trials with a frequency of more than 10 percent included: somnolence, headache, vomiting, extrapyramidal disorder, fatigue, increased appetite, 205 insomnia, nausea, nasopharyngitis, and weight increased. Review documents and adverse event data for quetiapine extended release were not included. Adverse events referenced in the warnings/precautions section of the package insert include: suicidal thoughts and behaviors, neuroleptic malignant syndrome, hyperglycemia, dyslipidemia, weight gain, tardive dyskinesia, hypotension, increased blood pressure, leukopenia, neutropenia and agranulocytosis, cataracts, hypothyroidism, hyperprolactinemia, and cognitive motor 207 impairment. Because divalproex was given as add on therapy with stimulants, many of the reported adverse effects such as anxiety, nail biting, and appetite suppression were attributed to stimulant use. Trends toward a higher rate of treatment emergent sadness (divalproex: 3 of 14, 20%; placebo: 0 of 13, 0%; p=0. The only adverse effects reported were gastrointestinal upset (n = 1) and sleepiness (n = 6) (Table 42). Harms reported in studies of divalproex Author, Year Group [Dose] (N at Harms in Treatment Group, n Harms in Comparison Group, a Design (Quality) Final Analysis) (%) N (%) 178 Blader et al. Harms reported in studies of divalproex (continued) Author, Year Group [Dose] (N at a Harms in Treatment Group, n (%) Design (Quality) Final Analysis) 184 Steiner et al. Package Insert Data 210 the safety information for divalproex sodium was obtained from the package insert. Adverse events referenced in the warnings/precautions section of the package insert include: suicidal behavior or ideation, thrombocytopenia, hyperammonemia, hyperammonemic encephalopathy, hypothermia, 211 hepatotoxicity, and pancreatitis. It is important to note that there is an increased risk of 210 developing fatal hepatotoxicity in patients less than two years of age. Specifically in pediatric clinical trials, consisting of 76 patients aged 10-17 years taking divalproex extended release for mania and 231 patients aged 12 to 17 years taking divalproex extended release for migraine, common adverse events (reported >5% and twice the rate of placebo) included: nausea, upper 210 abdominal pain, somnolence, increased ammonia, gastritis and rash. According to the package insert, divalproex safety and tolerability in pediatric patients is 210 similar to what has been observed in adults. Therefore, the adverse events reported below were not specified for pediatric patients but are included due to the similarity in pediatric safety response. The following adverse events were reported 89 patients being treated with divalproex for mania: nausea (22%), somnolence (19%), dizziness (12%), vomiting (12%), accidental injury 210 (11%), asthenia (10%), abdominal pain (9%), dyspepsia (9%), and rash (6%). Adverse events occurring at an incidence rate of greater than 1 percent (no more that 5%), in patients taking divalproex included: chest pain, chills, chills and fever, fever, neck pain, neck rigidity, hypertension, hypotension, palpitations, postural hypotension, tachycardia, vasodilation, anorexia, fecal incontinence, flatulence, gastroenteritis, glossitis, periodontal abscess, ecchymosis, edema, peripheral edema, arthralgia, arthrosis, leg cramps, twitching, abnormal dreams, abnormal gait, agitation, ataxia, catatonic reaction, confusion, depression, diplopia, dysarthria, hallucinations, hypertonia, hypokinesia, insomnia, paresthesia, reflexes increased, tardive dyskinesia, thinking abnormalities, vertigo, dyspnea, rhinitis, alopecia, discoid lupus erythematosus, dry skin, furunculosis, maculopapular rash, seborrhea, amblyopia, conjunctivitis, deafness, dry eyes, ear pain, eye pain, tinnitus, dysmenorrhea, dysuria, and urinary 210 incontinence. The clinical trials used to gather the following adverse events included patients on other 210 antiepilepsy medications. Therefore, it is impossible to clearly state if the following reactions 210 are due to divalproex alone in patients with epilepsy. In a controlled trial assessing the use of high dose divalproex (n = 131) as monotherapy for the treatment of complex partial seizures the following adverse events were reported: asthenia (21%), nausea (34%), diarrhea (23%), vomiting (23%), abdominal pain (12%), anorexia (11%), dyspepsia (11%), thrombocytopenia (24%), ecchymosis (5%), weight gain (9%), peripheral edema (8%), tremor (57%), somnolence (30%), dizziness (18%), insomnia (15%), nervousness (11%), amnesia (7%), nystagmus (7%), depression (5%), infection (20%), pharyngitis (8%), 210 dyspnea (5%), alopecia (24%), amblyopia/blurred vision (8%), and tinnitus (7%).
A retro Effects of radioiodine treatment on salivary gland spective analysis of facial fracture etiologies purchase lumigan 3 ml on line treatment room. Ann function in patients with differentiated thyroid car Plast Surg 2008; 60: 398–403 buy lumigan 3ml online symptoms 13dpo. Benign bone-forming lesions: osteoma buy 3 ml lumigan visa xerogenic medications, expert consensus approach in the Sjogren’s osteoid osteoma trusted lumigan 3ml treatment xanthoma, and osteoblastoma. Clinical, ima International Collaborative Clinical Alliance ging, pathologic, and differential considerations. Langerhans review of the literature, and discussion of its rela cell histiocytosis: oral/periodontal involvement in tionship to osteoid osteoma of the jaws. Oral Med Oral Pathol Oral Radiol Endod 2006; 102: Campanacci M, Baldini N, Boriani S, et al. Osteosarcoma inci oropharyngeal carcinoma treated with intensity dence and survival rates from 1973 to 2004: data modulated radiotherapy. Mucormycosis of maxilla diagnostic accuracy of the tongue blade test: still following tooth extraction in immunocompetent useful as a screening tool for mandibular fractures There is, however, no logical myofascial pain, other suggestions include, for exam necessity for positive signs not to be found in other ple, persistent orofacial muscle pain. The restriction to temporalis and fascial orofacial pain as an overarching label, adhering masseter does exclude individuals who may have highly to the term myofascial in recognition of the lack of localized myalgia in other masticatory muscles, and to concrete evidence linking pain to specic structures or the clinician this will also appear as a needless restric tissues in the muscle. Several studies have headache: that is, similar criteria regarding episode fre addressed the issue of additional dynamic/static testing quency have been introduced. More research is needed before myofascial pains (occurring less than once a month), such testing can be recommended for general inclusion. Future studies using the proposed temporal descriptions, emphasizing the importance of standardi distinction between myofascial pains may reveal thera zation of palpation force and duration: 1 kg for 2 sec peutic implications. The same could be Based on the International Association for the Study of applied to other muscles in the orofacial region. In primary pain conditions, and accepted that both acute and chronic types of pain the specic aetiology or cause cannot be determined – in the orofacial muscles may be associated with this that is, they are idiopathic, although substantial knowl clinical phenomenon. The pathophysiological signicance of this secondary to, or caused by, another known medical remains unclear, as do the therapeutic implications; condition or cause. Consequently, all diagnoses of myofascial pain Disorders are used for the underlying disorders (tendo are subcategorized according to the presence or absence nitis, myositis and muscle spasm). International Headache Society 2020 182 Cephalalgia 40(2) jaw, chewing and/or yawning, etc. Pain in masticatory muscles, with or without functional impairment, not attributable to another disorder. Modied by jaw movement, function or parafunc primary myofascial orofacial pain, and criterion B tion. Episodes may be single or recurrent within any day, pain each lasting at least 30 minutes and with a total duration within the day of at least 2 hours. Report of pain at a site beyond the boundary of the b) provoked by palpation of the aected ten muscle (temporalis or masseter) being palpated. No report of pain at a site beyond the boundary of Description: the muscle (temporalis or masseter) being palpated. Pain of tendon origin, aected by jaw movement, func tion or parafunction and replicated by provocation testing of the relevant masticatory tendon. The temporalis tendon is a common site of Diagnostic criteria: tendonitis and may refer pain to the teeth and other nearby structures. Report of pain at a site beyond the boundary of the muscle (temporalis or masseter) being palpated. Evidence of causation demonstrated by at least two Myofascial pain caused by an underlying disorder of the following: (inammation, infection or muscle spasm). An underlying disorder known to be able to cause lution of the tendonitis 1 myofascial pain has been diagnosed D. Such that the patient describes a step-change in in the aected muscle(s) or tendon(s) intensity. Muscle spasm in one or more masticatory muscles 1,2 tion or infection: oedema, erythema and/or increased has been diagnosed temperature. Evidence of causation demonstrated by at least two trauma of the muscle or from infection, or chronically of the following: with autoimmune disease. Limited range of jaw movement in a direction that of the following: elongates the aected muscle(s) is diagnostic: for 1. Comorbidity negatively influences the outcomes of creatine kinase), markers of inammation and the diagnostic tests for musculoskeletal pain in the oro presence of autoimmune diseases. Pain caused by sudden, involuntary, reversible tonic J Oral Rehabil 2018; 45: 185–190. Acute malocclusion may taxonomy of the diagnostic criteria for temporoman be present. The many accuracy of temporomandibular disorder pain faces of persistent orofacial muscle pain. Nevertheless, these subforms must, General comments: today, be regarded only as research topics and not con Like the myofascial pains, the temporomandibular joint sidered for clinical use. In primary pain conditions, the specic within the boundary of the aected anatomical struc aetiology or cause cannot be determined.
M More explicitly generic lumigan 3ml medicine 93 3109, this term involves pairing the distribution f on the diagonal in M2 with the delta-function on the diagonal cheap 3 ml lumigan medications 4 less. As U varies cheap lumigan 3 ml line medicine hat weather, this construction produces a factorization algebra on M lumigan 3 ml on-line medications migraine headaches, which we call the factorization algebra of classical observables associated to the action functional S. This result is a version of elliptic regularity, and we prove it later, in Chapter 4. If S is the action functional for the free eld theory, then we have a factorization algebra of classical observables. First, we can deform it into the factorization algebra of quantum observables for a free theory. Second, we can deform it into the factorization algebra of classical observables for an interacting eld theory. To construct the observables of an interacting eld theory, we use the renormal ization technique of Costello (2011b). In Costello (2011b), the rst author gives a denition of a quantum eld theory and a cohomological method for constructing eld theories. These divergence operators, for varying L, can be conjugated to each other by continuous linear isomorphisms of Vect (M)) and P(C(M)). However, roughly speaking, for L very small, the op c erator Div[L] only increases the support of a vector eld in Vect (c U)) by a small amount outside of U. This property turns out to be enough to dene the factorization algebra of quantum ob servables. This construction of quantum observables for an interacting eld theory is given in Volume 2, which is the most technically dicult part of this work. With the denition in hand, we proceed to examine several examples that arise naturally in mathematics. In particular, we explain how associative algebras can be viewed as prefactorization algebras on the real line, and when the converse holds. We also explain how to construct a prefactorization algebra from a sheaf of Lie algebras on a manifold M. This construction is called the factorization envelope, and it is related to the universal enveloping algebra of a Lie algebra as well as to Beilinson-Drinfeld’s notion of a chiral envelope. Although the factorization envelope construction is very simple, it plays an important role in eld theory. For example, the factorization algebra for any free theories is a factorization envelope, as is the factorization algebra corresponding to the Kac-Moody vertex algebra. More generally, factorization envelopes play an important role in our formulation of Noether’s theorem for quantum eld theories. Finally, when the manifold M is equipped with an action of a group G, we describe what a G-equivariant prefactorization algebra is. We will use this notion later in studying translation-invariant eld theories (see Section 8) and holomor phically translation-invariant eld theories (see Chapter 5). Prefactorization algebras In this section we will give a formal denition of the notion of a prefactor ization algebra, starting concretely and then generalizing. In the rst subsection, using plain language, we describe a prefactorization algebra taking values in vec tor spaces. The reader is free to generalize by replacing “vector space” and “linear map” with “object of a symmetric monoidal category C” and “morphism in C. In the nal subsections, we describe the category (and multicategory) of such prefactorization algebras. A pref actorization algebra F on M, taking values in vector spaces, is a rule that assigns a vector space F (U) to each open set U M along with the following maps and compatibilities. Un • the maps are compatible in the obvious way, so that if Ui,1t· · ·tUi,ni Vi and V1 t · · · t Vk W, the following diagram commutes. Nk Nn Nk i i=1 j=1F (U j) i=1F (Vi) F (W) Thus F resembles a precosheaf, except that we tensor the vector spaces rather than take their direct sum. V1 U1,2 W F (U1,1) F (U1,2) F (U2,1) U1,1 V2 F (V1) F (V2) F (W) U2,1 the case of k = n1 = 2, n2 = 1 these axioms imply that F is a commutative algebra. We say that F is a unital prefactorization algebra if F is a unital commutative algebra. In this case, F (U) is a pointed vector space by the image of the unit 1 F under the structure map F > F (U). Every associative algebra A denes a prefactorization algebra Afact on R, as follows. To any open set U = j Ij, N where each Ij is an open interval, we set F (U) = j A. The prefactorization algebra Afact of an associative al gebra A tensor products of unital algebras, as follows. Given an innite set I, consider the poset of nite subsets of I, ordered by inclusion. The structure maps F(U) > F(U t V) and F(U) > F(U t V) induce a canonical map F(U) F(V) > F(U t V), and so we obtain a natural map Sym(F(U) F(V)) > Sym F(U t V). But F (U) F (V) Sym(F(U)) Sym(F(V)) Sym(F(U) F(V)), so there is a natural map F (U) F (V) > F (U t V). In a similar way, one can provide all the structure maps to make F a prefactoriza tion algebra. We now pro vide a succinct and general denition of a prefactorization algebra using the e cient language of multicategories. If the U are pairwise disjoint and all are contained in V, then the set of maps is a single point.
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