The canonical basis and the quantum Frobenius morphism

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• mémoire - matière potentielle : olivia rossi - doria

The canonical basis and the quantum Frobenius morphism Pierre Baumann Abstract The first goal of this paper is to study the amount of compatibility between two important constructions in the theory of quantized enveloping algebras, namely the canonical basis and the quantum Frobenius morphism. The second goal is to study orders with which the Kashiwara crystal B(∞) of a symmetrizable Kac-Moody algebra can be endowed; these orders are defined so that the transition matrices between bases naturally indexed by B(∞) are lower triangular.

• convex polytope
• algebra uq
• φi
• b′
• highest weight vector
• algebra
• elements
• basis
• weight
• sequence

Sujets

• Weight
• Elements
• Álgebra
• Polytope
• Vector
• Finite

Informations

• Publié par: romek
• Nombre de lectures: 32
• Langue: English

Extrait

Lesson Plan Teacher’s Name......................................................................................Date.......................................................................... Grade..................................Class(es)....................................................Date(s)...............................................M Tu W Th F Lesson 12.1 tives SE/TWE pp. 451–45 Adjec FOCUS Objectives:To identify predicate adjectives and a correctly to describe nouns and pronouns

Bellringer and Motivating Activity, TWE p.

TEACH Teaching Tip, TWE p. 451 Crossreference: Grammar, TWE p. 451 Exploring Language, TWE p. 452

PRACTICE AND ASSESS Answers: Exercises 1 and 2, TWE p. 452

CLOSE Close activity, TWE p. 452

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n 12.2 ives Lesso Articles and Proper Adject SE FOCUS Objectives:To recognize definite and indefinite a proper adjectives and understand how to form an

Bellringer and Motivating Activity, TWE p.

TEACH Grammar Tip, TWE p. 453 Crossreference: Mechanics, TWE p. 453 Critical Thinking, TWE p. 454

PRACTICE AND ASSESS Answers: Exercises 3, 4, 5, TWE p. 454

Additional Resources Bellringer Activities Transparencies Daily Language Practice Transparencies Grammar Practice,p. 20

CLOSE Close activity, TWE p. 454

96 Writer’s Choice:Lesson Plans,Grade

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Assessment Options Tests with Answer Key & Rubrics Unit 12 Pretest, pp. 45–46 Testmaker Unit 12 Pretest

Lesson Plan Teacher’s Name......................................................................................Date.......................................................................... Grade..................................Class(es).....................................................Date(s)...............................................M Tu W Th F Lesson 12.3 – Comparative and Superlative Ad FOCUS Objectives:To recognize and use comparative and superlativ purpose; to use comparative and superlative adjectives correc

Bellringer and Motivating Activity, TWE p. 455 TEACH Grammar Tip, TWE p. 455 TwoMinute Skill Drill, TWE p. 456 English Language Learners, TWE p. 456 PRACTICE AND ASSESS Additional Resources Bellringer Activities Transparencies Answers: Exercise 6, TWE p. 456 Daily Language Practice Transparencies TwoMinute Skill Drill Transparencies CLOSE Grammar Practice,p. 21 Close activity, TWE p. 456 Grammar Reteaching,p. 24 Grammar Enrichment,p. 21 Grammar Workbook,Lesson 30

Les parative and Superlative Adjectives son 12.4 More Com SE/TWE pp. 457–458 FOCUS Objectives:To identify irregular comparative and superlative negative comparative and superlative adjectives correctly; to forms properly in speaking and writing

Bellringer and Motivating Activity, TWE p. 457 TEACH Grammar Tip, TWE p. 457 TwoMinute Skill Drill, TWE p. 458 Critical Thinking, TWE p. 458 PRACTICE AND ASSESS Answers: Exercises 7 and 8, TWE p. 458 Additional Resources Bellringer Activities Transparencies Grammar Practice,p. 21 Daily Language Practice Transparencies Grammar Reteaching,p. 24 TwoMinute Skill Drill Transparencies Grammar Enrichment,p. 21 Grammar Workbook,Lesson 31 CLOSE Close activity, TWE p. 458

Lesson Plan Teacher’s Name......................................................................................Date.......................................................................... Grade..................................Class(es)....................................................Date(s)...............................................M Tu W Th F

Lesson 12.5 Demonstratives SE/TWE pp. 459–46 FOCUS Objectives:To identify demonstratives and to determine pronouns; to use demonstratives correctly in sentences

Bellringer and Motivating Activity, TWE p. 459

TEACH Teaching Tip, TWE p. 459 Writing in the Real World, TWE p. 460

PRACTICE AND ASSESS Answers: Exercises 9 and 10, TWE p. 460

Additional Resources Bellringer Activities Transparencies Daily Language Practice Transparencies Grammar Practice,p. 22

CLOSE Close activity, TWE p. 460

Grammar Reteaching,p. 25 Grammar Enrichment,p. 22 Grammar Workbook,Lesson 32

Lesson 1 Adver E pp. 461–462 2.6 bs SE/TW FOCUS Objectives:To identify adverbs and the words that they modify; to use adverbs correctly

Bellringer and Motivating Activity, TWE p. 461

TEACH Critical Thinking, TWE p. 461 Critical Thinking, TWE p. 462

PRACTICE AND ASSESS Answers: Exercises 11 and 12, TWE p. 462

Additional Resources Bellringer Activities Transparencies Daily Language Practice Transparencies Grammar Practice,p. 23

CLOSE Close activity, TWE p. 462

98 Writer’s Choice:Lesson Plans,Grade

Grammar Reteaching,p. 26 Grammar Enrichment,p. 23 Grammar Workbook,Lesson 33

Lesson Plan Teacher’s Name......................................................................................Date.......................................................................... Grade..................................Class(es).....................................................Date(s)...............................................M Tu W Th F Lesson 12.7 WE pp. 463–46 Intensifiers SE/T FOCUS Objectives:To identify intensifiers—adverbs that m intensifiers correctly

Bellringer and Motivating Activity, TWE p. 4

TEACH Grammar Tip, TWE p. 463 Exploring Language, TWE p. 464

PRACTICE AND ASSESS Answers: Exercises 13 and 14, TWE p. 464

Additional Resources Bellringer Activities Transparencies Daily Language Practice Transparencies Grammar Practice,p. 23

CLOSE Close activity, TWE p. 464

Grammar Reteaching,p. 26 Grammar Enrichment,p. 24 Grammar Workbook,Lesson 34

Lesson 12.8 Comparative and Superlative Adverbs SE/TWE pp. 465–466 FOCUS Objectives:To identify comparative and superlative comparative and superlative adverbs correctly

Bellringer and Motivating Activity, TWE p. 4

TEACH Teaching Tip, TWE p. 465 Cooperative Learning, TWE p. 465 TwoMinute Skill Drill, TWE p. 466 Cultural Connections, TWE p. 466

PRACTICE AND ASSESS Answers: Exercise 15, TWE p. 466

Additional Resources Bellringer Activities Transparencies Daily Language Practice Transparencies

CLOSE Close activity, TWE p. 466

TwoMinute Skill Drill Transparencies Grammar Workbook,Lesson 35

Lesson Plan Teacher’s Name......................................................................................Date.......................................................................... Grade..................................Class(es)....................................................Date(s)...............................................M Tu W Th F 12.9 Lesson Using Adverbs and Adjectives SE/– FOCUS Objectives:To distinguish between adverbs that follow verbs and predicate adjectives to add detail and to clarify writing a

Bellringer and Motivating Activity, TWE p. 467

TEACH Teaching Tip, TWE p. 467 Exploring Language,TWE p. 468

PRACTICE AND ASSESS Answers: Exercises 16 and 17, TWE p. 468

Additional Resources Bellringer Activities Transparencies Daily Language Practice Transparencies Grammar Practice,p. 25

CLOSE Close activity, TWE p. 468

Lesson 12.10

Grammar Reteaching,p. 27 Grammar Enrichment,p. 25 Grammar Workbook,Lesson 36

Avoiding Double Negatives SE/ TWE pp. 469–470

FOCUS Objectives:To identify negative words, including contracted correctly and to avoid double negatives in speaking and writi

Bellringer and Motivating Activity, TWE p. 469

TEACH Critical Thinking, TWE p. 469 LessProficient Readers, TWE p. 470 PRACTICE AND ASSESS Answers: Exercises 18 and 19, TWE p. 470 Additional Resources Bellringer Activities Transparencies Daily Language Practice Transparencies Grammar Practice,p. 26 CLOSE Close activity, TWE p. 470 100 Writer’Lesson Plans,s Choice:Grade

Grammar Reteaching,p. 28 Grammar Enrichment,p. 26 Grammar Workbook,Lesson 37

Lesson Plan Teacher’s Name......................................................................................Date.......................................................................... Grade..................................Class(es).....................................................Date(s)...............................................M Tu W Th F Unit 12 mmar Review SE/TWE pp. 471–47 Gra

TEACH About the Literature, TWE p. 471 Linking Grammar and Literature, TWE p. 471 Teaching Tip, TWE p. 471 Critical Thinking, TWE p. 471 Viewing the Art, TWE p. 475 PRACTICE AND ASSESS Answers: Exercises 1–8, TWE pp. 472–476 Additional Resources Grammar Workbook, Lessons 28–37;Unit 5 Review; Cumulative Review: Units 1–5 CLOSE Close activity, TWE p. 476 Writing Application p. 477 Adjectives in Writing, TWE p. 477 Techniques with Adjectives, TWE p. 477 Practice, TWE p. 477

Assessment Options Tests with Answer Key & Rubrics Unit 12 Mastery Test, pp. 47–48 Testmaker Unit 12 Mastery Test MindJogger Videoquizzes Interactive Tutor: Self Assessment,Unit 12