Algebras with Involution and Classical Groups
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The Book of Involutions
The Book of Involutions Max-Albert Knus Alexander Sergejvich Merkurjev Herbert Markus Rost Jean-Pierre Tignol @ @ @ @ @ @ @ @ The Book of Involutions Max-Albert Knus Alexander Merkurjev Markus Rost Jean-Pierre Tignol Author address: Dept. Mathematik, ETH-Zentrum, CH-8092 Zurich,¨ Switzerland E-mail address: [email protected] URL: http://www.math.ethz.ch/~knus/ Dept. of Mathematics, University of California at Los Angeles, Los Angeles, California, 90095-1555, USA E-mail address: [email protected] URL: http://www.math.ucla.edu/~merkurev/ NWF I - Mathematik, Universitat¨ Regensburg, D-93040 Regens- burg, Germany E-mail address: [email protected] URL: http://www.physik.uni-regensburg.de/~rom03516/ Departement´ de mathematique,´ Universite´ catholique de Louvain, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium E-mail address: [email protected] URL: http://www.math.ucl.ac.be/tignol/ Contents Pr´eface . ix Introduction . xi Conventions and Notations . xv Chapter I. Involutions and Hermitian Forms . 1 1. Central Simple Algebras . 3 x 1.A. Fundamental theorems . 3 1.B. One-sided ideals in central simple algebras . 5 1.C. Severi-Brauer varieties . 9 2. Involutions . 13 x 2.A. Involutions of the first kind . 13 2.B. Involutions of the second kind . 20 2.C. Examples . 23 2.D. Lie and Jordan structures . 27 3. Existence of Involutions . 31 x 3.A. Existence of involutions of the first kind . 32 3.B. Existence of involutions of the second kind . 36 4. Hermitian Forms . 41 x 4.A. Adjoint involutions . 42 4.B. Extension of involutions and transfer . -
Introduction to Linear Logic
Introduction to Linear Logic Beniamino Accattoli INRIA and LIX (École Polytechnique) Accattoli ( INRIA and LIX (École Polytechnique)) Introduction to Linear Logic 1 / 49 Outline 1 Informal introduction 2 Classical Sequent Calculus 3 Sequent Calculus Presentations 4 Linear Logic 5 Catching non-linearity 6 Expressivity 7 Cut-Elimination 8 Proof-Nets Accattoli ( INRIA and LIX (École Polytechnique)) Introduction to Linear Logic 2 / 49 Outline 1 Informal introduction 2 Classical Sequent Calculus 3 Sequent Calculus Presentations 4 Linear Logic 5 Catching non-linearity 6 Expressivity 7 Cut-Elimination 8 Proof-Nets Accattoli ( INRIA and LIX (École Polytechnique)) Introduction to Linear Logic 3 / 49 Quotation From A taste of Linear Logic of Philip Wadler: Some of the best things in life are free; and some are not. Truth is free. You may use a proof of a theorem as many times as you wish. Food, on the other hand, has a cost. Having baked a cake, you may eat it only once. If traditional logic is about truth, then Linear Logic is about food Accattoli ( INRIA and LIX (École Polytechnique)) Introduction to Linear Logic 4 / 49 Informally 1 Classical logic deals with stable truths: if A and A B then B but A still holds) Example: 1 A = ’Tomorrow is the 1st october’. 2 B = ’John will go to the beach’. 3 A B = ’If tomorrow is the 1st october then John will go to the beach’. So if tomorrow) is the 1st october, then John will go to the beach, But of course tomorrow will still be the 1st october. Accattoli ( INRIA and LIX (École Polytechnique)) Introduction to Linear Logic 5 / 49 Informally 2 But with money, or food, that implication is wrong: 1 A = ’John has (only) 5 euros’. -
Arxiv:1809.02384V2 [Math.OA] 19 Feb 2019 H Ruet Eaefloigaecmlctd N Oi Sntcerin Clear Not Is T It Saying So Worth and Is Complicated, It Are Following Hand
INVOLUTIVE OPERATOR ALGEBRAS DAVID P. BLECHER AND ZHENHUA WANG Abstract. Examples of operator algebras with involution include the op- erator ∗-algebras occurring in noncommutative differential geometry studied recently by Mesland, Kaad, Lesch, and others, several classical function alge- bras, triangular matrix algebras, (complexifications) of real operator algebras, and an operator algebraic version of the complex symmetric operators stud- ied by Garcia, Putinar, Wogen, Zhu, and others. We investigate the general theory of involutive operator algebras, and give many applications, such as a characterization of the symmetric operator algebras introduced in the early days of operator space theory. 1. Introduction An operator algebra for us is a closed subalgebra of B(H), for a complex Hilbert space H. Here we study operator algebras with involution. Examples include the operator ∗-algebras occurring in noncommutative differential geometry studied re- cently by Mesland, Kaad, Lesch, and others (see e.g. [26, 25, 10] and references therein), (complexifications) of real operator algebras, and an operator algebraic version of the complex symmetric operators studied by Garcia, Putinar, Wogen, Zhu, and many others (see [21] for a survey, or e.g. [22]). By an operator ∗-algebra we mean an operator algebra with an involution † † N making it a ∗-algebra with k[aji]k = k[aij]k for [aij ] ∈ Mn(A) and n ∈ . Here we are using the matrix norms of operator space theory (see e.g. [30]). This notion was first introduced by Mesland in the setting of noncommutative differential geometry [26], who was soon joined by Kaad and Lesch [25]. In several recent papers by these authors and coauthors they exploit operator ∗-algebras and involutive modules in geometric situations. -
Algebraic Groups I. Homework 8 1. Let a Be a Central Simple Algebra Over a field K, T a K-Torus in A×
Algebraic Groups I. Homework 8 1. Let A be a central simple algebra over a field k, T a k-torus in A×. (i) Adapt Exercise 5 in HW5 to make an ´etalecommutative k-subalgebra AT ⊆ A such that (AT )ks is × generated by T (ks), and establish a bijection between the sets of maximal k-tori in A and maximal ´etale commutative k-subalgebras of A. Deduce that SL(A) is k-anisotropic if and only if A is a division algebra. (ii) For an ´etalecommutative k-subalgebra C ⊆ A, prove ZA(C) is a semisimple k-algebra with center C. × (iv) If T is maximal as a k-split subtorus of A prove T is the k-group of units in AT and that the (central!) simple factors Bi of BT := ZA(AT ) are division algebras. (v) Fix A ' EndD(V ) for a right module V over a central division algebra D, so V is a left A-module and Q × V = Vi with nonzero left Bi-modules Vi. If T is maximal as a k-split torus in A , prove Vi has rank 1 × × over Bi and D, so Bi ' D. Using D-bases, deduce that all maximal k-split tori in A are A (k)-conjugate. 2. For a torus T over a local field k (allow R, C), prove T is k-anisotropic if and only if T (k) is compact. 3. Let Y be a smooth separated k-scheme locally of finite type, and T a k-torus with a left action on Y . -
Orthogonal Involutions and Totally Singular Quadratic Forms In
Orthogonal involutions and totally singular quadratic forms in characteristic two A.-H. Nokhodkar Abstract We associate to every central simple algebra with involution of ortho- gonal type in characteristic two a totally singular quadratic form which reflects certain anisotropy properties of the involution. It is shown that this quadratic form can be used to classify totally decomposable algebras with orthogonal involution. Also, using this form, a criterion is obtained for an orthogonal involution on a split algebra to be conjugated to the transpose involution. Mathematics Subject Classification: 16W10, 16K20, 11E39, 11E04. 1 Introduction The theory of algebras with involution is closely related to the theory of bilinear forms. Assigning to every symmetric or anti-symmetric regular bilinear form on a finite-dimensional vector space V its adjoint involution induces a one- to-one correspondence between the similarity classes of bilinear forms on V and involutions of the first kind on the endomorphism algebra EndF (V ) (see [5, p. 1]). Under this correspondence several basic properties of involutions reflect analogous properties of bilinear forms. For example a symmetric or anti- symmetric bilinear form is isotropic (resp. anisotropic, hyperbolic) if and only if its adjoint involution is isotropic (resp. anisotropic, hyperbolic). Let (A, σ) be a totally decomposable algebra with orthogonal involution over a field F of characteristic two. In [3], a bilinear Pfister form Pf(A, σ) was associated to (A, σ), which determines the isotropy behaviour of (A, σ). It was shown that for every splitting field K of A, the involution σK on AK becomes adjoint to the bilinear form Pf(A, σ)K (see [3, (7.5)]). -
Subfields of Central Simple Algebras
The Brauer Group Subfields of Algebras Subfields of Central Simple Algebras Patrick K. McFaddin University of Georgia Feb. 24, 2016 Patrick K. McFaddin University of Georgia Subfields of Central Simple Algebras The Brauer Group Subfields of Algebras Introduction Central simple algebras and the Brauer group have been well studied over the past century and have seen applications to class field theory, algebraic geometry, and physics. Since higher K-theory defined in '72, the theory of algebraic cycles have been utilized to study geometric objects associated to central simple algebras (with involution). This new machinery has provided a functorial viewpoint in which to study questions of arithmetic. Patrick K. McFaddin University of Georgia Subfields of Central Simple Algebras The Brauer Group Subfields of Algebras Central Simple Algebras Let F be a field. An F -algebra A is a ring with identity 1 such that A is an F -vector space and α(ab) = (αa)b = a(αb) for all α 2 F and a; b 2 A. The center of an algebra A is Z(A) = fa 2 A j ab = ba for every b 2 Ag: Definition A central simple algebra over F is an F -algebra whose only two-sided ideals are (0) and (1) and whose center is precisely F . Examples An F -central division algebra, i.e., an algebra in which every element has a multiplicative inverse. A matrix algebra Mn(F ) Patrick K. McFaddin University of Georgia Subfields of Central Simple Algebras The Brauer Group Subfields of Algebras Why Central Simple Algebras? Central simple algebras are a natural generalization of matrix algebras. -
I-Rings with Involution
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOL-SAL OF ALGEBRA 38, 85-92 (1976) I-Rings with Involution Department of Mathematics, l.~&e~sity of Southern California, Los Angeles, California 90007 Communicated ly I. :V. Herstein Received September 18, 1974 The conditions of van Keumann regukity, x-reguiarity, and being an I-ring are placed on symmetric subrings of a ring with involution, and n-e determine when the whole ring must satisfy the same property. It is shown that any symmetric subring must satisfy any one of these properties if the whole ring does. In this paper, we continue our investigations begun in [6], in which we determined conditions on the symmetric elements of a ring witn involution that force the ring to be an I-ring. In the spirit of [S], the conditions of van Neumann reguhxrity, r-regularity, and being an I-ring are assumed for certain subrings generated by symmetric elements, to see if these conditions are then implied for the whole ring. Any of these conditions on the ring wiil restrict to the subrings under consideration, but we can only show that the I-ring condition extends up. In addition, we prove that r-regularity extends to the whole ring in the presence of a polynomial identity, and that the regularity condition cannot in general, be extended to the whole ring. Throughout this work, R will denote an associative ring with involution *. Let S = {.x E R i x:” = x> be the symmetric elements of R, 2” = {x L .x* i x E R] I the set of traces, and A’ = (xxx 1 x E RI, the set of norms. -
Lecture 8: Counting
Lecture 8: Counting Nitin Saxena ? IIT Kanpur 1 Generating functions (Fibonacci) P i 1 We had obtained φ(t) = i≥0 Fit = 1−t−t2 . We can factorize the polynomial, 1 − t − t2 = (1 − αt)(1 − βt), where α; β 2 C. p p 1+ 5 1− 5 Exercise 1. Show that α = 2 ; β = 2 . We can simplify the formula a bit more, 1 c c φ(t) = = 1 + 2 = c (1 + αt + α2t2 + ··· ) + c (1 + βt + β2t2 + ··· ) : 1 − t − t2 1 − αt 1 − βt 1 2 We got the second equality by putting c = p1 α and c = − p1 β. The final expression gives us an explicit 1 5 2 5 formula for the Fibonacci sequence, 1 n+1 n+1 Fn = p (α − β ) : 5 This is actually a very strong method and can be used for solving linear recurrences of the kind, Sn = a1Sn−1 + ··· + akSn−k. Here k; a1; a2; ··· ; ak are constants. Suppose φ(t) is the recurrence for Sn, then by the above method, k−1 b1 + b2t + ··· + bkt c1 ck φ(t) = 2 k = + ··· + : 1 − a1t − a2t − · · · − akt 1 − α1t 1 − αkt k k−1 Where α1; ··· ; αk are the roots (assumed distinct) of polynomial x − a1x − · · · − ak = 0 (replace t 1 by x ). This is known as the characteristic polynomial of the recurrence. Exercise 2. How are the coefficients b1; ··· ; bk or c1; ··· ; ck determined? nta conditions. Initial n n So we get Sn = c1α1 + ··· + ckαk . Note 1. For a linear recurrence if Fn and Gn are solutions then their linear combinations aFn + bGn are n n also solutions. -
Differential Central Simple Algebras and Picard–Vessiot
Differential Central Simple Algebras and Picard–Vessiot Representations Lourdes Juan Department of Mathematics Texas Tech University Lubbock TX 79409 Andy R. Magid Department of Mathematics University of Oklahoma Norman OK 73019 March 7, 2007 Abstract A differential central simple algebra, and in particular a differential matrix algebra, over a differential field K with constants C can be trivial- ized by a Picard–Vessiot (differential Galois) extension E. In the matrix algebra case, there is a correspondence between K algebras trivialized by E and representations of the Galois group of E over K in P GLn(C), which can be interpreted as cocyles equivalent up to coboundaries. 1 Introduction Let K be a differential field with algebraically closed characteristic zero field of constants C and derivation D = DK . By a differential central simple al- gebra (DCSA) over K we mean a pair (A, D) where A is a central simple K algebra and D is a derivation of A extending the derivation D of its center K. Given two such pairs (A1, D1) and (A2, D2), we can form the differential algebra (A1 ⊗ A2, D1 ⊗ D2), where the derivation D1 ⊗ D2 is given by D1 ⊗ 1 + 1 ⊗ D2. Morphisms of DCSA’s over K are K algebra homomorphisms preserving deriva- tions. We denote the isomorphism class of (A, D) by [A, D]. The above tensor product defines a monoid operation on the isomorphism classes (the identity being the class [K, DK ]), and hence we can form the corresponding universal diff group K0 Az(K). (A central simple algebra (CSA) over a field is a special case of Azumaya, or central separable, algebras over rings; Az is an abbre- viation for Azumaya.) The corresponding object for K when derivations are not considered is the familiar K0Az(K) as defined by Bass in [1, Chapter III]. -
The Involution Tool for Accurate Digital Timingand Power Analysis Daniel Ohlinger, Jürgen Maier, Matthias Függer, Ulrich Schmid
The Involution Tool for Accurate Digital Timingand Power Analysis Daniel Ohlinger, Jürgen Maier, Matthias Függer, Ulrich Schmid To cite this version: Daniel Ohlinger, Jürgen Maier, Matthias Függer, Ulrich Schmid. The Involution Tool for Accu- rate Digital Timingand Power Analysis. PATMOS 2019 - 29th International Symposium on Power and Timing Modeling, Optimization and Simulation, Jul 2019, Rhodes, Greece. 10.1109/PAT- MOS.2019.8862165. hal-02395242 HAL Id: hal-02395242 https://hal.inria.fr/hal-02395242 Submitted on 5 Dec 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Involution Tool for Accurate Digital Timing and Power Analysis Daniel Ohlinger¨ ∗,Jurgen¨ Maier∗, Matthias Fugger¨ y, Ulrich Schmid∗ ∗ECS Group, TU Wien [email protected], jmaier, s @ecs.tuwien.ac.at f g yCNRS & LSV, ENS Paris-Saclay, Universite´ Paris-Saclay & Inria [email protected] Abstract—We introduce the prototype of a digital timing sim- in(t) ulation and power analysis tool for integrated circuit (Involution Tool) which employs the involution delay model introduced by t Fugger¨ et al. at DATE’15. Unlike the pure and inertial delay out(t) models typically used in digital timing analysis tools, the involu- tion model faithfully captures pulse propagation. -
On a Class of Inverse Problems for a Heat Equation with Involution Perturbation
On a class of inverse problems for a heat equation with involution perturbation Nasser Al-Salti∗y, Mokhtar Kiranez and Berikbol T. Torebekx Abstract A class of inverse problems for a heat equation with involution pertur- bation is considered using four different boundary conditions, namely, Dirichlet, Neumann, periodic and anti-periodic boundary conditions. Proved theorems on existence and uniqueness of solutions to these problems are presented. Solutions are obtained in the form of series expansion using a set of appropriate orthogonal basis for each problem. Convergence of the obtained solutions is also discussed. Keywords: Inverse Problems, Heat Equation, Involution Perturbation. 2000 AMS Classification: 35R30; 35K05; 39B52. 1. Introduction Differential equations with modified arguments are equations in which the un- known function and its derivatives are evaluated with modifications of time or space variables; such equations are called in general functional differential equa- tions. Among such equations, one can single out equations with involutions [?]. 1.1. Definition. [?, ?] A function α(x) 6≡ x that maps a set of real numbers, Γ, onto itself and satisfies on Γ the condition α (α(x)) = x; or α−1(x) = α(x) is called an involution on Γ: Equations containing involution are equations with an alternating deviation (at x∗ < x being equations with advanced, and at x∗ > x being equations with delay, where x∗ is a fixed point of the mapping α(x)). Equations with involutions have been studied by many researchers, for exam- ple, Ashyralyev [?, ?], Babbage [?], Przewoerska-Rolewicz [?, ?, ?, ?, ?, ?, ?], Aftabizadeh and Col. [?], Andreev [?, ?], Burlutskayaa and Col. [?], Gupta [?, ?, ?], Kirane [?], Watkins [?], and Wiener [?, ?, ?, ?]. Spectral problems and inverse problems for equations with involutions have received a lot of attention as well, see for example, [?, ?, ?, ?, ?], and equations with a delay in the space variable have been the subject of many research papers, see for example, [?, ?]. -
Lecture 2 Central Simple Algebras Patricio Quiroz
Lecture 2 Central Simple Algebras Patricio Quiroz Wesleyan University 04.08.2014 Remember that the theory of spinor genera is the \abelian part" in the problem of study the difference between local and global information, that is: gen(Λ) = GA:Λ j 0 spn(Λ) = GGA:Λ jj If G is not compact at some archimedean place. cls(Λ) = G:Λ where ∗ #fspinor genera in gen(Λ)g = jJK =K HA(Λ)j = [ΣΛ : K]; 1 n and HA(Λ) is the \image" of the spinor norm ΘA : GA ! JK =JK . This theory can be used to study representation problems as: Given a lattice M ⊂ Λ (Λ fixed of maximal rank), how many classes (orbits G:Λ) in the genus of Λ (in the orbit GA:Λ) contain an isomorphic copy of M? We will focus on the case of orders (lattices with additional structure) in central simple algebras (CSA's). So, we have to have some background material about CSA's and orders inside them. Motivation. Central simple algebras and orders appear in different places, for instance: 1. Number theory. Brauer groups play a central role in class field theory. 2. Theory of hyperbolic varieties. Arithmetical Kleinian and Fuchsian groups can be described in terms of maximal orders2. 1n is 2 in the cases of forms and the degree of the central simple algebra in that case. 2See for example, L.E. Arenas-Carmona, Representation fields for cyclic orders, Acta arithmetica, 156.2 (2012) 1 3. Theory of modular forms. Studying maximal orders in CSA's is one way to gen- eralize the classic theory of modular forms3.