AUTONOMOUS CATEGORIES 1. Introduction
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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
Pr´e-Publica¸c~oesdo Departamento de Matem´atica Universidade de Coimbra Preprint Number 19{32 CATEGORICAL ASPECTS OF CONGRUENCE DISTRIBUTIVITY MARINO GRAN, DIANA RODELO AND IDRISS TCHOFFO NGUEFEU Abstract: We give new characterisations of regular Mal'tsev categories with dis- tributive lattice of equivalence relations through variations of the so-called Tri- angular Lemma and Trapezoid Lemma in universal algebra. We then give new characterisations of equivalence distributive Goursat categories (which extend 3- permutable varieties) through variations of the Triangular and Trapezoid Lemmas involving reflexive and positive (compatible) relations. Keywords: Mal'tsev categories, Goursat categories, congruence modular varieties, congruence distributive varieties, Shifting Lemma, Triangular Lemma, Trapezoid Lemma. Math. Subject Classification (2010): 08C05, 08B05, 08A30, 08B10, 18C05, 18B99, 18E10. Introduction Regular Mal'tsev categories [7] extend 2-permutable varieties of univer- sal algebras, also including many examples which are not necessarily vari- etal, such as topological groups, compact groups, torsion-free groups and C∗-algebras, for instance. These categories have the property that any pair of (internal) equivalence relations R and S on the same object permute: RS = SR (see [6], for instance, and the references therein). It is well known that regular Mal'tsev categories have the property that the lattice of equiva- lence relations on any object is modular, so that they satisfy (the categorical version of) Gumm's Shifting Lemma [16]. More generally, this is the case for Goursat categories [8], which are those regular categories for which the composition of equivalence relations on the same object is 3-permutable: RSR = SRS. In [13] we proved that, for a regular category, the property of being a Mal'tsev category, or of being a Goursat category, can be both characterised Received 23rd September 2019. -
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. -
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)]). -
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. -
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. -
ROTA-BAXTER CATEGORIES Edmundo Castillo and Rafael Dıaz
International Electronic Journal of Algebra Volume 5 (2009) 27-57 ROTA-BAXTER CATEGORIES Edmundo Castillo and Rafael D´ıaz Received: 11 March 2008; Revised: 12 October 2008 Communicated by A. C¸i˘gdem Ozcan¨ Abstract. We introduce Rota-Baxter categories and construct examples of such structures. Mathematics Subject Classification (2000): 05A30, 18A99, 81Q30 Keywords: Rota-Baxter algebras, categorification of rings, categorical inte- gration. 1. Introduction This work takes part in the efforts to understand the categorification of rings and other related algebraic structures. The idea of categorification of algebraic structures has been around for several decades and has gradually become better appreciated and understood. The expanding scope and applications of the notion of categorification has been greatly influenced by the works of Baez-Dolan [2,3], Crane-Frenkel [10], Crane-Yetter [11], Khovanov [24], among others. The basic idea is that it is worthwhile to look at the categorical foundations of set theoretical structures. Often sets arise as the equivalences classes of objects in a category. Going from a category to the set of equivalences classes of its objects is the pro- cess of decategorification. Categorification goes in the reverse direction, uncovering categories whose set of equivalences classes of objects reproduces a given set. Cat- egorifications always exist but are no unique. Thus two general problems arise: the classification of categorifications and the extraction of information regarding a given set theoretical construction from its categorical counterpart. Our approach to the categorification of rings, reviewed in Section 2, was first discussed in [17] with a view towards the categorification of the ring of functions on non-commutative spaces and the categorification of the algebra of annihilation and creation operators. -
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, [?, ?]. -
An Intuitive Approach to Normal Operators
An Intuitive Approach to Normal Operators Benoit Dherin, Dan Sparks The question In this short document I will transcribe and slightly modify an e-mail sent to me by Benoit Dherin, which explains how one would be lead to the definition of a normal operator. The usual definitions are: the adjoint T ∗ of T is the unique operator satisfying hTv, wi = hv, T ∗wi for all v, w. A normal operator is one such that T ∗T = TT ∗. The question is, where on earth does this definition come from? Consider the following answers: (1) Because complex numbers satisfy zz = zz. (2) Because operators with an orthonormal eigenbasis satisfy TT ∗ = T ∗T. The reason that these answers do not address the question is because they are inherently unnatural. The point is that complex numbers, and operators with orthonormal eigenbases, have many proper- ties. Not every one of those properties is worth centering a definition around, so there should be a reason to look at normal operators in particular. Groundwork; discovering self-adjoint operators Let V be a finite dimensional complex inner product space. The most natural way to think about L(V) is as a noncommutative ring without unit (or as an analyst might simply say, a ring). The prototype of this type of ring is the set of n × n matrices, which L(V) “is,” up to a non-canonical isomorphism. (For each basis of V, there is an isomorphism L(V) Mn×n(C). Better, for each orthonormal basis of V, there is an isomorphism L(V) Mn×n(C) which respects the norm nicely (c.f. -
Solutions for Week Four
CS103A Handout 04S Winter 2020 Solutions for Week Four Problem One: Concept Checks i. Give two examples of binary relations over the set ℕ. One option is the the = relation. Another is ≤. There are tons of other ones with nice names (≡2, ≡3, the relation “divides,” etc.), and lots more that you could only specify with a picture. ii. What three properties must a binary relation have to have in order to be an equivalence relation? Give the first-order definitions of each of those properties. For each definition of a property, explain how you would write a proof that a binary relation R has that property. An equivalence relation is a relation that's reflexive, symmetric, and transitive. A binary relation R over a set A is reflexive if ∀a ∈ A. aRa. To prove that R is reflexive, we can choose an arbitrary a ∈ A and prove that aRa holds. The relation R is symmetric if ∀a ∈ A. ∀b ∈ A. (aRb → bRa). To prove that R is symmetric, we can choose an arbitrary a and b in A where aRb, then prove that bRa also holds. The relation R is transitive if ∀a ∈ A. ∀b ∈ A. ∀c ∈ A. (aRb ∧ bRc → aRc). To prove that R is transitive, we can choose an arbitrary a, b, and c in A where aRb and bRc, then prove that aRc also holds. iii. If R is an equivalence relation over a set A and a is an element of A, what does the notation [a]R mean? Intuitively, what does it represent? This is the equivalence class of a with respect to R.