Artin–Tits group, the Glossary
In the mathematical area of group theory, Artin groups, also known as Artin–Tits groups or generalized braid groups, are a family of infinite discrete groups defined by simple presentations.[1]
Table of Contents
45 relations: Advances in Mathematics, Annals of Mathematics, Arrangement of hyperplanes, Automatic group, Braid group, CAT(k) space, Commentarii Mathematici Helvetici, Conjugacy problem, Coxeter group, Cyclic group, Direct product of groups, Egbert Brieskorn, Elementary abelian group, Emil Artin, Finiteness properties of groups, Free abelian group, Free group, Free product, Fundamental group, Garside element, Geometriae Dedicata, Graph (discrete mathematics), Graph of groups, Group (mathematics), Group cohomology, Group theory, HNN extension, Inventiones Mathematicae, Jacques Tits, Journal of Algebra, Journal of Pure and Applied Algebra, Journal of Topology, Kenneth Appel, Kyoji Saito, London Mathematical Society, Mathematische Annalen, Monoid, Non-commutative cryptography, Pierre Deligne, Presentation of a group, Subgroup series, Symmetric matrix, Torsion (algebra), Trace monoid, Word problem for groups.
- Braid groups
Advances in Mathematics
Advances in Mathematics is a peer-reviewed scientific journal covering research on pure mathematics.
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Annals of Mathematics
The Annals of Mathematics is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
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Arrangement of hyperplanes
In geometry and combinatorics, an arrangement of hyperplanes is an arrangement of a finite set A of hyperplanes in a linear, affine, or projective space S. Questions about a hyperplane arrangement A generally concern geometrical, topological, or other properties of the complement, M(A), which is the set that remains when the hyperplanes are removed from the whole space.
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Automatic group
In mathematics, an automatic group is a finitely generated group equipped with several finite-state automata.
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Braid group
In mathematics, the braid group on strands (denoted B_n), also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation (see); and in monodromy invariants of algebraic geometry. Artin–Tits group and braid group are braid groups.
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CAT(k) space
In mathematics, a \mathbf(k) space, where k is a real number, is a specific type of metric space.
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The Commentarii Mathematici Helvetici is a quarterly peer-reviewed scientific journal in mathematics.
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Conjugacy problem
In abstract algebra, the conjugacy problem for a group G with a given presentation is the decision problem of determining, given two words x and y in G, whether or not they represent conjugate elements of G. That is, the problem is to determine whether there exists an element z of G such that The conjugacy problem is also known as the transformation problem.
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Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors).
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Cyclic group
In abstract algebra, a cyclic group or monogenous group is a group, denoted Cn (also frequently \Zn or Zn, not to be confused with the commutative ring of p-adic numbers), that is generated by a single element.
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Direct product of groups
In mathematics, specifically in group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted.
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Egbert Brieskorn
Egbert Valentin Brieskorn (7 July 1936 in Rostock – 11 July 2013 in Bonn) was a German mathematician who introduced Brieskorn spheres and the Brieskorn–Grothendieck resolution.
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Elementary abelian group
In mathematics, specifically in group theory, an elementary abelian group is an abelian group in which all elements other than the identity have the same order.
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Emil Artin
Emil Artin (March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
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Finiteness properties of groups
In mathematics, finiteness properties of a group are a collection of properties that allow the use of various algebraic and topological tools, for example group cohomology, to study the group.
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Free abelian group
In mathematics, a free abelian group is an abelian group with a basis.
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Free group
In mathematics, the free group FS over a given set S consists of all words that can be built from members of S, considering two words to be different unless their equality follows from the group axioms (e.g. st.
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Free product
In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties, in the sense that any two homomorphisms from G and H into a group K factor uniquely through a homomorphism from to K.
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Fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space.
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Garside element
In mathematics, a Garside element is an element of an algebraic structure such as a monoid that has several desirable properties.
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Geometriae Dedicata
Geometriae Dedicata is a mathematical journal, founded in 1972, concentrating on geometry and its relationship to topology, group theory and the theory of dynamical systems.
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Graph (discrete mathematics)
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related".
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Graph of groups
In geometric group theory, a graph of groups is an object consisting of a collection of groups indexed by the vertices and edges of a graph, together with a family of monomorphisms of the edge groups into the vertex groups.
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Group (mathematics)
In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.
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Group cohomology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology.
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Group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
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HNN extension
In mathematics, the HNN extension is an important construction of combinatorial group theory.
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Inventiones Mathematicae
Inventiones Mathematicae is a mathematical journal published monthly by Springer Science+Business Media.
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Jacques Tits
Jacques Tits (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry.
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Journal of Algebra
Journal of Algebra (ISSN 0021-8693) is an international mathematical research journal in algebra.
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Journal of Pure and Applied Algebra
The Journal of Pure and Applied Algebra is a monthly peer-reviewed scientific journal covering that part of algebra likely to be of general mathematical interest: algebraic results with immediate applications, and the development of algebraic theories of sufficiently general relevance to allow for future applications.
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Journal of Topology
The Journal of Topology is a peer-reviewed scientific journal which publishes papers of high quality and significance in topology, geometry, and adjacent areas of mathematics.
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Kenneth Appel
Kenneth Ira Appel (October 8, 1932 – April 19, 2013) was an American mathematician who in 1976, with colleague Wolfgang Haken at the University of Illinois at Urbana–Champaign, solved one of the most famous problems in mathematics, the four-color theorem.
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Kyoji Saito
Kyōji Saitō (齋藤 恭司, Saitō Kyōji; born 25 December 1944) is a Japanese mathematician, specializing in algebraic geometry and complex analytic geometry.
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London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS).
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Mathematische Annalen
Mathematische Annalen (abbreviated as Math. Ann. or, formerly, Math. Annal.) is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann.
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Monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element.
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Non-commutative cryptography
Non-commutative cryptography is the area of cryptology where the cryptographic primitives, methods and systems are based on algebraic structures like semigroups, groups and rings which are non-commutative.
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Pierre Deligne
Pierre René, Viscount Deligne (born 3 October 1944) is a Belgian mathematician.
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Presentation of a group
In mathematics, a presentation is one method of specifying a group.
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Subgroup series
In mathematics, specifically group theory, a subgroup series of a group G is a chain of subgroups: where 1 is the trivial subgroup.
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Symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose.
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Torsion (algebra)
In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring.
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Trace monoid
In computer science, a trace is a set of strings, wherein certain letters in the string are allowed to commute, but others are not.
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Word problem for groups
In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element of G. The word problem is a well-known example of an undecidable problem.
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See also
Braid groups
- Affine braid group
- Artin–Tits group
- Braid group
- Braids, Links, and Mapping Class Groups
- Burau representation
- Dehornoy order
- Double affine braid group
- Group-based cryptography
- Lawrence–Krammer representation
- Loop braid group
- Matsumoto's theorem (group theory)
- Spherical braid group
References
[1] https://en.wikipedia.org/wiki/Artin–Tits_group
Also known as Artin group, Artin group of finite type, Artin-Tits groups, Free partially commutative group, Graph group, Locally free group, Right angled Artin group, Right-angled Artin group, Semifree group, Trace group.