1.
E8 (mathematics)
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The E8 algebra is the largest and most complicated of these exceptional cases. Wilhelm Killing discovered the complex Lie algebra E8 during his classification of simple compact Lie algebras, though he did not prove its existence, Cartan determined that a complex simple Lie algebra of type E8 admits three real forms. Each of them rise to a simple Lie group of dimension 248. Chevalley introduced algebraic groups and Lie algebras of type E8 over other fields, for example, the Lie group E8 has dimension 248. Its rank, which is the dimension of its maximal torus, is 8, therefore, the vectors of the root system are in eight-dimensional Euclidean space, they are described explicitly later in this article. The Weyl group of E8, which is the group of symmetries of the maximal torus which are induced by conjugations in the group, has order 21435527 =696729600. There is a Lie algebra Ek for every integer k ≥3, there is a unique complex Lie algebra of type E8, corresponding to a complex group of complex dimension 248. The complex Lie group E8 of complex dimension 248 can be considered as a simple real Lie group of real dimension 496 and this is simply connected, has maximal compact subgroup the compact form of E8, and has an outer automorphism group of order 2 generated by complex conjugation. The split form, EVIII, which has maximal compact subgroup Spin/, EIX, which has maximal compact subgroup E7×SU/, fundamental group of order 2 and has trivial outer automorphism group. For a complete list of forms of simple Lie algebras. Over finite fields, the Lang–Steinberg theorem implies that H1=0, meaning that E8 has no twisted forms, the characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula. There are two non-isomorphic irreducible representations of dimension 8634368000, the fundamental representations are those with dimensions 3875,6696000,6899079264,146325270,2450240,30380,248 and 147250. The values at 1 of the Lusztig–Vogan polynomials give the coefficients of the matrices relating the standard representations with the irreducible representations. These matrices were computed after four years of collaboration by a group of 18 mathematicians and computer scientists, led by Jeffrey Adams, the most difficult case is the split real form of E8, where the largest matrix is of size 453060×453060. The Lusztig–Vogan polynomials for all other simple groups have been known for some time. The announcement of the result in March 2007 received extraordinary attention from the media, the representations of the E8 groups over finite fields are given by Deligne–Lusztig theory. One can construct the E8 group as the group of the corresponding e8 Lie algebra. This algebra has a 120-dimensional subalgebra so generated by Jij as well as 128 new generators Qa that transform as a Weyl–Majorana spinor of spin and it is then possible to check that the Jacobi identity is satisfied
2.
Cyclic group
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In algebra, a cyclic group or monogenous group is a group that is generated by a single element. Each element can be written as a power of g in multiplicative notation and this element g is called a generator of the group. Every infinite cyclic group is isomorphic to the group of Z. Every finite cyclic group of n is isomorphic to the additive group of Z/nZ. Every cyclic group is a group, and every finitely generated abelian group is a direct product of cyclic groups. A group G is called if there exists an element g in G such that G = ⟨g⟩ =. Since any group generated by an element in a group is a subgroup of that group, for example, if G = is a group of order 6, then g6 = g0, and G is cyclic. In fact, G is essentially the same as the set with addition modulo 6, for example,1 +2 ≡3 corresponds to g1 · g2 = g3, and 2 +5 ≡1 corresponds to g2 · g5 = g7 = g1, and so on. One can use the isomorphism χ defined by χ = i, the name cyclic may be misleading, it is possible to generate infinitely many elements and not form any literal cycles, that is, every gn is distinct. A group generated in this way is called a cyclic group. The French mathematicians known as Nicolas Bourbaki referred to a group as a monogenous group. The set of integers, with the operation of addition, forms a group and it is an infinite cyclic group, because all integers can be written as a finite sum or difference of copies of the number 1. In this group,1 and −1 are the only generators, every infinite cyclic group is isomorphic to this group. For every positive n, the set of integers modulo n, again with the operation of addition, forms a finite cyclic group. An element g is a generator of this group if g is relatively prime to n, thus, the number of different generators is φ, where φ is the Euler totient function, the function that counts the number of numbers modulo n that are relatively prime to n. Every finite cyclic group is isomorphic to a group Z/n, where n is the order of the group, the integer and modular addition operations, used to define the cyclic groups, are both the addition operations of commutative rings, also denoted Z and Z/n. If p is a prime, then Z/p is a finite field, every field with p elements is isomorphic to this one. For every positive n, the subset of the integers modulo n that are relatively prime to n, with the operation of multiplication
3.
Lorentz group
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In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz, in general relativity physics is that of special relativity in small enough regions of spacetime. The Lorentz group is a subgroup of the Poincaré group—the group of all isometries of Minkowski spacetime, Lorentz transformations are, precisely, isometries that leave the origin fixed. Thus, the Lorentz group is a subgroup of the isometry group of Minkowski spacetime. For this reason, the Lorentz group is called the homogeneous Lorentz group while the Poincaré group is sometimes called the inhomogeneous Lorentz group. Lorentz transformations are examples of linear transformations, general isometries of Minkowski spacetime are affine transformations. Mathematically, the Lorentz group may be described as the orthogonal group O. This quadratic form is, when put on matrix form, interpreted in physics as the tensor of Minkowski spacetime. The Lorentz group is a six-dimensional noncompact non-abelian real Lie group that is not connected, the four connected components are not simply connected, but rather doubly connected. The identity component of the Lorentz group is itself a group, and is called the restricted Lorentz group. The restricted Lorentz group consists of those Lorentz transformations that preserve the orientation of space, the restricted Lorentz group has often been presented through a facility of biquaternion algebra. The restricted Lorentz group arises in other ways in pure mathematics, for example, it arises as the point symmetry group of a certain ordinary differential equation. This fact also has physical significance, because it is a Lie group, the Lorentz group O is both a group and admits a topological description as a smooth manifold. As a manifold, it has four connected components, intuitively, this means that it consists of four topologically separated pieces. The subgroup of transformations is often denoted O+. Those that preserve orientation are called proper, and as linear transformations they have determinant +1, the subgroup of proper Lorentz transformations is denoted SO. The subgroup of all Lorentz transformations preserving both orientation and direction of time is called the proper, orthochronous Lorentz group or restricted Lorentz group, and is denoted by SO+. The set of the four connected components can be given a structure as the quotient group O/SO+
4.
Group theory
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In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra, linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is central to public key cryptography. The first class of groups to undergo a systematic study was permutation groups, given any set X and a collection G of bijections of X into itself that is closed under compositions and inverses, G is a group acting on X. If X consists of n elements and G consists of all permutations, G is the symmetric group Sn, in general, an early construction due to Cayley exhibited any group as a permutation group, acting on itself by means of the left regular representation. In many cases, the structure of a group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for n ≥5 and this fact plays a key role in the impossibility of solving a general algebraic equation of degree n ≥5 in radicals. The next important class of groups is given by matrix groups, here G is a set consisting of invertible matrices of given order n over a field K that is closed under the products and inverses. Such a group acts on the vector space Kn by linear transformations. In the case of groups, X is a set, for matrix groups. The concept of a group is closely related with the concept of a symmetry group. The theory of groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, the groups themselves may be discrete or continuous. Most groups considered in the first stage of the development of group theory were concrete, having been realized through numbers, permutations, or matrices. It was not until the nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by generators and relations, a significant source of abstract groups is given by the construction of a factor group, or quotient group, G/H, of a group G by a normal subgroup H. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory
5.
Modular group
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In mathematics, the modular group is the projective special linear group PSL of 2 x 2 matrices with integer coefficients and unit determinant. The matrices A and -A are identified, the group operation is function composition. This group of transformations is isomorphic to the special linear group PSL. In other words, PSL consists of all matrices where a, b, c, and d are integers, ad − bc =1, the group operation is the usual multiplication of matrices. Some authors define the group to be PSL, and still others define the modular group to be the larger group SL. Some mathematical relations require the consideration of the group GL of matrices with determinant plus or minus one, similarly, PGL is the quotient group GL/. A2 ×2 matrix with unit determinant is a matrix, and thus SL = Sp. The unit determinant of implies that the fractions a/b, a/c, c/d and b/d are all irreducible, more generally, if p/q is an irreducible fraction, then a p + b q c p + d q is also irreducible. Elements of the group provide a symmetry on the two-dimensional lattice. Let ω1 and ω2 be two numbers whose ratio is not real. Then the set of points Λ = is a lattice of parallelograms on the plane, a different pair of vectors α1 and α2 will generate exactly the same lattice if and only if = for some matrix in GL. It is for this reason that doubly periodic functions, such as elliptic functions, the action of the modular group on the rational numbers can most easily be understood by envisioning a square grid, with grid point corresponding to the fraction p/q. An irreducible fraction is one that is visible from the origin, the action of the group on a fraction never takes a visible to a hidden one. If p n −1 / q n −1 and p n / q n are two successive convergents of a fraction, then the matrix belongs to GL. In particular, if bc − ad =1 for positive integers a, b, c and d with a < b and c < d then a/b, important special cases of continued fraction convergents include the Fibonacci numbers and solutions to Pells equation. In both cases, the numbers can be arranged to form a subset of the modular group. Geometrically, S represents inversion in the unit followed by reflection with respect to the imaginary axis. The generators S and T obey the relations S2 =1 and 3 =1
6.
Dihedral group
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In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of groups, and they play an important role in group theory, geometry. The notation for the group of order n differs in geometry. In geometry, Dn or Dihn refers to the symmetries of the n-gon, in abstract algebra, Dn refers to the dihedral group of order n. The geometric convention is used in this article, a regular polygon with n sides has 2 n different symmetries, n rotational symmetries and n reflection symmetries. Usually, we take n ≥3 here. The associated rotations and reflections make up the dihedral group D n, if n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides, in either case, there are n axes of symmetry and 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes, as with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity, r1 and r2 denote counterclockwise rotations by 120° and 240° respectively, for example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°. The order of elements denoting the composition is right to left, the composition operation is not commutative. In all cases, addition and subtraction of subscripts are to be performed using modular arithmetic with modulus n, if we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of Dn as matrices, with composition being matrix multiplication and this is an example of a group representation. For example, the elements of the group D4 can be represented by the eight matrices. In general, the matrices for elements of Dn have the following form, rk is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk/n. Sk is a reflection across a line makes an angle of πk/n with the x-axis. Further equivalent definitions of Dn are, D1 is isomorphic to Z2, D2 is isomorphic to K4, the Klein four-group. D1 and D2 are exceptional in that, D1 and D2 are the only abelian dihedral groups, Dn is a subgroup of the symmetric group Sn for n ≥3