Examples
- The partial group algebra is isomorphic to the direct sum: [1]
In mathematics, a partial group algebra is an associative algebra related to the partial representations of a group.
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.
In mathematics, especially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset of the quaternions under multiplication. It is given by the group presentation
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices.
In mathematics the spin group, denoted Spin(n), is a Lie group whose underlying manifold is the double cover of the special orthogonal group SO(n) = SO(n, R), such that there exists a short exact sequence of Lie groups (when n ≠ 2)
In algebraic geometry, motives is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically, a "motif" is the "cohomology essence" of a variety.
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theory. They also give rise to very interesting examples of Riemannian manifolds and hence are objects of interest in differential geometry and topology. Finally, these two topics join in the theory of automorphic forms which is fundamental in modern number theory.
In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition:
In algebra, a split complex number is based on a hyperbolic unitj satisfying A split-complex number has two real number components x and y, and is written The conjugate of z is Since the product of a number z with its conjugate is an isotropic quadratic form.
In theoretical physics and mathematics, a Wess–Zumino–Witten (WZW) model, also called a Wess–Zumino–Novikov–Witten model, is a type of two-dimensional conformal field theory named after Julius Wess, Bruno Zumino, Sergei Novikov and Edward Witten. A WZW model is associated to a Lie group, and its symmetry algebra is the affine Lie algebra built from the corresponding Lie algebra. By extension, the name WZW model is sometimes used for any conformal field theory whose symmetry algebra is an affine Lie algebra.
In mathematics, the Grothendieck group, or group of differences, of a commutative monoid M is a certain abelian group. This abelian group is constructed from M in the most universal way, in the sense that any abelian group containing a homomorphic image of M will also contain a homomorphic image of the Grothendieck group of M. The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation.
In mathematics, the Picard group of a ringed space X, denoted by Pic(X), is the group of isomorphism classes of invertible sheaves (or line bundles) on X, with the group operation being tensor product. This construction is a global version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds.
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more elementary kind of structure, the abelian group. The direct sum of two abelian groups and is another abelian group consisting of the ordered pairs where and . To add ordered pairs, we define the sum to be ; in other words addition is defined coordinate-wise. For example, the direct sum , where is real coordinate space, is the Cartesian plane, . A similar process can be used to form the direct sum of two vector spaces or two modules.
In differential geometry, a discipline within mathematics, a distribution on a manifold is an assignment of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle .
In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise statements to be made and eliminates many exceptional cases.
In mathematics, a super vector space is a -graded vector space, that is, a vector space over a field with a given decomposition of subspaces of grade and grade . The study of super vector spaces and their generalizations is sometimes called super linear algebra. These objects find their principal application in theoretical physics where they are used to describe the various algebraic aspects of supersymmetry.
In mathematics, the projective unitary groupPU(n) is the quotient of the unitary group U(n) by the right multiplication of its center, U(1), embedded as scalars. Abstractly, it is the holomorphic isometry group of complex projective space, just as the projective orthogonal group is the isometry group of real projective space.
In the mathematical surgery theory the surgery exact sequence is the main technical tool to calculate the surgery structure set of a compact manifold in dimension . The surgery structure set of a compact -dimensional manifold is a pointed set which classifies -dimensional manifolds within the homotopy type of .
In mathematics, the Satake isomorphism, introduced by Ichirō Satake, identifies the Hecke algebra of a reductive group over a local field with a ring of invariants of the Weyl group. The geometric Satake equivalence is a geometric version of the Satake isomorphism, proved by Ivan Mirković and Kari Vilonen.
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