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1. Representation theory of semisimple Lie algebras: In mathematics, the representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory was worked out mainly by E. [100%] 2024-06-03 [Lie algebras]

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1. Representation theory of Hopf algebras: In abstract algebra, a representation of a Hopf algebra is a representation of its underlying associative algebra. That is, a representation of a Hopf algebra H over a field K is a K-vector space V with an action H ... [87%] 2023-11-22 [Hopf algebras] [Representation theory]...
2. Representation of a Lie algebra: in a vector space $ V $ A homomorphism $ \rho $ of a Lie algebra $ L $ over a field $ k $ into the algebra $ \mathfrak g \mathfrak l ( V) $ of all linear transformations of $ V $ over $ k $. Two representations $ \rho _ {1} : L \rightarrow ... (Mathematics) [83%] 2023-10-26
3. Lie algebra, algebraic: The Lie algebra of an algebraic subgroup (see Algebraic group) of the general linear group of all automorphisms of a finite-dimensional vector space $V$ over a field $k$. If $\mathfrak g$ is an arbitrary subalgebra of the Lie algebra ... (Mathematics) [82%] 2023-10-17
4. Algebra representation: In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. [79%] 2023-12-11 [Algebras] [Module theory]...
5. Representation theory: Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more ... (Branch of mathematics that studies abstract algebraic structures) [79%] 2023-12-16 [Representation theory]
6. Representation theory: A theory studying homomorphisms of semi-groups (in particular, groups), algebras or other algebraic systems into corresponding endomorphism systems of suitable structures. Most often one considers linear representations, i.e. (Mathematics) [79%] 2023-05-22
7. Lie-Algebra: Eine Lie-Algebra (auch Liesche Algebra), benannt nach Sophus Lie, ist eine algebraische Struktur, die mit einer Lie-Klammer versehen ist, d. h., es existiert eine antisymmetrische Verknüpfung, die die Jacobi-Identität erfüllt. [70%] 2024-01-20
8. Lie algebra: In mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space \displaystyle{ \mathfrak g }[/math] together with an operation called the Lie bracket, an alternating bilinear map \displaystyle{ \mathfrak g \times \mathfrak g \rightarrow \mathfrak g }[/math], that satisfies the ... (Algebraic structure used in analysis) [70%] 2024-01-01 [Lie groups] [Lie algebras]...
9. Lie- algebra: In mathematics, a Lie-* algebra is a D-module with a Lie* bracket. They were introduced by Alexander Beilinson and Vladimir Drinfeld ((Beilinson Drinfeld)), and are similar to the conformal algebras discussed by (Kac 1998) and to vertex Lie algebras. [70%] 2023-01-25 [Lie algebras]
10. Lie algebra: A Lie algebra is an easy example of an algebraic structure that is not associative. Lie algebras describe infinitesimal symmetries or transformations. [70%] 2023-07-30 [Physics] [Mathematics]...
11. Lie algebra: A Lie algebra is a unitary $k$-module $L$ over a commutative ring $k$ with a unit that is endowed with a bilinear mapping $(x,y)\mapsto [x,y]$ of $L\times L$ into $L$ having the following two properties ... (Mathematics) [70%] 2024-01-01
12. Weight of a representation of a Lie algebra: in a vector space $ V $ A linear mapping $ \alpha $ from the Lie algebra $ L $ into its field of definition $ k $ for which there exists a non-zero vector $ x $ of $ V $ such that for the representation $ \rho $ one has $$ ( \rho ... (Mathematics) [69%] 2023-10-22
13. Emulation theory of representation: The emulation theory of representation postulates that there are multiple internal modeling circuitries in the brain referred to as emulators. These emulators mimic the input-output patterns of many cognitive operations including action, perception, and imagery. [68%] 2024-01-08 [Neuroscience]
14. Glossary of representation theory: This is a glossary of representation theory in mathematics. The term "module" is often used synonymously for a representation; for the module-theoretic terminology, see also glossary of module theory. (none) [68%] 2023-12-19 [Glossaries of mathematics] [Representation theory]...
15. Lie algebra of an algebraic group: l0584801.png ~/encyclopedia/old_files/data/L058/L.0508480 104 0 104 The analogue of the Lie algebra of an analytic group, which relates to the case of affine algebraic groups. As in the analytic case, the Lie algebra of an ... (Mathematics) [68%] 2023-12-15
16. Representation theorem: In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to another (abstract or concrete) structure. (Proof that every structure with certain properties is isomorphic to another structure) [67%] 2023-12-14 [Mathematical theorems]
17. Algebraic algebra: An algebra with associative powers (in particular, an associative algebra) over a field in which all elements are algebraic: an element $a$ of the algebra $A$ is called algebraic over the field $F$ if the subalgebra $F$ generated by $a ... (Mathematics) [66%] 2023-10-13
18. Lie algebras, variety of: over a ring $ k $ A class $ \mathfrak V $ of Lie algebras (cf. Lie algebra) over $ k $ that satisfy a fixed system of identities. (Mathematics) [66%] 2024-01-01
19. Glossary of Lie algebras: This is a glossary for the terminology applied in the mathematical theories of Lie algebras. The statements in this glossary mainly focus on the algebraic sides of the concepts, without referring to Lie groups or other related subjects. [66%] 2024-01-01 [Glossaries of mathematics] [Lie algebras]...