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Weyl's Complete Reducibility Theorem
In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations (specifically in the representation theory of semisimple Lie algebras). Let \mathfrak be a semisimple Lie algebra over a field of characteristic zero. The theorem states that every finite-dimensional module over \mathfrak is semisimple as a module (i.e., a direct sum of simple modules.) The enveloping algebra is semisimple Weyl's theorem implies (in fact is equivalent to) that the enveloping algebra of a finite-dimensional representation is a semisimple ring in the following way. Given a finite-dimensional Lie algebra representation \pi: \mathfrak \to \mathfrak(V), let A \subset \operatorname(V) be the associative subalgebra of the endomorphism algebra of ''V'' generated by \pi(\mathfrak g). The ring ''A'' is called the enveloping algebra of \pi. If \pi is semisimple, then ''A'' is semisimple. (Proof: Since ''A'' is a finite-dimensional algebra, it is an Artini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra Representation
In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space V together with a collection of operators on V satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators. The notion is closely related to that of a representation of a Lie group. Roughly speaking, the representations of Lie algebras are the differentiated form of representations of Lie groups, while the representations of the universal cover of a Lie group are the integrated form of the representations of its Lie algebra. In the study of representations of a Lie algebra, a particular ring, called the universal enveloping algebra, associated with the Lie algebra plays an important role. The univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Casimir Element
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group. The Casimir element is named after Hendrik Casimir, who identified them in his description of rigid body dynamics in 1931. Definition The most commonly-used Casimir invariant is the quadratic invariant. It is the simplest to define, and so is given first. However, one may also have Casimir invariants of higher order, which correspond to homogeneous symmetric polynomials of higher order. Quadratic Casimir element Suppose that \mathfrak is an n-dimensional Lie algebra. Let ''B'' be a nondegenerate bilinear form on \mathfrak that is invariant under the adjoint action of \mathfrak on itself, meaning that B(\operatorname_XY, Z) + B(Y, \operatorname_X Z) = 0 for all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nathan Jacobson
Nathan Jacobson (October 5, 1910 – December 5, 1999) was an American mathematician. Biography Born Nachman Arbiser in Warsaw, Jacobson emigrated to America with his family in 1918. He graduated from the University of Alabama in 1930 and was awarded a doctorate in mathematics from Princeton University in 1934. While working on his thesis, ''Non-commutative polynomials and cyclic algebras'', he was advised by Joseph Wedderburn. Jacobson taught and researched at Bryn Mawr College (1935–1936), the University of Chicago (1936–1937), the University of North Carolina at Chapel Hill (1937–1943), and Johns Hopkins University (1943–1947) before joining Yale University in 1947. He remained at Yale until his retirement. He was a member of the National Academy of Sciences and the American Academy of Arts and Sciences. He served as president of the American Mathematical Society from 1971 to 1973, and was awarded their highest honour, the Leroy P. Steele prize for lifetime achievemen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chevalley Generators
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such a Lie algebra \mathfrak g, if nonzero, the following conditions are equivalent: *\mathfrak g is semisimple; *the Killing form, κ(x,y) = tr(ad(''x'')ad(''y'')), is non-degenerate; *\mathfrak g has no non-zero abelian ideals; *\mathfrak g has no non-zero solvable ideals; * the radical (maximal solvable ideal) of \mathfrak g is zero. Significance The significance of semisimplicity comes firstly from the Levi decomposition, which states that every finite dimensional Lie algebra is the semidirect product of a solvable ideal (its radical) and a semisimple algebra. In particular, there is no nonzero Lie algebra that is both solvable and semisimple. Semisimple Lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie's Theorem
In mathematics, specifically the theory of Lie algebras, Lie's theorem states that, over an algebraically closed field of characteristic zero, if \pi: \mathfrak \to \mathfrak(V) is a finite-dimensional representation of a solvable Lie algebra, then there's a flag V = V_0 \supset V_1 \supset \cdots \supset V_n = 0 of invariant subspaces of \pi(\mathfrak) with \operatorname V_i = i, meaning that \pi(X)(V_i) \subseteq V_i for each X \in \mathfrak and ''i''. Put in another way, the theorem says there is a basis for ''V'' such that all linear transformations in \pi(\mathfrak) are represented by upper triangular matrices. This is a generalization of the result of Frobenius that commuting matrices are simultaneously upper triangularizable, as commuting matrices generate an abelian Lie algebra, which is a fortiori solvable. A consequence of Lie's theorem is that any finite dimensional solvable Lie algebra over a field of characteristic 0 has a nilpotent derived algebra (see #Consequen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Subalgebra
In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra \mathfrak is a maximal solvable subalgebra. The notion is named after Armand Borel. If the Lie algebra \mathfrak is the Lie algebra of a complex Lie group, then a Borel subalgebra is the Lie algebra of a Borel subgroup. Borel subalgebra associated to a flag Let \mathfrak g = \mathfrak(V) be the Lie algebra of the endomorphisms of a finite-dimensional vector space ''V'' over the complex numbers. Then to specify a Borel subalgebra of \mathfrak g amounts to specify a flag of ''V''; given a flag V = V_0 \supset V_1 \supset \cdots \supset V_n = 0, the subspace \mathfrak b = \ is a Borel subalgebra, and conversely, each Borel subalgebra is of that form by Lie's theorem. Hence, the Borel subalgebras are classified by the flag variety of ''V''. Borel subalgebra relative to a base of a root system Let \mathfrak g be a complex semisimple Lie algebra, \mathfrak h a Cartan subalgebra and ''R'' th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Submodule
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals contained between ''I'' and ''R''. Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields. In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one sided maximal ideal ''A'' is not necessarily two-sided, the quotient ''R''/''A'' is not necessarily a ring, but it is a simple module over ''R''. If ''R'' has a unique maximal right ideal, then ''R'' is known as a local ring, and the maximal right ideal is also the unique maxi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Verma Module
Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics. Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra. Specifically, although Verma modules themselves are infinite dimensional, quotients of them can be used to construct finite-dimensional representations with highest weight \lambda, where \lambda is dominant and integral. Their homomorphisms correspond to invariant differential operators over flag manifolds. Informal construction We can explain the idea of a Verma module as follows. Let \mathfrak be a semisimple Lie algebra (over \mathbb, for simplicity). Let \mathfrak be a fixed Cartan subalgebra of \mathfrak and let R be the associated root system. Let R^+ be a fixed set of positive roots. For each \alpha\in R^+, choose a nonzero element X_\alpha for the corresponding root space \mathfrak_\alpha and a nonzero element Y_\alpha in the root ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur's Lemma
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations of a group ''G'' and ''φ'' is a linear map from ''M'' to ''N'' that commutes with the action of the group, then either ''φ'' is invertible, or ''φ'' = 0. An important special case occurs when ''M'' = ''N'', i.e. ''φ'' is a self-map; in particular, any element of the center of a group must act as a scalar operator (a scalar multiple of the identity) on ''M''. The lemma is named after Issai Schur who used it to prove the Schur orthogonality relations and develop the basics of the representation theory of finite groups. Schur's lemma admits generalisations to Lie groups and Lie algebras, the most common of which are due to Jacques Dixmier and Daniel Quillen. Representation theory of groups Representation theory is the study of homomo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Enveloping Algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representation theory of Lie groups and Lie algebras. For example, Verma modules can be constructed as quotients of the universal enveloping algebra. In addition, the enveloping algebra gives a precise definition for the Casimir operators. Because Casimir operators commute with all elements of a Lie algebra, they can be used to classify representations. The precise definition also allows the importation of Casimir operators into other areas of mathematics, specifically, those that have a differential algebra. They also play a central role in some recent developments in mathematics. In particular, their dual provides a commutative example of the objects studied in non-commutative geometry, the quantum groups. This dual can be shown, by the Gelfand ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homological Algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through both their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences. It has played a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Cartan and H. Weyl and because of that, the theory is also known as the Cartan–Weyl theory. The theory gives the structural description and classification of a finite-dimensional representation of a semisimple Lie algebra (over \mathbb); in particular, it gives a way to parametrize (or classify) irreducible finite-dimensional representations of a semisimple Lie algebra, the result known as the theorem of the highest weight. There is a natural one-to-one correspondence between the finite-dimensional representations of a simply connected compact Lie group ''K'' and the finite-dimensional representations of the complex semisimple Lie algebra \mathfrak g that is the complexification of the Lie algebra of ''K'' (this fact is essentially a special case of the Lie group–Lie algebra correspondence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
