Quantum enveloping algebra

Last updated August 11, 2019

In mathematics, a quantum or quantized enveloping algebra is a q-analog of a universal enveloping algebra.^[1] Given a Lie algebra ${\mathfrak {g}}$ , the quantum enveloping algebra is typically denoted as $U_{q}({\mathfrak {g}})$ . Among the applications, studying the $q\to 0$ limit led to the discovery of crystal bases.

In mathematics, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as $q \to 1$ . Typically, mathematicians are interested in q-analogs that arise naturally, rather than in arbitrarily contriving q-analogs of known results. The earliest q-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century.

In mathematics, a universal enveloping algebra is the most general algebra that contains all representations of a Lie algebra.

Lie algebra A vector space with an alternating binary operation satisfying the Jacobi identity.

In mathematics, a Lie algebra is a vector space $together with a non-associative operation called the Lie bracket, an alternating bilinear map, satisfying the Jacobi identity.$

The case of ${\mathfrak {sl}}_{2}$

Michio Jimbo considered the algebras with three generators related by the three commutators

Michio Jimbo is a Japanese mathematician, currently a professor at the Rikkyo University. He is a grandson of the linguist Kaku Jimbo.

[h,e]=2e,\ [h,f]=-2f,\ [e,f]=\sinh(\eta h)/\sinh \eta .

When $\eta \to 0$ , these reduce to the commutators that define the special linear Lie algebra ${\mathfrak {sl}}_{2}$ . In contrast, for nonzero $\eta$ , the algebra defined by these relations is not a Lie algebra but instead an associative algebra that can be regarded as a deformation of the universal enveloping algebra of ${\mathfrak {sl}}_{2}$ .^[2]

Special linear Lie algebra Lie algebra of traceless linear transformations; Lie algebra of the special linear group

In mathematics, the special linear Lie algebra of order n is the Lie algebra of $matrices with trace zero and with the Lie bracket . This algebra is well studied and understood, and is often used as a model for the study of other Lie algebras. The Lie group that it generates is the special linear group.$

In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication, and a scalar multiplication by elements in some field. The addition and multiplication operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a vector space over K. In this article we will also use the term K-algebra to mean an associative algebra over the field K. A standard first example of a K-algebra is a ring of square matrices over a field K, with the usual matrix multiplication.

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In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.

In mathematics, the Baker–Campbell–Hausdorff formula is the solution for $to the equation$

Lie algebra representation homomorphism of Lie algebras whose codomain is the endomorphism algebra of a vector space

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 in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space $together with a collection of operators on satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.$

In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebras with additional structure. In general, a quantum group is some kind of Hopf algebra. There is no single, all-encompassing definition, but instead a family of broadly similar objects.

In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form. Cartan connections describe the geometry of manifolds modelled on homogeneous spaces.

In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in.

In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A natural example of a differential field is the field of rational functions C(t) in one variable, over the complex numbers, where the derivation is the differentiation with respect to t.

The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear transformations, or unitary operators on some Hilbert space; it has a variety of representations. In any relativistically invariant physical theory, these representations must enter in some fashion; physics itself must be made out of them. Indeed, special relativity together with quantum mechanics are the two physical theories that are most thoroughly established, and the conjunction of these two theories is the study of the infinite-dimensional unitary representations of the Lorentz group. These have both historical importance in mainstream physics, as well as connections to more speculative present-day theories.

In differential geometry, a Lie algebra-valued form is a differential form with values in a Lie algebra. Such forms have important applications in the theory of connections on a principal bundle as well as in the theory of Cartan connections.

In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions, the superconformal algebra is infinite-dimensional. In higher dimensions, superconformal algebras are finite-dimensional and generate the superconformal group.

In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it's a set with a Lie algebra and a Lie coalgebra structure which are compatible.

In conformal field theory and representation theory, a W-algebra is an algebra that generalizes the Virasoro algebra. W-algebras were introduced by Alexander Zamolodchikov, and the name "W-algebra" comes from the fact that Zamolodchikov used the letter W for one of the elements of one of his examples.

Classical group groups representable as matrix groups over a real division associative algebra (reals, complexes, or quaternions) that preserve a certain bilinear form (symmetric, skew-symmetric, Hermitian, skew-Hermitian, etc.)

In mathematics, the classical groups are defined as the special linear groups over the reals $R$ , the complex numbers $C$ and the quaternions $H$ together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups.

In mathematics, the Littelmann path model is a combinatorial device due to Peter Littelmann for computing multiplicities without overcounting in the representation theory of symmetrisable Kac–Moody algebras. Its most important application is to complex semisimple Lie algebras or equivalently compact semisimple Lie groups, the case described in this article. Multiplicities in irreducible representations, tensor products and branching rules can be calculated using a coloured directed graph, with labels given by the simple roots of the Lie algebra.

In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G. The key examples are the matrix coefficients of the spherical principal series, the irreducible representations appearing in the decomposition of the unitary representation of G on L²(G/K). In this case the commutant of G is generated by the algebra of biinvariant functions on G with respect to K acting by right convolution. It is commutative if in addition G/K is a symmetric space, for example when G is a connected semisimple Lie group with finite centre and K is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the spectrum of the corresponding C* algebra generated by the biinvariant functions of compact support, often called a Hecke algebra. The spectrum of the commutative Banach *-algebra of biinvariant L¹ functions is larger; when G is a semisimple Lie group with maximal compact subgroup K, additional characters come from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series.

In mathematics, a quantum affine algebra is a Hopf algebra that is a q-deformation of the universal enveloping algebra of an affine Lie algebra. They were introduced independently by Drinfeld (1985) and Jimbo (1985) as a special case of their general construction of a quantum group from a Cartan matrix. One of their principal applications has been to the theory of solvable lattice models in quantum statistical mechanics, where the Yang-Baxter equation occurs with a spectral parameter. Combinatorial aspects of the representation theory of quantum affine algebras can be described simply using crystal bases, which correspond to the degenerate case when the deformation parameter q vanishes and the Hamiltonian of the associated lattice model can be explicitly diagonalized.

In algebra, the Nichols algebra of a braided vector space is a braided Hopf algebra which is denoted by $and named after the mathematician Warren Nichols. It takes the role of quantum Borel part of a pointed Hopf algebra such as a quantum groups and their well known finite-dimensional truncations. Nichols algebras can immediately be used to write down new such quantum groups by using the Radford biproduct.$

In the theory of Lie groups, Lie algebras and their representation theory, a Lie algebra extension $e$ is an enlargement of a given Lie algebra $g$ by another Lie algebra $h$ . Extensions arise in several ways. There is the trivial extension obtained by taking a direct sum of two Lie algebras. Other types are the split extension and the central extension. Extensions may arise naturally, for instance, when forming a Lie algebra from projective group representations. Such a Lie algebra will contain central charges.

References

↑ Kassel, Christian (1995), Quantum groups, Graduate Texts in Mathematics, 155, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94370-1, MR 1321145
↑ Jimbo, Michio (1985), "A $q$ -difference analogue of $U({\mathfrak {g}})$ and the Yang–Baxter equation", Letters in Mathematical Physics , 10 (1): 63–69, Bibcode:1985LMaPh..10...63J, doi:10.1007/BF00704588

Drinfel'd, V. G. (1987), "Quantum Groups", Proceedings of the International Congress of Mathematicians 986, American Mathematical Society, 1: 798–820

Vladimir Gershonovich Drinfeld, surname also romanized as Drinfel'd, is a Soviet-American mathematician, currently working at the University of Chicago.

The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.

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[kassel-1] Kassel, Christian (1995), Quantum groups, Graduate Texts in Mathematics, 155, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94370-1, MR 1321145

[jimbo-2] Jimbo, Michio (1985), "A $q$ -difference analogue of $U({\mathfrak {g}})$ and the Yang–Baxter equation", Letters in Mathematical Physics , 10 (1): 63–69, Bibcode:1985LMaPh..10...63J, doi:10.1007/BF00704588

Quantum enveloping algebra

Contents

The case of sl2{\displaystyle {\mathfrak {sl}}_{2}}

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References

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The case of ${\mathfrak {sl}}_{2}$