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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, the term quantum group denotes one of a few different kinds of
noncommutative algebra In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not ...
s with additional structure. These include Drinfeld–Jimbo type quantum groups (which are
quasitriangular Hopf algebra In mathematics, a Hopf algebra, ''H'', is quasitriangularMontgomery & Schneider (2002), p. 72 if there exists an invertible element, ''R'', of H \otimes H such that :*R \ \Delta(x)R^ = (T \circ \Delta)(x) for all x \in H, where \Delta is the cop ...
s), compact matrix quantum groups (which are structures on unital separable C*-algebras), and bicrossproduct quantum groups. Despite their name, they do not themselves have a natural group structure, though they are in some sense 'close' to a group. The term "quantum group" first appeared in the theory of quantum integrable systems, which was then formalized by
Vladimir Drinfeld Vladimir Gershonovich Drinfeld ( uk, Володи́мир Ге́ршонович Дрінфельд; russian: Влади́мир Ге́ршонович Дри́нфельд; born February 14, 1954), surname also romanized as Drinfel'd, is a renowne ...
and
Michio Jimbo is a Japanese mathematician working in mathematical physics and is a professor of mathematics at Rikkyo University. He is a grandson of the linguist . Career After graduating from the University of Tokyo in 1974, he studied under Mikio Sato at t ...
as a particular class of Hopf algebra. The same term is also used for other Hopf algebras that deform or are close to classical
Lie groups In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
or
Lie algebras In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
, such as a "bicrossproduct" class of quantum groups introduced by
Shahn Majid Shahn Majid (born 1960 in Patna, Bihar, India) is an English pure mathematician and theoretical physicist, trained at Cambridge University and Harvard University and, since 2001, a Professor of Mathematics at the School of Mathematical Sciences, ...
a little after the work of Drinfeld and Jimbo. In Drinfeld's approach, quantum groups arise as Hopf algebras depending on an auxiliary parameter ''q'' or ''h'', which become
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 represent ...
s of a certain Lie algebra, frequently
semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...
or
affine Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine comb ...
, when ''q'' = 1 or ''h'' = 0. Closely related are certain dual objects, also Hopf algebras and also called quantum groups, deforming the algebra of functions on the corresponding semisimple
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...
or a
compact Lie group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
.


Intuitive meaning

The discovery of quantum groups was quite unexpected since it was known for a long time that compact groups and semisimple Lie algebras are "rigid" objects, in other words, they cannot be "deformed". One of the ideas behind quantum groups is that if we consider a structure that is in a sense equivalent but larger, namely a group algebra or a
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 represent ...
, then a group or enveloping algebra can be "deformed", although the deformation will no longer remain a group or enveloping algebra. More precisely, deformation can be accomplished within the category of Hopf algebras that are not required to be either
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
or
cocommutative In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagram ...
. One can think of the deformed object as an algebra of functions on a "noncommutative space", in the spirit of the
noncommutative geometry Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some g ...
of
Alain Connes Alain Connes (; born 1 April 1947) is a French mathematician, and a theoretical physicist, known for his contributions to the study of operator algebras and noncommutative geometry. He is a professor at the , , Ohio State University and Vand ...
. This intuition, however, came after particular classes of quantum groups had already proved their usefulness in the study of the quantum
Yang–Baxter equation In physics, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve the ...
and
quantum inverse scattering method In quantum physics, the quantum inverse scattering method is a method for solving integrable models in 1+1 dimensions, introduced by L. D. Faddeev in 1979. The quantum inverse scattering method relates two different approaches: #the Bethe an ...
developed by the Leningrad School (
Ludwig Faddeev Ludvig Dmitrievich Faddeev (also ''Ludwig Dmitriyevich''; russian: Лю́двиг Дми́триевич Фадде́ев; 23 March 1934 – 26 February 2017) was a Soviet and Russian mathematical physicist. He is known for the discovery of the ...
, Leon Takhtajan, Evgeny Sklyanin,
Nicolai Reshetikhin Nicolai Yuryevich Reshetikhin (russian: Николай Юрьевич Решетихин, born October 10, 1958 in Leningrad, Soviet Union) is a mathematical physicist, currently a professor of mathematics at Tsinghua University, China and a prof ...
and
Vladimir Korepin Vladimir E. Korepin (born 1951) is a professor at the C. N. Yang Institute of Theoretical Physics of the Stony Brook University. Korepin made research contributions in several areas of mathematics and physics. Educational background Korepin c ...
) and related work by the Japanese School. The intuition behind the second, bicrossproduct, class of quantum groups was different and came from the search for self-dual objects as an approach to quantum gravity.


Drinfeld–Jimbo type quantum groups

One type of objects commonly called a "quantum group" appeared in the work of Vladimir Drinfeld and Michio Jimbo as a deformation of the
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 represent ...
of a
semisimple Lie algebra 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 i ...
or, more generally, a
Kac–Moody algebra In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a g ...
, in the category of Hopf algebras. The resulting algebra has additional structure, making it into a
quasitriangular Hopf algebra In mathematics, a Hopf algebra, ''H'', is quasitriangularMontgomery & Schneider (2002), p. 72 if there exists an invertible element, ''R'', of H \otimes H such that :*R \ \Delta(x)R^ = (T \circ \Delta)(x) for all x \in H, where \Delta is the cop ...
. Let ''A'' = (''aij'') be the
Cartan matrix In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Ki ...
of the Kac–Moody algebra, and let ''q'' ≠ 0, 1 be a complex number, then the quantum group, ''Uq''(''G''), where ''G'' is the Lie algebra whose Cartan matrix is ''A'', is defined as the unital
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
with generators ''kλ'' (where ''λ'' is an element of the
weight lattice In the mathematical field of representation theory, a weight of an algebra ''A'' over a field F is an algebra homomorphism from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a multipli ...
, i.e. 2(λ, α''i'')/(α''i'', α''i'') is an integer for all ''i''), and ''ei'' and ''fi'' (for
simple root Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
s, α''i''), subject to the following relations: :\begin k_0 &= 1 \\ k_\lambda k_\mu &= k_ \\ k_\lambda e_i k_\lambda^ &= q^ e_i \\ k_\lambda f_i k_\lambda^ &= q^ f_i \\ \left _i, f_j \right &= \delta_ \frac && k_i = k_, q_i = q^ \\ \end And for ''i'' ≠ ''j'' we have the ''q''-Serre relations, which are deformations of the Serre relations: :\begin \sum_^ (-1)^n \frac e_i^n e_j e_i^ &= 0 \\ pt\sum_^ (-1)^n \frac f_i^n f_j f_i^ &= 0 \end where the
q-factorial In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer sym ...
, the
q-analog 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 . Typically, mathematicians are interested in ''q' ...
of the ordinary
factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \t ...
, is defined recursively using q-number: :\begin _! &= 1 \\ _! &= \prod_^n , && = \frac \end In the limit as ''q'' → 1, these relations approach the relations for the universal enveloping algebra ''U''(''G''), where :k_ \to 1, \qquad \frac \to t_\lambda and ''tλ'' is the element of the Cartan subalgebra satisfying (''tλ'', ''h'') = ''λ''(''h'') for all ''h'' in the Cartan subalgebra. There are various coassociative coproducts under which these algebras are Hopf algebras, for example, : \begin \Delta_1(k_\lambda) = k_\lambda \otimes k_\lambda & \Delta_1(e_i) = 1 \otimes e_i + e_i \otimes k_i & \Delta_1(f_i) = k_i^ \otimes f_i + f_i \otimes 1 \\ \Delta_2(k_\lambda) = k_\lambda \otimes k_\lambda & \Delta_2(e_i) = k_i^ \otimes e_i + e_i \otimes 1 & \Delta_2(f_i) = 1 \otimes f_i + f_i \otimes k_i \\ \Delta_3(k_\lambda) = k_\lambda \otimes k_\lambda & \Delta_3(e_i) = k_i^ \otimes e_i + e_i \otimes k_i^ & \Delta_3(f_i) = k_i^ \otimes f_i + f_i \otimes k_i^ \end where the set of generators has been extended, if required, to include ''kλ'' for ''λ'' which is expressible as the sum of an element of the weight lattice and half an element of the
root lattice In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation ...
. In addition, any Hopf algebra leads to another with reversed coproduct ''T'' o Δ, where ''T'' is given by ''T''(''x'' ⊗ ''y'') = ''y'' ⊗ ''x'', giving three more possible versions. The counit on ''U''''q''(''A'') is the same for all these coproducts: ''ε''(''kλ'') = 1, ''ε''(''ei'') = ''ε''(''fi'') = 0, and the respective
antipodes In geography, the antipode () of any spot on Earth is the point on Earth's surface diametrically opposite to it. A pair of points ''antipodal'' () to each other are situated such that a straight line connecting the two would pass through Ear ...
for the above coproducts are given by : \begin S_1(k_\lambda) = k_ & S_1(e_i) = - e_i k_i^ & S_1(f_i) = - k_i f_i \\ S_2(k_\lambda) = k_ & S_2(e_i) = - k_i e_i & S_2(f_i) = - f_i k_i^ \\ S_3(k_\lambda) = k_ & S_3(e_i) = - q_i e_i & S_3(f_i) = - q_i^ f_i \end Alternatively, the quantum group ''U''''q''(''G'') can be regarded as an algebra over the field C(''q''), the field of all
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
s of an indeterminate ''q'' over C. Similarly, the quantum group ''U''''q''(''G'') can be regarded as an algebra over the field Q(''q''), the field of all
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
s of an indeterminate ''q'' over Q (see below in the section on quantum groups at ''q'' = 0). The center of quantum group can be described by quantum determinant.


Representation theory

Just as there are many different types of representations for Kac–Moody algebras and their universal enveloping algebras, so there are many different types of representation for quantum groups. As is the case for all Hopf algebras, ''Uq''(''G'') has an
adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
on itself as a module, with the action being given by :\mathrm_x \cdot y = \sum_ x_ y S(x_), where :\Delta(x) = \sum_ x_ \otimes x_.


Case 1: ''q'' is not a root of unity

One important type of representation is a weight representation, and the corresponding
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
is called a weight module. A weight module is a module with a basis of weight vectors. A weight vector is a nonzero vector ''v'' such that ''kλ'' · ''v'' = ''dλv'' for all ''λ'', where ''dλ'' are complex numbers for all weights ''λ'' such that :d_0 = 1, :d_\lambda d_\mu = d_, for all weights ''λ'' and ''μ''. A weight module is called integrable if the actions of ''ei'' and ''fi'' are locally nilpotent (i.e. for any vector ''v'' in the module, there exists a positive integer ''k'', possibly dependent on ''v'', such that e_i^k.v = f_i^k.v = 0 for all ''i''). In the case of integrable modules, the complex numbers ''d''''λ'' associated with a weight vector satisfy d_\lambda = c_\lambda q^, where ''ν'' is an element of the weight lattice, and ''cλ'' are complex numbers such that :*c_0 = 1, :*c_\lambda c_\mu = c_, for all weights ''λ'' and ''μ'', :*c_ = 1 for all ''i''. Of special interest are highest-weight representations, and the corresponding highest weight modules. A highest weight module is a module generated by a weight vector ''v'', subject to ''k''''λ'' · ''v'' = ''dλv'' for all weights ''μ'', and ''ei'' · ''v'' = 0 for all ''i''. Similarly, a quantum group can have a lowest weight representation and lowest weight module, ''i.e.'' a module generated by a weight vector ''v'', subject to ''kλ'' · ''v'' = ''dλv'' for all weights ''λ'', and ''fi'' · ''v'' = 0 for all ''i''. Define a vector ''v'' to have weight ''ν'' if k_\lambda\cdot v = q^ v for all ''λ'' in the weight lattice. If ''G'' is a Kac–Moody algebra, then in any irreducible highest weight representation of ''U''''q''(''G''), with highest weight ν, the multiplicities of the weights are equal to their multiplicities in an irreducible representation of ''U''(''G'') with equal highest weight. If the highest weight is dominant and integral (a weight ''μ'' is dominant and integral if ''μ'' satisfies the condition that 2 (\mu,\alpha_i)/(\alpha_i,\alpha_i) is a non-negative integer for all ''i''), then the weight spectrum of the irreducible representation is invariant under the
Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections th ...
for ''G'', and the representation is integrable. Conversely, if a highest weight module is integrable, then its highest weight vector ''v'' satisfies k_\lambda\cdot v = c_\lambda q^ v, where ''c''''λ'' · ''v'' = ''d''''λ''''v'' are complex numbers such that :*c_0 = 1, :*c_\lambda c_\mu = c_, for all weights ''λ'' and ''μ'', :*c_ = 1 for all ''i'', and ''ν'' is dominant and integral. As is the case for all Hopf algebras, the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
of two modules is another module. For an element ''x'' of ''Uq(G)'', and for vectors ''v'' and ''w'' in the respective modules, ''x'' ⋅ (''v'' ⊗ ''w'') = Δ(''x'') ⋅ (''v'' ⊗ ''w''), so that k_\lambda\cdot(v \otimes w) = k_\lambda\cdot v \otimes k_\lambda.w, and in the case of coproduct Δ1, e_i\cdot(v \otimes w) = k_i\cdot v \otimes e_i\cdot w + e_i\cdot v \otimes w and f_i\cdot(v \otimes w) = v \otimes f_i\cdot w + f_i\cdot v \otimes k_i^\cdot w. The integrable highest weight module described above is a tensor product of a one-dimensional module (on which ''k''λ = ''c''''λ'' for all ''λ'', and ''ei'' = ''fi'' = 0 for all ''i'') and a highest weight module generated by a nonzero vector ''v''0, subject to k_\lambda\cdot v_0 = q^ v_0 for all weights ''λ'', and e_i\cdot v_0 = 0 for all ''i''. In the specific case where ''G'' is a finite-dimensional Lie algebra (as a special case of a Kac–Moody algebra), then the irreducible representations with dominant integral highest weights are also finite-dimensional. In the case of a tensor product of highest weight modules, its decomposition into submodules is the same as for the tensor product of the corresponding modules of the Kac–Moody algebra (the highest weights are the same, as are their multiplicities).


Case 2: ''q'' is a root of unity


Quasitriangularity


Case 1: ''q'' is not a root of unity

Strictly, the quantum group ''U''''q''(''G'') is not quasitriangular, but it can be thought of as being "nearly quasitriangular" in that there exists an infinite formal sum which plays the role of an ''R''-matrix. This infinite formal sum is expressible in terms of generators ''ei'' and ''fi'', and Cartan generators ''t''''λ'', where ''kλ'' is formally identified with ''q''''t''''λ''. The infinite formal sum is the product of two factors, :q^ and an infinite formal sum, where ''λ''''j'' is a basis for the dual space to the Cartan subalgebra, and ''μ''''j'' is the dual basis, and ''η'' = ±1. The formal infinite sum which plays the part of the ''R''-matrix has a well-defined action on the tensor product of two irreducible highest weight modules, and also on the tensor product of two lowest weight modules. Specifically, if ''v'' has weight ''α'' and ''w'' has weight ''β'', then :q^\cdot(v \otimes w) = q^ v \otimes w, and the fact that the modules are both highest weight modules or both lowest weight modules reduces the action of the other factor on ''v'' ⊗ ''W'' to a finite sum. Specifically, if ''V'' is a highest weight module, then the formal infinite sum, ''R'', has a well-defined, and
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
, action on ''V'' ⊗ ''V'', and this value of ''R'' (as an element of End(''V'' ⊗ ''V'')) satisfies the
Yang–Baxter equation In physics, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve the ...
, and therefore allows us to determine a representation of the
braid group A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strande ...
, and to define quasi-invariants for
knots A knot is a fastening in rope or interwoven lines. Knot may also refer to: Places * Knot, Nancowry, a village in India Archaeology * Knot of Isis (tyet), symbol of welfare/life. * Minoan snake goddess figurines#Sacral knot Arts, entertainme ...
, links and
braids A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strande ...
.


Case 2: ''q'' is a root of unity


Quantum groups at ''q'' = 0

Masaki Kashiwara is a Japanese mathematician. He was a student of Mikio Sato at the University of Tokyo. Kashiwara made leading contributions towards algebraic analysis, microlocal analysis, D-module, ''D''-module theory, Hodge theory, sheaf theory and represent ...
has researched the limiting behaviour of quantum groups as ''q'' → 0, and found a particularly well behaved base called a
crystal base A crystal base for a representation of a quantum group on a \Q(v)-vector space is not a base of that vector space but rather a \Q-base of L/vL where L is a \Q(v)-lattice in that vector spaces. Crystal bases appeared in the work of and also in the ...
.


Description and classification by root-systems and Dynkin diagrams

There has been considerable progress in describing finite quotients of quantum groups such as the above ''Uq''(g) for ''qn'' = 1; one usually considers the class of pointed
Hopf algebras Hopf is a German surname. Notable people with the surname include: * Eberhard Hopf (1902–1983), Austrian mathematician * Hans Hopf (1916–1993), German tenor *Heinz Hopf (1894–1971), German mathematician *Heinz Hopf (actor) (1934–2001), Swed ...
, meaning that all subcoideals are 1-dimensional and thus there sum form a group called coradical: * In 2002 H.-J. Schneider and N. Andruskiewitsch finished their classification of pointed Hopf algebras with an abelian co-radical group (excluding primes 2, 3, 5, 7), especially as the above finite quotients of ''Uq''(g) decompose into ''E''′s (Borel part), dual ''F''′s and ''K''′s (Cartan algebra) just like ordinary
Semisimple Lie algebra 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 i ...
s: ::\left(\mathfrak(V)\otimes k mathbf^notimes\mathfrak(V^*)\right)^\sigma :Here, as in the classical theory ''V'' is a braided vector space of dimension ''n'' spanned by the ''E''′s, and ''σ'' (a so-called cocylce twist) creates the nontrivial linking between ''E''′s and ''F''′s. Note that in contrast to classical theory, more than two linked components may appear. The role of the quantum Borel algebra is taken by a
Nichols algebra In algebra, the Nichols algebra of a braided vector space (with the braiding often induced by a finite group) is a braided Hopf algebra which is denoted by \mathfrak(V) and named after the mathematician Warren Nichols. It takes the role of quantum ...
\mathfrak(V) of the braided vectorspace. * A crucial ingredient was I. Heckenberger's classification of finite Nichols algebras for abelian groups in terms of generalized
Dynkin diagram In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
s. When small primes are present, some exotic examples, such as a triangle, occur (see also the Figure of a rank 3 Dankin diagram). * Meanwhile, Schneider and Heckenberger have generally proven the existence of an arithmetic
root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representati ...
also in the nonabelian case, generating a PBW basis as proven by Kharcheko in the abelian case (without the assumption on finite dimension). This can be usedHeckenberger, Schneider: Right coideal subalgebras of Nichols algebras and the Duflo order of the Weyl grupoid, 2009. on specific cases ''Uq''(g) and explains e.g. the numerical coincidence between certain coideal subalgebras of these quantum groups and the order of the
Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections th ...
of the
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
g.


Compact matrix quantum groups

S. L. Woronowicz Stanisław Lech Woronowicz (born 22 July 1941, Ukmergė, Lithuania) is a Polish mathematician and physicist. He is affiliated with the University of Warsaw and is a member of the Polish Academy of Sciences. Research Woronowicz and Erling Størmer ...
introduced compact matrix quantum groups. Compact matrix quantum groups are abstract structures on which the "continuous functions" on the structure are given by elements of a C*-algebra. The geometry of a compact matrix quantum group is a special case of a
noncommutative geometry Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some g ...
. The continuous complex-valued functions on a compact Hausdorff topological space form a commutative C*-algebra. By the Gelfand theorem, a commutative C*-algebra is isomorphic to the C*-algebra of continuous complex-valued functions on a compact Hausdorff topological space, and the topological space is uniquely determined by the C*-algebra up to
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
. For a compact
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
, ''G'', there exists a C*-algebra homomorphism Δ: ''C''(''G'') → ''C''(''G'') ⊗ ''C''(''G'') (where ''C''(''G'') ⊗ ''C''(''G'') is the C*-algebra tensor product - the completion of the algebraic tensor product of ''C''(''G'') and ''C''(''G'')), such that Δ(''f'')(''x'', ''y'') = ''f''(''xy'') for all ''f'' ∈ ''C''(''G''), and for all ''x'', ''y'' ∈ ''G'' (where (''f'' ⊗ ''g'')(''x'', ''y'') = ''f''(''x'')''g''(''y'') for all ''f'', ''g'' ∈ ''C''(''G'') and all ''x'', ''y'' ∈ ''G''). There also exists a linear multiplicative mapping ''κ'': ''C''(''G'') → ''C''(''G''), such that ''κ''(''f'')(''x'') = ''f''(''x''−1) for all ''f'' ∈ ''C''(''G'') and all ''x'' ∈ ''G''. Strictly, this does not make ''C''(''G'') a Hopf algebra, unless ''G'' is finite. On the other hand, a finite-dimensional representation of ''G'' can be used to generate a *-subalgebra of ''C''(''G'') which is also a Hopf *-algebra. Specifically, if g \mapsto (u_(g))_ is an ''n''-dimensional representation of ''G'', then for all ''i'', ''j'' ''uij'' ∈ ''C''(''G'') and :\Delta(u_) = \sum_k u_ \otimes u_. It follows that the *-algebra generated by ''uij'' for all ''i, j'' and ''κ''(''uij'') for all ''i, j'' is a Hopf *-algebra: the counit is determined by ε(''uij'') = δ''ij'' for all ''i, j'' (where ''δ''''ij'' is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
), the antipode is ''κ'', and the unit is given by :1 = \sum_k u_ \kappa(u_) = \sum_k \kappa(u_) u_.


General definition

As a generalization, a compact matrix quantum group is defined as a pair (''C'', ''u''), where ''C'' is a C*-algebra and u = (u_)_ is a matrix with entries in ''C'' such that :*The *-subalgebra, ''C''0, of ''C'', which is generated by the matrix elements of ''u'', is dense in ''C''; :*There exists a C*-algebra homomorphism called the comultiplication Δ: ''C'' → ''C'' ⊗ ''C'' (where ''C'' ⊗ ''C'' is the C*-algebra tensor product - the completion of the algebraic tensor product of ''C'' and ''C'') such that for all ''i, j'' we have: :::\Delta(u_) = \sum_k u_ \otimes u_ :*There exists a linear antimultiplicative map κ: ''C''0 → ''C''0 (the coinverse) such that ''κ''(''κ''(''v''*)*) = ''v'' for all ''v'' ∈ ''C''0 and :::\sum_k \kappa(u_) u_ = \sum_k u_ \kappa(u_) = \delta_ I, where ''I'' is the identity element of ''C''. Since κ is antimultiplicative, then ''κ''(''vw'') = ''κ''(''w'') ''κ''(''v'') for all ''v'', ''w'' in ''C''0. As a consequence of continuity, the comultiplication on ''C'' is coassociative. In general, ''C'' is not a bialgebra, and ''C''0 is a Hopf *-algebra. Informally, ''C'' can be regarded as the *-algebra of continuous complex-valued functions over the compact matrix quantum group, and ''u'' can be regarded as a finite-dimensional representation of the compact matrix quantum group.


Representations

A representation of the compact matrix quantum group is given by a corepresentation of the Hopf *-algebra (a corepresentation of a counital coassociative coalgebra ''A'' is a square matrix v = (v_)_ with entries in ''A'' (so ''v'' belongs to M(''n'', ''A'')) such that :\Delta(v_) = \sum_^n v_ \otimes v_ for all ''i'', ''j'' and ''ε''(''vij'') = δ''ij'' for all ''i, j''). Furthermore, a representation ''v'', is called unitary if the matrix for ''v'' is unitary (or equivalently, if κ(''vij'') = ''v*ij'' for all ''i'', ''j'').


Example

An example of a compact matrix quantum group is SUμ(2), where the parameter μ is a positive real number. So SUμ(2) = (C(SUμ(2)), ''u''), where C(SUμ(2)) is the C*-algebra generated by α and γ, subject to :\gamma \gamma^* = \gamma^* \gamma, :\alpha \gamma = \mu \gamma \alpha, :\alpha \gamma^* = \mu \gamma^* \alpha, :\alpha \alpha^* + \mu \gamma^* \gamma = \alpha^* \alpha + \mu^ \gamma^* \gamma = I, and :u = \left( \begin \alpha & \gamma \\ - \gamma^* & \alpha^* \end \right), so that the comultiplication is determined by ∆(α) = α ⊗ α − γ ⊗ γ*, ∆(γ) = α ⊗ γ + γ ⊗ α*, and the coinverse is determined by κ(α) = α*, κ(γ) = −μ−1γ, κ(γ*) = −μγ*, κ(α*) = α. Note that ''u'' is a representation, but not a unitary representation. ''u'' is equivalent to the unitary representation :v = \left( \begin \alpha & \sqrt \gamma \\ - \frac \gamma^* & \alpha^* \end \right). Equivalently, SUμ(2) = (C(SUμ(2)), ''w''), where C(SUμ(2)) is the C*-algebra generated by α and β, subject to :\beta \beta^* = \beta^* \beta, :\alpha \beta = \mu \beta \alpha, :\alpha \beta^* = \mu \beta^* \alpha, :\alpha \alpha^* + \mu^2 \beta^* \beta = \alpha^* \alpha + \beta^* \beta = I, and :w = \left( \begin \alpha & \mu \beta \\ - \beta^* & \alpha^* \end \right), so that the comultiplication is determined by ∆(α) = α ⊗ α − μβ ⊗ β*, Δ(β) = α ⊗ β + β ⊗ α*, and the coinverse is determined by κ(α) = α*, κ(β) = −μ−1β, κ(β*) = −μβ*, κ(α*) = α. Note that ''w'' is a unitary representation. The realizations can be identified by equating \gamma = \sqrt \beta. When μ = 1, then SUμ(2) is equal to the algebra ''C''(SU(2)) of functions on the concrete compact group SU(2).


Bicrossproduct quantum groups

Whereas compact matrix pseudogroups are typically versions of Drinfeld-Jimbo quantum groups in a dual function algebra formulation, with additional structure, the bicrossproduct ones are a distinct second family of quantum groups of increasing importance as deformations of solvable rather than semisimple Lie groups. They are associated to Lie splittings of Lie algebras or local factorisations of Lie groups and can be viewed as the cross product or Mackey quantisation of one of the factors acting on the other for the algebra and a similar story for the coproduct Δ with the second factor acting back on the first. The very simplest nontrivial example corresponds to two copies of R locally acting on each other and results in a quantum group (given here in an algebraic form) with generators ''p'', ''K'', ''K''−1, say, and coproduct : , Kh K(K-1) :\Delta p=p\otimes K+1\otimes p :\Delta K=K\otimes K where ''h'' is the deformation parameter. This quantum group was linked to a toy model of Planck scale physics implementing Born reciprocity when viewed as a deformation of the
Heisenberg algebra In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ::\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ...
of quantum mechanics. Also, starting with any compact real form of a semisimple Lie algebra g its complexification as a real Lie algebra of twice the dimension splits into g and a certain solvable Lie algebra (the
Iwasawa decomposition In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (QR decomposition, a cons ...
), and this provides a canonical bicrossproduct quantum group associated to g. For su(2) one obtains a quantum group deformation of the Euclidean group E(3) of motions in 3 dimensions.


See also

* Hopf algebra *
Lie bialgebra In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible. It is a bialgebra where the multiplication is skew-symmetric and satisfies a dual Jacob ...
*
Poisson–Lie group In mathematics, a Poisson–Lie group is a Poisson manifold that is also a Lie group, with the group multiplication being compatible with the Poisson algebra structure on the manifold. The infinitesimal counterpart of a Poisson–Lie group is a L ...
*
Quantum affine algebra In mathematics, a quantum affine algebra (or affine quantum group) is a Hopf algebra that is a ''q''-deformation of the universal enveloping algebra of an affine Lie algebra. They were introduced independently by and as a special case of their g ...


Notes


References

* * * * * * * * * {{DEFAULTSORT:Quantum Group Mathematical quantization