Braided Lie Algebras Research Articles

Atiyah sequences of braided Lie algebras and their splittings

Given an equivariant noncommutative principal bundle, we construct an Atiyah sequence of braided derivations whose splittings give connections on the bundle. Vertical braided derivations act as infinitesimal gauge transformations on connections. In the case of the principal \mathrm{SU}(2) -bundle over the sphere S^{4}_{\theta} an equivariant splitting of the Atiyah sequence recovers the instanton connection. An infinitesimal action of the braided conformal Lie algebra \mathrm{so}_{\theta}(5,1) yields a five parameter family of splittings. On the principal \mathrm{SO}_\theta(2n,\mathbb{R}) -bundle of orthonormal frames over the sphere S^{2n}_\theta , a splitting of the sequence leads to the Levi-Civita connection for the ‘round’ metric on S^{2n}_\theta . The corresponding Riemannian geometry of S^{2n}_{\theta} is worked out.

Braided Hopf Algebras and Gauge Transformations II: *-Structures and Examples

We consider noncommutative principal bundles which are equivariant under a triangular Hopf algebra. We present explicit examples of infinite dimensional braided Lie and Hopf algebras of infinitesimal gauge transformations of bundles on noncommutative spheres. The braiding of these algebras is implemented by the triangular structure of the symmetry Hopf algebra. We present a systematic analysis of compatible *-structures, encompassing the quasitriangular case.

Braided varvec{L_{infty }}-algebras, braided field theory and noncommutative gravity

We define a new homotopy algebraic structure, that we call a braided L_infty -algebra, and use it to systematically construct a new class of noncommutative field theories, that we call braided field theories. Braided field theories have gauge symmetries which realize a braided Lie algebra, whose Noether identities are inhomogeneous extensions of the classical identities, and which do not act on the solutions of the field equations. We use Drinfel’d twist deformation quantization techniques to generate new noncommutative deformations of classical field theories with braided gauge symmetries, which we compare to the more conventional theories with star-gauge symmetries. We apply our formalism to introduce a braided version of general relativity without matter fields in the Einstein–Cartan–Palatini formalism. In the limit of vanishing deformation parameter, the braided theory of noncommutative gravity reduces to classical gravity without any extensions.

Schur–Weyl quasi-duality and (co)triangular Hopf quasigroups

Schur–Weyl duality relates the representation theories of general linear and symmetric groups in defining characteristic and plays a central role in many parts of algebraic Lie theory. In this paper, we will introduce the notion of Schur–Weyl quasi-duality and study it. For this, generally, we consider a braided vector space (V,c) and its braided Lie algebra Endk(V)(−). Then, we can construct its braided enveloping algebra U(Endk(V)(−)), which is a connected braided c-cocommutative Hopf algebra. Let H be a triangular Hopf quasigroup with bijective antipode and B be a cotriangular Hopf quasigroup with bijective antipode. Let V be any finite dimensional vector space in the category LQ(H,R)(B,σ) of generalized Long quasimodules. We show that (U((EndkV)(−))⋆H⋆B,kSn,V⊗n) is a Schur–Weyl quasi-duality under suitable conditions.

RELATIONSHIP BETWEEN GRAPHS AND NICHOLS BRAIDED LIE ALGEBRAS

In this paper we give the relationship between the connected components of pure generalized Dynkin graphs and Nichols braided Lie algebras.

On rigidity of Nichols algebras

We study deformations of graded braided bialgebras using cohomological methods. In particular, we show that many examples of Nichols algebras, including the finite-dimensional ones arising in the Andruskiewitsch–Schneider program of classification of pointed Hopf algebras, are rigid. This result can be regarded as nonexistence of “braided Lie algebras” with nontrivial bracket.

Braided groups and quantum groupoids

Let H be a quasitriangular weak Hopf algebra. It is proved that the centralizer subalgebra of its source subalgebra in H is a braided group (or Hopf algebra in the category of left H-modules), which is cocommutative and also a left braided Lie algebra in the sense of Majid.

A Milnor–Moore type theorem for primitively generated braided bialgebras

A braided bialgebra is called primitively generated if it is generated as an algebra by its space of primitive elements. We prove that any primitively generated braided bialgebra is isomorphic to the universal enveloping algebra of its infinitesimal braided Lie algebra, notions hereby introduced. This result can be regarded as a Milnor–Moore type theorem for primitively generated braided bialgebras and leads to the introduction of a concept of braided Lie algebra for an arbitrary braided vector space.

Generalized Lie Algebras and Cocycle Twists

We use the results of Etingof and Gelaki on the classification of (co)triangular Hopf algebras to extend Scheunert's “discoloration” technique to Lie algebras in the category of (co)modules. As an application, we prove a PBW-type theorem for such Lie algebras. We also discuss the relationship between Lie algebras in the category of (co)modules and symmetric braided Lie algebras introduced by Gurevich. Finally, we construct examples of symmetric braided Lie algebras that are essentially different from Lie coloralgebras.

Braided symmetric and exterior algebras and quantizations of braided lie algebras

We study quantizations of braided symmetric and exterior algebras of graded vector spaces and of braided derivations on these algebras. We find quantizations of braided Lie algebras by considering quantizations of derivations on their braided exterior algebra.

Classical and Exceptional Root Systems and Quantizations

We present a method for quantizing semisimple Lie algebras. In Huru (Russ. Math. [2007]) we defined quantizations of the braided Lie algebra structure on a finite dimensional graded vector space V by quantizations of braided derivations on the exterior algebra of V*. We find quantizations of semisimple Lie algebras in this setting using the grading by their roots and shall go through all root systems, classical and exceptional.

Braided m-Lie Algebras

Braided m-Lie algebras induced by multiplication are introduced, which generalize Lie algebras, Lie color algebras and quantum Lie algebras. The necessary and sufficient conditions for the braided m-Lie algebras to be strict Jacobi braided Lie algebras are given. Two classes of braided m-Lie algebras are given, which are generalized matrix braided m-Lie algebras and braided m-Lie subalgebras of $End_F M$, where $M$ is a Yetter-Drinfeld module over $B$ with dim $B< \infty $ . In particular, generalized classical braided m-Lie algebras $sl_{q, f}(GM_G(A), F)$ and $osp_{q, t} (GM_G(A), M, F)$ of generalized matrix algebra $GM_G(A)$ are constructed and their connection with special generalized matrix Lie superalgebra $sl_{s, f}(GM_{{\bf Z}_2}(A^s), F)$ and orthosymplectic generalized matrix Lie super algebra $osp_{s, t} (GM_{{\bf Z}_2}(A^s), M^s, F)$ are established. The relationship between representations of braided m-Lie algebras and their associated algebras are established.

Braided Lie algebras and bicovariant differential calculi over co-quasitriangular Hopf algebras

We show that if g Γ is the quantum tangent space (or quantum Lie algebra in the sense of Woronowicz) of a bicovariant first-order differential calculus over a co-quasitriangular Hopf algebra (A, r ) , then a certain extension of it is a braided Lie algebra in the category of A -comodules. This is used to show that the Woronowicz quantum universal enveloping algebra U( g Γ ) is a bialgebra in the braided category of A -comodules. We show that this algebra is quadratic when the calculus is inner. Examples with this unexpected property include finite groups and quantum groups with their standard differential calculi. We also find a quantum Lie functor for co-quasitriangular Hopf algebras, which has properties analogous to the classical one. This functor gives trivial results on standard quantum groups O q (G) , but reasonable ones on examples closer to the classical case, such as the cotriangular Jordanian deformations. In addition, we show that split braided Lie algebras define ‘generalized-Lie algebras’ in a different sense of deforming the adjoint representation. We construct these and their enveloping algebras for O q (SL(n)) , recovering the Witten algebra for n =2.

Riemannian Geometry of Quantum Groups¶and Finite Groups with Nonuniversal Differentials

We construct noncommutative `Riemannian manifold' structures on dual quasitriangular Hopf algebras such as $C_q[SU_2]$ with its standard bicovariant differential calculus, using the quantum frame bundle formalism introduced previously. The metric is provided by the braided-Killing form on the braided-Lie algebra on the tangent space and the $n$-bein by the Maurer-Cartan form. We also apply the theory to finite sets and in particular to finite group function algebras $C[G]$ with differential calculi and Killing forms determined by a conjugacy class. The case of the permutation group $C[S_3]$ is worked out in full detail and a unique torsion free and cotorsion free or `Levi-Civita' connection is obtained with noncommutative Ricci curvature essentially proportional to the metric (an Einstein space). We also construct Dirac operators in the metric background, including on finite groups such as $S_3$. In the process we clarify the construction of connections from gauge fields with nonuniversal calculi on quantum principal bundles of tensor product form.

SOLUTIONS OF THE YANG-BAXTER EQUATIONS FROM BRAIDED-LIE ALGEBRAS AND BRAIDED GROUPS

We obtain an R-matrix or matrix representation of the Artin braid group acting in a canonical way on the vector space of every (super)-Lie algebra or braided-Lie algebra. The same result applies for every (super)-Hopf algebra or braided-Hopf algebra. We recover some known representations such as those associated to racks. We also obtain new representations such as a non-trivial one on the ring k[x] of polynomials in one variable, regarded as a braided-line. Representations of the extended Artin braid group for braids in the complement of S1 are also obtained by the same method.

Quantum Koszul complexes for Majid’s braided Lie algebras

A Yang–Baxter operator was canonically associated to any cocommutative braided Lie algebra as defined by Majid. Applying previous results, we construct quantum Koszul complexes for braided Lie algebras with respect to this operator. In the usual case, the complexes reduce to the Chevalley–Eilenberg complexes for classical Lie algebras.

Quantum and braided-Lie algebras

We introduce the notion of a braided-Lie algebra consisting of a finite-dimensional vector space L equipped with a bracket [ , ]: L ⊗ L → L and Yang-Baxter operator Ψ: L ⊗ L → L ⊗ L obeying some axioms. We show that such an object has an enveloping braided-bialgebra U (L). We show that every generic R-matrix leads to such a braided-Lie algebra with [ , ] given by structure constants c IJ K determined from R. In this case U(L) = B(R) the braided matrices introduced previously. We also introduce the basic theory of these braided-Lie algebras, including the natural right-regular action of a braided-Lie algebra L by braided vector fields, the braided-Killing form and the quadratic Casimir associated to L. These constructions recover the relevant notions for usual, colour and super-Lie algebras as special cases. In addition, the standard quantum deformations U q(g) are understood as the enveloping algebras of such underlying braided-Lie algebras with [ , ] on L ⊂ U q(g) given by the quantum adjoint action.

On the addition of quantum matrices

An addition law is introduced for the usual quantum matrices A(R) by means of a coaddition Δ_t=t⊗1+1⊗t. It supplements the usual comultiplication Δt=t⊗t and together they obey a codistributivity condition. The coaddition does not form a usual Hopf algebra but a braided one. The same remarks apply for rectangular m×n quantum matrices. As an application, left-invariant vector fields are constructed on A(R) and other quantum spaces. They close in the form of a braided Lie algebra. As another application, the wave functions in the lattice approximation of Kac–Moody algebras and other lattice fields can be added and functionally differentiated.