Hopf Superalgebra Research Articles

Two-parameter Quantum Superalgebras and PBW Theorem

A class of two-parameter quantum algebras [Formula: see text] is constructed. It is shown that [Formula: see text] is a Hopf superalgebra. Then the PBW basis of [Formula: see text] is described. For this purpose, some commutative relations of root vectors of [Formula: see text] are given.

The extension of a quantized Borcherds superalgebra by a Hopf algebra

A two-parameter quantum group is obtained from the usual enveloping algebra by adding two commutative grouplike elements. In this paper, we generalize this procession further by adding commutative grouplike elements bik, cik, gik, hik (i ∈ I, k = 1,..., mi) of a Hopf algebra H to the quantized enveloping algebra Uq(G) of a Borcherds superalgebra G defined by a symmetrizable integral Borcherds–Cartan matrix A = (aij)i,j∈I. Therefore, we define an extended Hopf superalgebra H ⊗′ Uq(G). We also discuss the basis and the grouplike elements of H ⊗′ Uq(G).

Cohomological finite-generation for finite supergroup schemes

In this paper we compute extension groups in the category of strict polynomial superfunctors and thereby exhibit certain “universal extension classes” for the general linear supergroup. Some of these classes restrict to the universal extension classes for the general linear group exhibited by Friedlander and Suslin, while others arise from purely super phenomena. We then use these extension classes to show that the cohomology ring of a finite supergroup scheme—equivalently, of a finite-dimensional cocommutative Hopf superalgebra—over a field is a finitely-generated algebra. Implications for the rational cohomology of the general linear supergroup are also discussed.

Integrable open spin-chains inAdS3/CFT2correspondences

We study integrable open boundary conditions for d(2,1;\alpha)^2 and psu(1,1|2)^2 spin-chains. Magnon excitations of these open spin-chains are mapped to massive excitations of type IIB open superstrings ending on D-branes in the AdS_3 x S^3 x S^3 x S^1 and AdS_3 x S^3 x T^4 supergravity geometries with pure R-R flux. We derive reflection matrix solutions of the boundary Yang-Baxter equation which intertwine representations of a variety of boundary coideal subalgebras of the bulk Hopf superalgebra. Many of these integrable boundaries are matched to D1- and D5-brane maximal giant gravitons.

A categorification of TEXTBACKSLASHmathboldTEXTBACKSLASHUT(𝔰𝔩(1|1)) and its tensor product representations

We define the Hopf superalgebra UT(sl(1|1)), which is a variant of the quantum supergroup Uq(sl(1|1)), and its representations V1⊗n for n>0. We construct families of DG algebras A, B and Rn, and consider the DG categories DGP(A), DGP(B) and DGP(Rn), which are full DG subcategories of the categories of DG A–, B– and Rn–modules generated by certain distinguished projective modules. Their 0 th homology categories HP(A), HP(B) and HP(Rn) are triangulated and give algebraic formulations of the contact categories of an annulus, a twice punctured disk and an n times punctured disk. Their Grothendieck groups are isomorphic to UT(sl(1|1)), UT(sl(1|1))⊗ℤUT(sl(1|1)) and V1⊗n, respectively. We categorify the multiplication and comultiplication on UT(sl(1|1)) to a bifunctor HP(A)× HP(A)→ HP(A) and a functor HP(A)→ HP(B), respectively. The UT(sl(1|1))–action on V1⊗n is lifted to a bifunctor HP(A)× HP(Rn)→ HP(Rn).

On Classification of Finite-Dimensional Superbialgebras and Hopf Superalgebras

The purpose of this paper is to investigate finite-dimensional superbialgebras and Hopf superalgebras. We study connected superbialgebras and provide a classification of non-trivial superbialgebras and Hopf superalgebras in dimension $n$ with $n\leq 4$.

Lusztig symmetries and Poincare-Birkhoff-Witt basis for $\mathfrak {wU}_{r, s}^{\mathrm{d}}({\rm osp}(1|2n))$wUr,sd( osp (1|2n))

We investigate a new kind of two-parameter weak quantized superalgebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {wU}_{r, s}^{\mathrm{d}}({\rm osp}(1|2n))$\end{document}wUr,sd( osp (1|2n)), which is a weak Hopf superalgebra. It has a homomorphic image which is isomorphic to the usual two-parameter quantum superalgebra \documentclass[12pt]{minimal}\begin{document}$\textrm {U}_{r,s}(\textrm {osp}(1|2n))$\end{document}Ur,s( osp (1|2n)) of \documentclass[12pt]{minimal}\begin{document}$\textrm {osp}(1|2n)$\end{document} osp (1|2n). We also discuss the basis of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {wU}_{r, s}^{\mathrm{d}}({\rm osp}(1|2n))$\end{document}wUr,sd( osp (1|2n)) by Lusztig's symmetries.

A Two-Parameter Quantum (2+1)-Superspace and its Deformed Derivation Algebra as Hopf Superalgebra

As is well known, a Hopf algebra setting is an efficient tool to study some geometric structures such as the Maurer-Cartan invariant forms and the corresponding vector fields on a noncommutative space. In this study we introduce a two-parameter quantum (2+1)-superspace with a Hopf superalgebra structure.We also define some derivation operators acting on this quantum superspace, and we show that the algebra of these derivations is a Hopf superalgebra. Furthermore it will be shown how the derivation operators lead to a bicovariant differential calculus on the two- parameter quantum (2+1)-superspace. In conclusion, based on the bicovariant differential calculus, the Maurer-Cartan right invariant differential forms and the corresponding quantum Lie superalgebra are given.

Drinfel'd Twist Deformations of the Super-Virasoro Algebras

In this article, we describe nonstandard quantum deformations of the centerless super-Virasoro algebras. Using the Drinfel'd twist quantization technique, we obtain the deformed coproduct and antipode. Hence we get a family of noncommutative and noncocommutative Hopf superalgebras. As a by-product, we also obtain two combinational identities.

Quantum superalgebras u q (sl(m|n)) at roots of unity

Finite dimensional Hopf superalgebras u q (sl(m|n)) corresponding to the Lie superalgebras sl(m|n) are constructed. The PBW type basis and the left and right integrals of u q (sl(m|n)) are obtained. Furthermore, the group of Hopf superalgebra automorphisms is described.

The Hopf algebra of odd symmetric functions

We consider a q-analogue of the standard bilinear form on the commutative ring of symmetric functions. The q=−1 case leads to a Z-graded Hopf superalgebra which we call the algebra of odd symmetric functions. In the odd setting, we describe counterparts of the elementary and complete symmetric functions, power sums, Schur functions, and combinatorial interpretations of associated change of basis relations.

Restricted forms of quantum supergroups

Assume that q is a root of unity. In the present paper, finite dimensional quantum supergroups OqGL(m|n)′ and OqSL(m|n)′ are constructed. They can be viewed as restricted forms of quantum supergroups OqGL(m|n) and OqSL(m|n), which are Hopf superalgebras. The PBW (Poincare-Birkhoff-Witt) bases for these supergroups are also established.

Orthosymplectic quantum function superalgebras OSP q (2l+1|2n)

By the R-matrix of orthosymplectic quantum superalgebra Uq(osp(2l+1|2n)) in the vector representation, we establish the corresponding quantum Hopf superalgebra OSPq(2l + 1|2n). Furthermore, it is shown that OSPq(2l + 1|2n) is coquasitriangular.

Poisson Deformed Super-Algebras and Moyal Quantization

The procedure to obtain the semi-classical limit of Lie-Hopf and quantum algebras introduced in preceding papers has been extended to Lie and quantum super-algebras. With a different definition of the limit the approach is the same: an ℏ-depending family of Hopf super-algebras is constructed starting from the super-bialgebra; then the singular limit ℏ → 0 allows to move from quantum to semi-classical framework in the spirit of Moyal quantization. The Poisson undeformed ospP(1|2) and deformed ospPq(1|2) super-algebras are exhibited, both as a paradigmatic example and because of their interest in applications.

The Hopf superalgebra of AdS/CFT

We describe the unconventional infinite-dimensional Hopf superalgebra related to the integrable S-matrix of the AdS/CFT correspondence, and discuss its typical and atypical representations.

HOPF ALGEBRA STRUCTURE OF (2+1)-DIMENSIONAL QUANTUM SUPERSPACE, ITS DUAL AND ITS DIFFERENTIAL CALCULUS

We consider a (2+1)-dimensional quantum superspace which has noncommuting coordinates in Manin sense and it was shown that this space has a Hopf algebra structure, i.e. the quantum supergroup, when it is extended by the inverse of the bosonic variable. Differential structures on this space were given by constructing the differential calculus in the sense of Woronowicz. Thus, we deduce that the corresponding quantum Lie superalgebra which as a commutation superalgebra appears classical, and as Hopf structure is non-cocommutative q-deformed. Finally, dual Hopf superalgebra was given.

Lax Operator for the Quantised Orthosymplectic Superalgebra U q [osp(m|n)

Representations of quantum superalgebras provide a natural framework in which to model supersymmetric quantum systems. Each quantum superalgebra, belonging to the class of quasi-triangular Hopf superalgebras, contains a universal R-matrix which automatically satisfies the Yang-Baxter equation. Applying the vector representation pi, which acts on the vector module V, to the left-hand side of a universal R-matrix gives a Lax operator. In this article a Lax operator is constructed for the quantised orthosymplectic superalgebras U (q) [osp(m|n)] for all m > 2, n >= 0 where n is even. This can then be used to find a solution to the Yang-Baxter equation acting on V circle times V circle times W, where W is an arbitrary U (q) [osp(m|n)] module. The case W = V is studied as an example.

The quantum double of the Yangian of the Lie superalgebra A(m, n) and computation of the universal R-matrix

The Yangian double DY(A(m, n)) of the Lie superalgebra A(m, n) is described in terms of generators and defining relations. We prove the triangular decomposition for Yangian Y(A(m, n)) and its quantum double DY(A(m, n)) as a corollary of the PBW theorem. We introduce normally ordered bases in the Yangian and its dual Hopf superalgebra in the quantum double. We calculate the pairing formulas between the elements of these bases. We obtain the formula for the universal R-matrix of the Yangian double. The formula for the universal R-matrix of the Yangian, which was introduced by V. Drinfel’d, is also obtained.

Weak quantum Borcherds superalgebras and their representations

We define a weak quantum Borcherds superalgebra nqd(G), which is a weak Hopf superalgebra. We also discuss the basis and the highest weight modules of nqd(G). Then we study the weak A-form and the classical limit of nqd(G).

Lax operator for the quantized orthosymplectic superalgebraUq[osp(2|n)

Each quantum superalgebra is a quasi-triangular Hopf superalgebra, so contains auniversal R-matrix in the tensor product algebra which satisfies theYang–Baxter equation. Applying the vector representationπ, which acts on thevector module V, toone side of a universal R-matrix gives a Lax operator. In this paper a Lax operator is constructed for theC-type quantumsuperalgebras Uq[osp(2|n)]. This is achieved without reference to the specific details of the universalR-matrix, but instead appealing to the co-product structure ofUq[osp(2|n)]. The result can in turn be used to find a solution to the Yang–Baxter equation acting on , where W isan arbitrary Uq[osp(2|n)] module. The case W = V is included here as an example.