Hopf Superalgebra Research Articles

Finite-dimensional Hopf superalgebras and graded pentagon equation

Suppose that H is an arbitrary finite-dimensional Hopf superalgebra. Let H(H) be the Heisenberg double of H and let R be the canonical matrix of H(H) that satisfies the graded pentagon equation R12R13R23=R23R12. It is established that H is isomorphic to the Hopf superalgebra P(H(H),R) of left coefficients of R. This result can be regarded as a generalisation of Militaru's result [10] from the non-super situation to the super situation.

Quotients in super-symmetry: formal supergroup case

We describe the structure of the quotient G / H of a formal supergroup G by its formal sub-supergroup H . This is a consequence which arises as a continuation of the authors’ work (partly with M. Hoshi) on algebraic/analytic supergoups. The results are presented and proved in terms of super-cocommutative Hopf superalgebras. The notion of co-free super-coalgebras plays a role, in particular.

Fermionization of fusion category symmetries in 1+1 dimensions

We discuss the fermionization of fusion category symmetries in two-dimensional topological quantum field theories (TQFTs). When the symmetry of a bosonic TQFT is described by the representation category Rep(H) of a semisimple weak Hopf algebra H, the fermionized TQFT has a superfusion category symmetry SRep(H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document}u), which is the supercategory of super representations of a weak Hopf superalgebra H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document}u. The weak Hopf superalgebra H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document}u depends not only on H but also on a choice of a non-anomalous ℤ2 subgroup of Rep(H) that is used for the fermionization. We derive a general formula for H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document}u by explicitly constructing fermionic TQFTs with SRep(H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document}u) symmetry. We also construct lattice Hamiltonians of fermionic gapped phases when the symmetry is non-anomalous. As concrete examples, we compute the fermionization of finite group symmetries, the symmetries of finite gauge theories, and duality symmetries. We find that the fermionization of duality symmetries depends crucially on F-symbols of the original fusion categories. The computation of the above concrete examples suggests that our fermionization formula of fusion category symmetries can also be applied to non-topological QFTs.

Quantization of a Class of Super W-Agebras

We study a class of super W-algebras whose even part is the Virasoro type Lie algebra [Formula: see text]. We quantize the Lie superbialgebra of the super W-algebra by the Drinfeld twist quantization technique and obtain a class of noncommutative and noncocommutative Hopf superalgebras.

Twisting Kuperberg invariants via Fox calculus and Reidemeister torsion

We study Kuperberg invariants for sutured manifolds in the case of a semidirect product of an involutory Hopf superalgebra $H$ with its automorphism group $\text{Aut}(H)$. These are topological invariants of balanced sutured 3-manifolds endowed with a homomorphism of the fundamental group into $\text{Aut}(H)$ and possibly with a $\text{Spin}^c$ structure and a homology orientation. We show that these invariants are computed via a form of Fox calculus and that, if $H$ is $\mathbb{N}$-graded, they can be extended in a canonical way to polynomial invariants. When $H$ is an exterior algebra, we show that this invariant specializes to a refinement of the twisted relative Reidemeister torsion of sutured 3-manifolds. We also give an explanation of our Fox calculus formulas in terms of a particular Hopf group-algebra.

Shifted quiver quantum toroidal algebra and subcrystal representations

Recently, new classes of infinite-dimensional algebras, quiver Yangian (QY) and shifted QY, were introduced, and they act on BPS states for non-compact toric Calabi-Yau threefolds. In particular, shifted QY acts on general subcrystals of the original BPS crystal. A trigonometric deformation called quiver quantum toroidal algebra (QQTA) was also proposed and shown to act on the same BPS crystal. Unlike QY, QQTA has a formal Hopf superalgebra structure which is useful in deriving representations.In this paper, we define the shifted QQTA and study a class of their representations. We define 1d and 2d subcrystals of the original 3d crystal by removing a few arrows from the original quiver diagram and show how the shifted QQTA acts on them. We construct the 2d crystal representations from the 1d crystal representations by utilizing a generalized coproduct acting on different shifted QQTAs. We provide a detailed derivation of subcrystal representations of ℂ3, ℂ3/ℤn(n ≥ 2), conifold, suspended pinch point, and ℂ3/(ℤ2× ℤ2).

Invariants and reduced Wigner coefficients for quasi-triangular Hopf superalgebras

We develop explicit formulae for the eigenvalues of various invariants for highest weight irreducible representations of the quantum supergroup U q [gl(m|n)]. The techniques employed make use of modified characteristic identity methods and allow for the evaluation of generator matrix elements and reduced Wigner coefficients.

A note on quiver quantum toroidal algebra

Recently, Li and Yamazaki proposed a new class of infinite-dimensional algebras, quiver Yangian, which generalizes the affine Yangian mathfrak{gl} 1. The characteristic feature of the algebra is the action on BPS states for non-compact toric Calabi-Yau threefolds, which are in one-to-one correspondence with the crystal melting models. These algebras can be bootstrapped from the action on the crystals and have various truncations.In this paper, we propose a q-deformed version of the quiver Yangian, referred to as the quiver quantum toroidal algebra (QQTA). We examine some of the consistency conditions of the algebra. In particular, we show that QQTA is a Hopf superalgebra with a formal super coproduct, like known quantum toroidal algebras. QQTA contains an extra central charge C. When it is trivial (C = 1), QQTA has a representation acting on the three-dimensional crystals, like Li-Yamazaki’s quiver Yangian. While we focus on the toric Calabi-Yau threefolds without compact 4-cycles, our analysis can likely be generalized to all toric Calabi-Yau threefolds.

A new quantum supergroup and its gauss decomposition

We get a new quantum supergroup (Hopf superalgebra), denoted by GL q (2|1), by using the standard method (the RTT-relation) with an R -matrix which is a solution of the quantum Yang-Baxter equation. We give the Gauss decomposition of a matrix in GL q (2|1) and considering the three matrices in the decomposition in pairs, we show that two superalgebras formed by matrix elements of these matrices are Hopf superalgebras.

Quantization of the Lie superalgebra of Witt type

In this paper, we use the quantization technique by a Drinfel’d twist to quantize the Lie superalgebra of Witt type, whose Lie super-bialgebra structures were recently determined by Yue and Zhu, and we obtain a new class of non-commutative and non-cocommutative Hopf superalgebras.

A Hennings type invariant of 3-manifolds from a topological Hopf superalgebra

We prove the unrolled superalgebra \mathcal{U}_{\xi}^{H}\mathfrak{sl}(2|1) has a completion which is a ribbon Hopf superalgebra in a topological sense where \xi is a root of unity of odd order. Using this ribbon Hopf superalgebra we construct its universal invariant of links. We use it to construct an invariant of 3-manifolds endowed with a first cohomology class of Hennings type.

Kuperberg invariants for balanced sutured 3-manifolds

AbstractWe construct quantum invariants of balanced sutured 3-manifolds with a ${\text {Spin}^c}$ structure out of an involutive (possibly nonunimodular) Hopf superalgebra H. If H is the Borel subalgebra of ${U_q(\mathfrak {gl}(1|1))}$ , we show that our invariant is computed via Fox calculus, and it is a normalization of Reidemeister torsion. The invariant is defined via a modification of a construction of Kuperberg, where we use the ${\text {Spin}^c}$ structure to take care of the nonunimodularity of H or $H^{*}$ .

Hopf-algebraic techniques applied to super Lie groups over a complete field

We show basic results on super-manifolds and super Lie groups over a complete field of characteristic ≠2, extensively using Hopf-algebraic techniques. The main results are two theorems. The first main theorem shows a category equivalence between super Lie groups and Harish-Chandra pairs, which is applied especially to construct the Hopf super-algebra of all analytic representative functions on a super Lie group. The second constructs homogeneous super-manifolds by a new Hopf-algebraic method, showing their remarkable property.

On support varieties for Lie superalgebras and finite supergroup schemes

We study the spectrum of the cohomology rings of cocommutative Hopf superalgebras, restricted and non-restricted Lie superalgebras, and finite supergroup schemes. We also investigate support varieties in these settings and demonstrate that they have the desirable properties of such a theory. We completely characterize support varieties for finite supergroup schemes over algebraically closed fields of characteristic zero, while for non-restricted Lie superalgebras in positive characteristic we obtain descriptions that are suggestive of varieties previously defined by Duflo and Serganova in characteristic zero but which have no known cohomological interpretation. Our computations for restricted Lie superalgebras and infinitesimal supergroup schemes provide natural generalizations of foundational results of Friedlander and Parshall and of Bendel, Friedlander, and Suslin in the classical setting.

Ribbon Hopf Superalgebras and Drinfel’d Double

Finite dimensional ribbon Hopf (super) algebras play an important role in constructing invariants of 3-manifolds. In the present paper, the authors give a necessary and sufficient condition for the Drinfel’d double of a finite dimensional Hopf superalgebra to have a ribbon element. The criterion can be seen as a generalization of Kauffman and Radford’s result in the non-super situation to the ℤ2-graded situation, however, the derivation of the result in the ℤ2-graded case will be much more complicated.

On functor points of affine supergroups

To construct an affine supergroup from a Harish-Chandra pair, Gavarini [2] invented a natural method, which first constructs a group functor and then proves that it is representable. We give a simpler and more conceptual presentation of his construction in a generalized situation, using Hopf superalgebras over a superalgebra. As an application of the construction, given a closed super-subgroup of an algebraic supergroup, we describe the normalizer and the centralizer, using Harish-Chandra pairs. We also prove a tensor product decomposition theorem for Hopf superalgebras, and describe explicitly by cocycle deformation, the difference which results from the two choices of dualities found in literature.

Lie superbialgebra structures on the Lie superalgebra C3+A and deformation of related integrable Hamiltonian systems

Admissible structure constants related to the dual Lie superalgebras of particular Lie superalgebra (C3+A) are found by straightforward calculations from the matrix form of super Jacobi and mixed super Jacobi identities which are obtained from adjoint representation. Then, by making use of the automorphism supergroup of the Lie superalgebra (C3+A), the Lie superbialgebra structures on the Lie superalgebra (C3+A) are obtained and classified into inequivalent 31 families. We also determine all corresponding coboundary and bi-r-matrix Lie superbialgebras. The quantum deformations associated with some Lie superbialgebras (C3+A) are obtained, together with the corresponding deformed Casimir elements. As an application of these quantum deformations, we construct a deformed integrable Hamiltonian system from the representation of the Hopf superalgebra Uλ(Cp=12,𝜖⊕A1,1)(C3+A).

Berry phase, entanglement entropy, and algebraic properties of ground states of BCS and BEC superfluids

By using Bogoliubov transformations to construct the ground states of fermionic Bardeen–Cooper–Schrieffer (BCS) superfluids and weakly-interacting Bose gases supporting Bose Einstein Condensation (BEC), their algebraic structures and implications can be analyzed in detail. Both ground states are generalized squeezed coherent states saturating a generalized Heisenberg uncertainty relation, and they acquire quantized Berry phases when the corresponding systems are transported along a closed path in their parameter spaces. While the Berry phase of the BCS ground state depends on the total particle number, the Berry phase of the BEC ground state depends only on the particles outside the BEC. The Berry phases are associated with magnetic monopoles in the parameter spaces and we found that the Dirac quantization condition is satisfied. Moreover, both ground states are entangled states of the fermion or boson quanta and we found the entanglement entropy quantifying the internal correlations. A fixed particle-number approach of fermionic superfluids does not saturate the generalized uncertainty relation, exhibits internal entanglement, and gives corresponding Berry phase. In addition, the algebraic structures of the ground states can be classified by the q-deformed Hopf algebra, ⨁khqk(1) for bosons and q-deformed Hopf superalgebra ⨁khqk(1|1) for fermions, respectively.

Solvability and nilpotency for algebraic supergroups

We study solvability, nilpotency and splitting property for algebraic supergroups over an arbitrary field K of characteristic charK≠2. Our first main theorem tells us that an algebraic supergroup G is solvable if the associated algebraic group Gev is trigonalizable. To prove it we determine the algebraic supergroups G such that dim⁡Lie(G)1=1; their representations are studied when Gev is diagonalizable. The second main theorem characterizes nilpotent connected algebraic supergroups. A super-analogue of the Chevalley Decomposition Theorem is proved, though it must be in a weak form. An appendix is given to characterize smooth Noetherian superalgebras as well as smooth Hopf superalgebras.

Lie super-bialgebra and quantization of the super Virasoro algebra

In this paper, we study Lie super-bialgebra and quantization of the super Virasoro algebra, whose even part is the centerless twisted Heisenberg-Virasoro algebra. By using some Liu-Pei-Zhu’s results, we prove that all Lie super-bialgebra structures on the super Virasoro algebra are triangular coboundary. Furthermore, we quantize the super Virasoro algebra by the Drinfel’d twist quantization technique and obtain a class of noncommutative and noncocommutative Hopf superalgebras.