Introduction to the Quantum Yang-Baxter Equation and Quantum Groups: An Algebraic Approach

Lambe, Larry A.; Radford, David E.

doi:10.1007/978-1-4615-4109-7

Cited by 96 publications

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“…Identifying H and H * * in the usual way, we see that (9) can be formulated r(p (1) ) * • T • (p (2) ) * = (p (2) ) * • T • r(p (1) ) * for all p ∈ H * which in turn is equivalent to (p (2) ) • T * • r(p (1) ) = r(p (1) ) • T * • (p (2) ) for all p ∈ H * . 2 Suppose that n > 1 and k contains a primitive nth root of unity ω.…”

Section: The Subalgebra Of Cocommutative Elements Of D(h ) *mentioning

confidence: 98%

Trace-like functionals on the double of the Taft Hopf algebra

Radford

Westreich

2006

Journal of Algebra

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The cocommutative elements of the dual of the double D(H ) * , which are the trace-like functionals on D(H ), of the Taft (Hopf) algebra H over a field k are studied in detail. The subalgebra of cocommutative elements of D(H ) * is isomorphic to the center of D(H ). Trace-like functionals on D(H ) determine regular isotopy invariants of oriented knots and links. Specific calculations made here suggest that further study of these invariants is definitely warranted.

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Section: The Subalgebra Of Cocommutative Elements Of D(h ) *mentioning

confidence: 98%

Trace-like functionals on the double of the Taft Hopf algebra

Radford

Westreich

2006

Journal of Algebra

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“…(10) Then (C,/3, S, G) is a twist quantum coalgebra over C. The bracket polynomial for knots is the expression computed according to (4) multiplied by -t2/(1 + t4), where c = e~ + e~ ~ M2(C)* = C is the trace function. The quantum coalgebra described above is generalized as follows.…”

Section: S(e~) = E 2 S(e 2) = El S(e~) = -T2e12 and S(e 2) = -T-zementioning

confidence: 99%

“…Background material needed for the coalgebra theory used in this paper is more than adequately covered in [1,10,11,14]. Throughout k is a field and k* is the set of nonzero elements of k.…”

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On a parameterized family of twist quantum coalgebras and the bracket polynomial

Radford

2000

Journal of Algebra

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Quantum coalgebras over a field k are structures that produce regular isotopy invariants of 1-1 tangles, and twist quantum coalgebras over k are structures that produce regular isotopy invariants of knots. These structures are introduced and studied in [9]. In [6] and [8] certain families of quantum coalgebras and their resulting invariants of 1-1 tangles are examined in detail. In this paper we use the simple coalgebra C,(k), the dual of the algebra Mn(k) of n x n matrices over k, as a starting point for constructing twist quantum coalgebras. Among the twist quantum coalgebras we construct is a parameterized family related to the Jones polynomial in an interesting way.The Jones polynomial can be computed by the (normalized) bracket polynomial [5, Theorem 5.2]. The bracket polynomial (for knots) in turn can be calculated from a quantum algebra structure on M2(C), which is described in [5, p. 580]. The dual of this quantum algebra is a certain twist quantum coalgebra structure on C2(C), which is discussed in [9, Sect. 8]. Thus the Jones polynomial for knots can be computed from a twist quantum coalgebra structure on C2(C).

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“…Throughout k will be a field and H a finite dimensional Hopf algebra over k. For general facts on Hopf algebras and related notions we refer the reader to [8], [14], and [17]. In this section we recall the construction of the Brauer group BM(k, H, R) of a finite dimensional quasi-triangular Hopf algebra (H, R) over a field k; see [2], [3].…”

Section: Preliminariesmentioning

confidence: 99%

The Brauer Group of the Dihedral Group

Carnovale¹,

Cuadra

2004

Glasgow Math. J.

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Abstract. Let p m be a power of a prime number p, ‫ބ‬ p m be the dihedral group of order 2p m and k be a field where p is invertible and containing a primitive 2p m -th root of unity. The aim of this paper is computing the Brauer group BM(k, ‫ބ‬ p m , R z ) of the group Hopf algebra of ‫ބ‬ p m with respect to the quasi-triangular structure R z arising from the group Hopf algebra of the cyclic group ‫ޚ‬ p m of order p m , for z coprime with p.

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Introduction to the Quantum Yang-Baxter Equation and Quantum Groups: An Algebraic Approach

Cited by 96 publications

References 0 publications

Trace-like functionals on the double of the Taft Hopf algebra

Trace-like functionals on the double of the Taft Hopf algebra

On a parameterized family of twist quantum coalgebras and the bracket polynomial

The Brauer Group of the Dihedral Group

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