View source for Quasi-triangular Hopf algebra

''dual quasi-triangular Hopf algebra, co-quasi-triangular Hopf algebra, quantum group''

A quantum group in the strict sense, i.e. a [[Hopf algebra|Hopf algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q1200701.png" /> equipped with a further (co-) quasi-triangular structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q1200702.png" /> obeying certain axioms such that the [[Category|category]] of (co-) modules of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q1200703.png" /> is a [[Braided category|braided category]] (cf. also [[Quantum groups|Quantum groups]]). This is arguably the key property behind the quantum group enveloping algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q1200704.png" /> or their associated quantum group coordinate rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q1200705.png" />.

More precisely, a quasi-triangular Hopf algebra is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q1200706.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q1200707.png" /> is a Hopf algebra over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q1200708.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q1200709.png" /> obeys

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007010.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007011.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007012.png" /> is the permutation operation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007015.png" /> are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007016.png" /> in the latter equations. One may show that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007017.png" /> then obeys

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007018.png" /></td> </tr></table>

which is an abstract form of the [[Yang–Baxter equation|Yang–Baxter equation]]. One denotes the Hopf algebra structure by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007019.png" /> for the co-product, and denotes by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007020.png" /> the co-unit and by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007021.png" /> the antipode.

==Examples.==


1) When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007022.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007023.png" />th root of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007024.png" />, the quantum group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007025.png" /> is given by the polynomial algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007026.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007027.png" /> (the group ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007028.png" />) with co-algebra, antipode and quasi-triangular structure

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007029.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007030.png" /></td> </tr></table>

It is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007031.png" /> is invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007032.png" />.

2) When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007033.png" /> is a primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007034.png" />th root of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007035.png" />, the finite-dimensional quantum group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007036.png" /> is the free associative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007037.png" /> modulo the relations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007038.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007039.png" /></td> </tr></table>

and the co-algebra, antipode and quasi-triangular structure

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007040.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007041.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007042.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007043.png" /></td> </tr></table>

is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007045.png" />-exponential with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007046.png" />-integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007047.png" /> in the factorial.

For general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007048.png" /> (an invertible element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007049.png" />), one has the infinite-dimensional Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007050.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007051.png" /> relations above are omitted. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007052.png" /> has to be described via some form of completion. One formulation is to work over the ring of formal power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007053.png" /> rather than over a field as above. If it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007055.png" />, one can reformulate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007056.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007057.png" /> as a generator and define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007058.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007059.png" /> in place of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007060.png" /> above (and now take an infinite sum in the exponential). It does not, however, live in the algebraic tensor product but in a completion of it. On the other hand, this formulation allows one to consider the structure to lowest order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007061.png" />. This is the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007062.png" />, the Lie co-bracket <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007063.png" /> (forming a Lie bi-algebra) and a Lie quasi-triangular structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007064.png" /> obeying the classical Yang–Baxter equations. It extends to a Poisson bracket (cf. [[Poisson brackets|Poisson brackets]]) on the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007065.png" />, making it a [[Poisson Lie group|Poisson Lie group]]. This means a Poisson bracket on the group such that the product mapping is a Poisson mapping from the direct product Poisson structure. There are similar quantum group enveloping algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007067.png" /> for all complex semi-simple Lie algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007068.png" />.

3) Every finite-dimensional Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007069.png" /> with invertible antipode can be  "doubled"  to obtain a quasi-triangular Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007070.png" />, called the quantum double of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007071.png" />. This contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007073.png" /> (the dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007074.png" /> with reversed product) as sub-Hopf algebras and the additional cross relations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007075.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007077.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007078.png" /> is a notation for the co-product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007079.png" />, etc., and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007080.png" /> denotes the evaluation pairing. The quasi-triangular structure is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007081.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007082.png" /> is a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007083.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007084.png" /> is a dual basis. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007085.png" /> can in fact be built explicitly on the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007086.png" /> as a double cross product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007087.png" />. The braided category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007088.png" />-modules can be identified with that of crossed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007089.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007090.png" /> by viewing a co-action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007091.png" /> as an action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007092.png" /> by evaluation.

Dually, i.e. reversing all arrows, one has the notion of a dual quasi-triangular or co-quasi-triangular Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007093.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007094.png" /> is a Hopf algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007095.png" /> obeys

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007096.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007097.png" /></td> </tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007098.png" />. One also requires <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q12007099.png" /> to be convolution-invertible in the sense

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070100.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070101.png" /></td> </tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070102.png" />.

A) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070103.png" /> be an [[Abelian group|Abelian group]] equipped with a bi-character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070104.png" /> (a function multiplicative in each input). The group algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070105.png" /> is the vector space with basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070106.png" /> and with product among basis elements the group product. The co-product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070107.png" /> and co-unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070108.png" /> extended linearly make <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070109.png" /> into a Hopf algebra. The bi-character makes this dual quasi-triangular, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070110.png" /> on basis elements, extended linearly.

B) The dual quasi-triangular Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070111.png" /> is the free associative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070112.png" /> modulo the relations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070113.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070114.png" /></td> </tr></table>

and the  "q-determinant relation" 

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070115.png" /></td> </tr></table>

The co-algebra has the matrix form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070116.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070117.png" /></td> </tr></table>

(matrix multiplication understood). The antipode is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070118.png" /></td> </tr></table>

and the dual quasi-triangular structure is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070119.png" /></td> </tr></table>

on a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070120.png" /> of generators. The extension to products is then determined. Here it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070121.png" /> has a square root in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070122.png" />. The quantum group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070123.png" /> is similar, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070124.png" /> inverted rather than set to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070125.png" />.

There are similar quantum group coordinate rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070126.png" /> for the standard families of simple Lie groups, known explicitly for the non-exceptional families. They are deformations of the classical coordinate rings and in fact quantize the Poisson Lie group structures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070127.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070128.png" />. They are dually paired as Hopf algebras with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070129.png" />.

C) More generally, given any invertible matrix solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070130.png" /> of the Yang–Baxter equations, there is a dual-quasi-triangular bi-algebra of quantum matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070131.png" /> as the free associative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070132.png" /> (on a matrix of generators) modulo the relations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070133.png" /></td> </tr></table>

(summation of repeated indices). The co-algebra has the matrix form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070134.png" /></td> </tr></table>

(summation over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070135.png" />). The dual-quasi-triangular structure is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070136.png" /></td> </tr></table>

extended to products by the quasi-triangularity axioms. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070137.png" /> need not obey the Yang–Baxter equations in order to have a bi-algebra: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070138.png" /> provides the dual quasi-triangular structure.

In between these formulations is an intermediate one, called a quasi-triangular dual pair. This is a pair of Hopf algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070139.png" />, a duality pairing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070140.png" /> between them, and a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070141.png" /> obeying certain axioms. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070142.png" /> are dually paired and there is a suitable mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120070/q120070143.png" />. This provides a way of working with quasi-triangular structures that includes quantum group enveloping algebras but avoids formal power series.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.G. Drinfel'd,   "Quantum groups"  A. Gleason (ed.) , ''Proc. Internat. Math. Congress'' , Amer. Math. Soc.  (1987)  pp. 798–820</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.D. Faddeev,   N.Yu. Reshetikhin,   L.A. Takhtajan,   "Quantization of Lie groups and Lie algebras"  ''Leningrad Math. J.'' , '''1'''  (1990)  pp. 193–225</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Majid,   "Quasitriangular Hopf algebras and Yang-Baxter equations"  ''Internat. J. Modern Physics A'' , '''5''' :  1  (1990)  pp. 1–91</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Majid,   "Foundations of quantum group theory" , Cambridge Univ. Press  (1995)</TD></TR></table>