BRAIN. Broad Research in Artificial Intelligence and Neuroscience,
ISSN 2067-3957, Volume 1, July 2010,
Special Issue on Complexity in Sciences and Artificial Intelligence,
Eds. B. Iantovics, D. Rǎdoiu, M. Mǎruşteri and M. Dehmer

ON THE (COLORED) YANG-BAXTER EQUATION

David Hobby, Barna Laszlo Iantovics and Florin F. Nichita

Abstract. The quantum Yang-Baxter equation first appeared in the-
oretical physics and statistical mechanics. Afterwards, it has proved to be
important also in knot theory, quantum groups, etc. This paper deals with
the (colored) Yang-Baxter equation and computational methods. A new result
about the set-theoretical Yang-Baxter equation is presented.

Keywords: Yang-Baxter equations, computational complexity, computa-
tionally hard problems, applications of mathematics.

2000 Mathematics Subject Classification: 16W30, 13B02.

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Hobby, Iantovics, Nichita - On the Yang-Baxter equations

1. Introduction and Preliminaries

The quantum Yang-Baxter equation first appeared in theoretical physics
and statistical mechanics. Afterwards, it has proved to be important also in
knot theory, quantum groups, the quantization of integrable non-linear evolu-
tion systems, etc.

Throughout this paper k is a field. All tensor products appearing in this
paper are defined over k. For V a k-space, we denote by τ : V ⊗ V → V ⊗V
the twist map defined by τ (v⊗w) = w⊗v, and by I : V → V the identity map
of the space V.

We use the following notation concerning the Yang-Baxter equation.
If R : V ⊗V → V ⊗V is a k-linear map, then R12 = R⊗I, R23 = I⊗R, R13 =

(I⊗τ )(R⊗I)(I⊗τ ).
Definition. An invertible k-linear map R : V ⊗V → V ⊗V is called a

Yang-Baxter operator if it satisfies the equation

R12 ◦ R23 ◦ R12 = R23 ◦ R12 ◦ R23 (1)

Remark. The equation (1) is usually called the braid equation. It is a
well-known fact that the operator R satisfies (1) if and only if R◦τ satisfies the
constant quantum Yang-Baxter equation (QYBE), if and only if τ ◦R satisfies
the constant QYBE:

R12 ◦ R13 ◦ R23 = R23 ◦ R13 ◦ R12 (2)

Remark (i) τ : V ⊗V → V ⊗V is an example of a Yang-Baxter operator.
(ii) An exhaustive list of invertible solutions for (2) in dimension 2 was

obtained in [5].
(iii) Finding all Yang-Baxter operators in dimension greater than 2 is an

unsolved problem.

Let A be an associative k-algebra, and α, β, γ ∈ k. We define the k-linear
map:

RAα,β,γ : A⊗A → A⊗A, RAα,β,γ(a⊗b) = αab⊗1 + β1⊗ab − γa⊗b

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Hobby, Iantovics, Nichita - On the Yang-Baxter equations

Theorem. (S. Dăscălescu and F. F. Nichita, [4]) Let A be an associative
k-algebra with dim A ≥ 2, and α, β, γ ∈ k. Then RAα,β,γ is a Yang-Baxter
operator if and only if one of the following holds:

(i) α = γ 6= 0, β 6= 0;
(ii) β = γ 6= 0, α 6= 0;
(iii) α = β = 0, γ 6= 0.
If so, we have (RAα,β,γ)

−1 = RA1
β

, 1
α

, 1
γ

in cases (i) and (ii), and (RA0,0,γ)
−1 =

RA
0,0, 1

γ

in case (iii).

Remark. In the next section, we will generalize the construction given in
the previous theorem, in order to solve another type of Yang-Baxter equation.

2. The main result

There are many versions of the Yang-Baxter equation. We present a lesser
known version of this equation (which is called the “colored Yang-Baxter equa-
tion” by some authors) in this paper.

We attempt to find solutions for it and to explain how computational meth-
ods could help us. In another paper, we will study its applications in theoretical
physics.

Formally, a colored Yang-Baxter operator is defined as a function

R : k × X × X → EndkV ⊗ V ,

where X is a set and V is a finite dimensional vector space over a field k.
Thus, for any x ∈ k, u, v ∈ X, R(x, u, v) : V ⊗ V → V ⊗ V is a linear

operator.
As in the previous section, we consider three operators acting on a triple

tensor product V ⊗ V ⊗ V , R12(x, u, v) = R(x, u, v) ⊗ I, R23(x, v, w) = I ⊗
R(x, v, w), and similarly R13(x, u, w) as an operator that acts non-trivially on
the first and third factor in V ⊗ V ⊗ V .

R is a colored Yang-Baxter operator if it satisfies the equation:

R12(x, u, v)R13(x+y, u, w)R23(y, v, w) = R23(y, v, w)R13(x+y, u, w)R12(x, u, v)
(3)

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Hobby, Iantovics, Nichita - On the Yang-Baxter equations

for all x, y ∈ k, u, v, w ∈ X.
We now apply our original method to find solutions for the equations (3).

We assume that X is equal to (a subset of) the ground field k. Our method
of constructing solutions to equation (3) is based on the ideas applied before.

The key point of the construction is to suppose that V = A is an associative
k-algebra, and then to derive a solution to equation (3) from the associativity
of the product in A.

Thus, we seek solutions to equation (3) of the following form

R(x, u, v)(a ⊗ b) = αx(u, v)1 ⊗ ab + βx(u, v)ab ⊗ 1 − γx(u, v)b ⊗ a, (4)
where αx, βx, γx are k-valued functions on X × X, for any x ∈ k.

Inserting this ansatz into equation (3), we obtained the following system
of equations (whose solutions produce colored Yang-Baxter operators):

(βy(v, w) − γy(v, w))(αx(u, v)βx+y(u, w) − αx+y(u, w)βx(u, v))
+(αx(u, v) − γx(u, v))(αy(v, w)βx+y(u, w) − αx+y(u, w)βy(v, w)) = 0 (5)

βy(v, w)(βx(u, v) − γx(u, v))(αx+y(u, w) − γx+y(u, w))
+(αy(v, w) − γy(v, w))(βx+y(u, w)γx(u, v) − βx(u, v)γx+y(u, w)) = 0 (6)

αx(u, v)βy(v, w)(αx+y(u, w) − γx+y(u, w)) + αy(v, w)γx+y(u, w)(γx(u, v) − αx(u, v))
+γy(v, w)(αx(u, v)γx+y(u, w) − αx+y(u, w)γx(u, v)) = 0 (7)

αx(u, v)βy(v, w)(βx+y(u, w) − γx+y(u, w)) + βy(v, w)γx+y(u, w)(γx(u, v) − βx(u, v))
+γy(v, w)(βx(u, v)γx+y(u, w) − βx+y(u, w)γx(u, v)) = 0 (8)

αx(u, v)(αy(v, w) − γy(v, w))(βx+y(u, w) − γx+y(u, w))
+(βx(u, v) − γx(u, v))(αx+y(u, w)γy(v, w) − αy(v, w)γx+y(u, w)) = 0 (9)

Remark.
(i) The system of equations (5–9) is rather non-trivial. It is an open problem

to classify its solutions.

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Hobby, Iantovics, Nichita - On the Yang-Baxter equations

(ii) That system has some remarkable symmetry properties which can be
used to find some solutions. For example, the equations (6) and (9) are in
some sense dual to each other. Likewise, (7) and (8) are in some sense dual to
each other.

(iii) If we look at the system of equations (5–9), one simplification that
produces many solutions is to let αx, βx and γx be functions that do not depend
on x. In this case we may call them α, β, and γ. Letting γ be an arbitrary
function and a an arbitrary element of the base field k, one verifies that setting
α = aγ and β = γ gives a solution. Similarly, letting γ and b ∈ k be arbitrary,
setting α = γ and β = bγ gives a solution.

3. The set-theoretical Yang-Baxter equation

If X is a set, let S : X × X → X × X be a function, S12 = S × I and
S23 = I × S.

Definition. Using the above notation, the set-theoretical Yang-Baxter
equation reads:

S12 ◦ S23 ◦ S12 = S23 ◦ S12 ◦ S23 (10)

We obtained the following theorem, which will be proved in another paper.

Theorem. Let X be a set and R ⊂ X × X a reflexive relation on X.
We define the function S = SR : X × X → X × X, by
S(u, v) = (u, v) if (u, v) ∈ R,
S(u, v) = (v, u) if (u, v) /∈ R.
Then, S satisfies (10) if and only if R ∪ Rop is an equivalence relation, and

R̄ is a strict partial order relation on each class of R ∪ Rop.
(We let Rop denote the opposite relation of R and let R̄ denote the comple-

mentary relation of R.)

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Hobby, Iantovics, Nichita - On the Yang-Baxter equations

4. Conclusions

Computational methods are very important in solving the Yang-Baxter
equations. For example, Hietarinta found all R-matrices in the case n=2 using
Grobner basis and computer calculations (see [5], [6]). These papers are very
much cited by many authors.

However, the computational methods are not powerful enough to fully clas-
sify the solutions for other small dimensions: a complete computer calculation
for n=3 is still out of reach at this time.

On the other hand, the computational methods can be very useful in ap-
proaching problems in Quantum Group Theory. For example, one can ex-
plicitly solve the set-theoretical Yang-Baxter equation (10) for small sets be
exhaustive search. Computational methods are helpful for solving other equa-
tions and problems in Quantum Group Theory.

In Section 3, we give a new theorem about the set-theoretical Yang-Baxter
equation which will be proved and studied in future papers. We anticipate
interesting connections of it with the constructions given in Section 2.

Finally, we propose the problem of finding algorithms for solving the system
of equations (5–9).

References

[1] R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Acad.
Press, London,1982.

[2] R.J. Baxter, Partition function for the eight-vertex lattice model, Ann.
Physics 70(1972), 193-228.

[3] T. Brzezinski and F. F. Nichita, Yang-Baxter Systems and Entwining
Structures,Communications in Algebra, vol. 33(4), 1083-1093, 2005.

[4] S. Dascalescu and F. Nichita, Yang-Baxter operators arising from (co)algebra
structures, Communications in Algebra, Vol. 27, Nr. 12, 5833-5845(1999).

[5] J. Hietarinta, All solutions to the constant quantum Yang-Baxter equa-
tion in two dimensions, Phys. Lett. A 165(1992), 245-251.

[6] J. Hietarinta, Solving the two-dimensional constant quantum Yang-
Baxter equation. J. Math. Phys. 34 (1993), no. 5, 1725–1756.

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Hobby, Iantovics, Nichita - On the Yang-Baxter equations

David Hobby
Department of Mathematics
SUNY of New Paltz, USA
email:hobbyd@newpaltz.edu

Barna Laszlo Iantovics
Petru Maior University of Targu Mures, Romania
email:ibarna@upm.ro

Florin Felix Nichita
Institute of Mathematics ”Simion Stoilow” of the Romanian Academy
Bucharest, Romania
email:Florin.Nichita@imar.ro

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