Yang-Baxter operators
In their most familiar form, Yang–Baxter operators are certain invertible linear endomorphisms which have applications to physics and topology. In physics these operators often provide solutions to the quantum Yang–Baxter equation, an equation which has its roots in statistical mechanics [a30], [a7], [a28], [a31], [a35] (cf. also Statistical mechanics, mathematical problems in). In topology quite often they can be used to construct invariants of knots, links or three-dimensional manifolds (cf. also Knot theory; Link; Three-dimensional manifold); cf. [a30], [a17], [a18], [a26], [a27], [a32], [a33].
Closely related to the quantum Yang–Baxter equation is the braid equation. There are natural categorical structures associated with the braid and quantum Yang–Baxter equations which play an important role in quantum groups and their applications [a21], [a23], [a5], [a34].
Yang–Baxter operators in the category of left modules over a commutative ring
are certain
-linear mappings
. Let
,
and
, where
is the "twist" mapping defined for
-modules
and
by
for all
and
. Then
satisfies the quantum Yang–Baxter equation in
if
![]() | (a1) |
Note that satisfies (a1) if and only if
satisfies the braid equation in
, which is
![]() | (a2) |
If is invertible and satisfies (a2), that is,
, then
is a Yang–Baxter operator in
(see [a5]). There are other formulations of the notion of Yang–Baxter operator in the context of modules; see, e.g., [a8] and [a9].
Observe that the quantum Yang–Baxter and braid equations have natural formulations in any category with a suitable notion of tensor product and in which the tensor product of morphisms is defined [a16], [a20], [a34]. The notion of quantum Yang–Baxter operator thus has a natural generalization to categories
with such additional structure; see, e.g., [a16], [a5].
A good source of solutions to (a1) in are certain elements
, where
is an algebra over
. For
and a
-module
, let
be defined by
for all
, where
is regarded as a left
-module under component multiplication. Then
is a solution to (a1) for all left
-modules
if and only if
![]() | (a3) |
When is the algebra of
-matrices over
, then an
which satisfies (a1), or equivalently (a3), is called an
-matrix. Suppose that
and
, where
and
is the standard basis for
. Then (a3) is equivalent to
![]() | (a4) |
for all , which is probably the most familiar form of the quantum Yang–Baxter equation. Ordinarily, (a4) is written using the Einstein summation convention (cf. also Einstein rule), that is, summation signs are omitted with the understanding that indices that appear as upper and lower indices are summed over their full range of values.
Certain -algebras
with an
which satisfies (a1) can be used to construct invariants. Quasi-triangular Hopf algebras, in particular quantum algebras (cf. also Quasi-triangular Hopf algebra), give rise to regular isotopy invariants of
-
tangles. Ribbon Hopf algebras give rise to regular isotopy invariants of knots and links, and under mild restrictions they give rise to invariants of three-dimensional manifolds. Let
be a field. In this case a finite-dimensional Hopf algebra
over
is closely linked to these structures. The Hopf algebra
is a subHopf algebra of the quantum double
of
, which is a quasi-triangular Hopf algebra [a1]. Every finite-dimensional quasi-triangular Hopf algebra over
is a subHopf algebra of a ribbon Hopf algebra; in particular,
is a subHopf algebra of a ribbon Hopf algebra [a6].
The classification of -matrices seems to be a very daunting task, and most work to date (1998) has involved symbolic computation. Suppose that
is the field of complex numbers. Then the
-matrices are completely classified in the
case [a13] and the classification of one basic family is known in the
case [a14].
Some of the more important examples of -matrices, those related to the quantized enveloping algebras, are formal infinite sums or belong to a completed tensor product. See [a4], [a1] for discussion of this important part of the theory.
There is a category with a pre-braiding structure, defined and studied in [a34], associated to a bi-algebra
over
which gives rise to Yang–Baxter operators. Here, the formal variant
is considered, whose objects are left
-modules and right
-comodules
which satisfy the condition
![]() |
for all and
, where
denotes the coproduct
applied to
. For an object
of
, define
by
for all
. Then
satisfies (a1). The pre-braiding structure on
is the collection of morphisms of the form
which are defined for all pairs of objects
,
by
for all
and
. Observe that
is a solution to the braid equation (a2). When
is a Hopf algebra, the morphism
is invertible, and the collection of all
is referred to as a braiding structure. When
is a field and
is a finite-dimensional Hopf algebra over
, the category
can be identified with
, the category of left modules of the quantum double [a23].
The FRT construction of [a11], [a10] has an interesting interpretation in light of the preceding paragraph. Suppose that is a field and
is a solution to (a1), where
is a finite-dimensional vector space over
. The FRT construction
is a certain bi-algebra over
associated with
. There is a natural way of turning
into an object of
such that
, described in [a24]. For a universal description of the FRT construction associated with certain Yang–Baxter operators, see [a5], [a21]. See also [a8] for a discussion of algebras associated with Yang–Baxter operators.
There is a certain quotient of
which is more closely tied to
from a computational point of view. If
, then it is never the case that
is a Hopf algebra, whereas
may very well be a Hopf algebra [a25]. Determining new families of solutions to (a1) of the type described in the preceding paragraph may very well involve using a combination of bi-algebra techniques involving
and computer methods [a3], [a25].
There are parametrized versions of (a1), and hence parametrized versions of Yang–Baxter operators. Let be a set,
be a function and suppose that
is a non-empty subset with a (multiplication) mapping
. Then
satisfies the
-parameter quantum Yang–Baxter equation if
![]() |
holds for all . There is an FRT construction for
-parameter families [a3]. A
-parameter family of solutions to the quantum Yang–Baxter equation is a function
which satisfies
![]() |
![]() |
for all . For examples and discussion, see [a1], [a12], [a11], [a15], [a19].
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Yang-Baxter operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Yang-Baxter_operators&oldid=15986