In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a quantum affine algebra (or affine quantum group) is a
Hopf algebra Hopf is a German surname. Notable people with the surname include:
*Eberhard Hopf (1902–1983), Austrian mathematician
*Hans Hopf (1916–1993), German tenor
*Heinz Hopf (1894–1971), German mathematician
*Heinz Hopf (actor) (1934–2001), Swedis ...
that is a ''q''-deformation of the
universal enveloping algebra of an
affine Lie algebra In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody a ...
. They were introduced independently by and as a special case of their general construction of a
quantum group
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras) ...
from a
Cartan matrix. One of their principal applications has been to the theory of
solvable lattice models in
quantum statistical mechanics
Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble (probability distribution over possible quantum states) is described by a density operator ''S'', which is a ...
, where the
Yang–Baxter equation occurs with a
spectral parameter. Combinatorial aspects of the representation theory of quantum affine algebras can be described simply using
crystal bases
A crystal base for a representation of a quantum group on a \Q(v)-vector space
is not a base of that vector space but rather a \Q-base of L/vL where L is a \Q(v)-lattice in that vector spaces. Crystal bases appeared in the work of and also in the ...
, which correspond to the degenerate case when the deformation parameter ''q'' vanishes and the
Hamiltonian of the associated lattice model can be explicitly diagonalized.
See also
*
Quantum enveloping algebra
*
Quantum KZ equations
*
Littelmann path model
*
Yangian
References
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Quantum groups
Representation theory
Exactly solvable models
Mathematical quantization
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