28 research outputs found
The formal shift operator on the Yangian double
Let be a symmetrizable Kac-Moody algebra with associated
Yangian and Yangian double
. An elementary result of fundamental importance
to the theory of Yangians is that, for each , there is an
automorphism of corresponding to the translation
of the complex plane. Replacing by a formal parameter
yields the so-called formal shift homomorphism from
to the polynomial algebra .
We prove that uniquely extends to an algebra homomorphism
from the Yangian double into the -adic
closure of the algebra of Laurent series in with coefficients in the
Yangian . This induces, via evaluation at any point , a homomorphism from into
the completion of the Yangian with respect to its grading. We show that each
such homomorphism gives rise to an isomorphism between completions of
and and, as a corollary,
we find that the Yangian can be realized as a
degeneration of the Yangian double . Using these
results, we obtain a Poincar\'{e}-Birkhoff-Witt theorem for
applicable when is of finite
type or of simply-laced affine type.Comment: 40 page
Vertex Representations for Yangians of Kac-Moody algebras
30 pagesUsing vertex operators, we build representations of the Yangian of a simply laced Kac-Moody algebra and of its double. As a corollary, we prove the PBW property for simply laced affine Yangians.Peer reviewe
Poles of finite-dimensional representations of Yangians
Let be a finite-dimensional simple Lie algebra over
, and let be the Yangian of
. In this paper, we initiate the study of the set of poles of the
rational currents defining the action of on an
arbitrary finite-dimensional vector space . We prove that this set is
completely determined by the eigenvalues of the commuting Cartan currents of
, and therefore encodes the singularities of the
components of the -character of . We obtain sufficient conditions for a
tensor product of finite-dimensional irreducible representations to be cyclic
and irreducible, in terms of these sets of poles. Furthermore, we show that
these sets are precisely the zeros of the Yangian analogue of the Baxter
polynomials defined by Frenkel and Hernandez. For an irreducible
representation, we obtain a concrete description of its sets of poles, in terms
of the zeroes of the underlying Drinfeld polynomials and the inverse of the
-Cartan matrix of . In particular, our results yield a
complete classification of the finite-dimensional irreducible representations
of the Yangian double.Comment: 61 pages. Significantly revised version. Updates include: modified
introduction; relation with Baxter polynomials; proof of the Conjecture (for
simply-laced types) from the previous version, valid for any type; relation
with denominator formulae; and more references to the literatur
The R-matrix of the affine Yangian
Let g be an affine Lie algebra with associated Yangian Y_hg. We prove the
existence of two meromorphic R-matrices associated to any pair of
representations of Y_hg in the category O. They are related by a unitary
constraint and constructed as products of the form R(s)=R^+(s)R^0(s)R^-(s),
where R^+(s) = R^-_{21}(-s)^{-1}. The factor R^0(s) is a meromorphic, abelian
R-matrix, with a WKB-type singularity in h, and R^-(s) is a rational twist. Our
proof relies on two novel ingredients. The first is an irregular, abelian,
additive difference equation whose difference operator is given in terms of the
q-Cartan matrix of g. The regularization of this difference equation gives rise
to R^0(s) as the exponentials of the two canonical fundamental solutions. The
second key ingredient is a higher order analogue of the adjoint action of the
affine Cartan subalgebra of g on Y_hg. This action has no classical
counterpart, and produces a system of linear equations from which R^-(s) is
recovered as the unique solution. Moreover, we show that both operators give
rise to the same rational R-matrix on the tensor product of any two
highest-weight representations.Comment: 57 pages. 5 figure
Representations of twisted Yangians of types B, C, D: II
We continue the study of finite-dimensional irreducible representations of twisted Yangians associated to symmetric pairs of types B, C and D, with focus on those of types BI, CII and DI. After establishing that, for all twisted Yangians of these types, the highest weight of such a module necessarily satisfies a certain set of relations, we classify the finite-dimensional irreducible representations of twisted Yangians for the pairs and
Twisted Yangians of small rank
We study quantized enveloping algebras called twisted Yangians associated with the symmetric pairs of types CI, BDI, and DIII (in Cartan’s classification) when the rank is small. We establish isomorphisms between these twisted Yangians and the well known Olshanskii’s twisted Yangians of types AI and AII, and also with the Molev-Ragoucy reflection algebras associated with symmetric pairs of type AIII. We also construct isomorphisms with twisted Yangians in Drinfeld’s original presentation
Representations of twisted Yangians of types B, C, D: I
We initiate a theory of highest weight representations for twisted Yangians of types B, C, D and we classify the finite-dimensional irreducible representations of twisted Yangians associated to symmetric pairs of types CI, DIII and BCD0