Tensor Product Of Fundamental Representations Research Articles

Web calculus and tilting modules in type $C_2$

Using Kuperberg’s web calculus (1996), and following Elias and Libedinsky, we describe a “light leaves” algorithm to construct a basis of morphisms between arbitrary tensor products of fundamental representations for \mathfrak{sp}_4 (and the associated quantum group). Our argument has very little dependence on the base field. As a result, we prove that when [2]_q\ne 0 , the Karoubi envelope of the C_2 web category is equivalent to the category of tilting modules for the divided powers quantum group U_q^{\mathcal{A}}(\mathfrak{sp}_4) .

Triple clasp formulas for C2 webs

Using the light ladder basis for Kuperberg's C2 webs, we derive triple clasp formulas for idempotents projecting to the top summand in each tensor product of fundamental representations. We then find explicit formulas for the coefficients occurring in the clasps, by computing these coefficients as local intersection forms. Our formulas provide further evidence for Elias's clasp conjecture, which was given for type A webs, and suggests how to generalize the conjecture to non-simply laced types.

ON CLASSICAL LIMITS OF BETHE SUBALGEBRAS IN YANGIANS

The Yangian Y (𝔤) of a simple Lie algebra 𝔤 can be regarded as a deformation of two different Hopf algebras: the universal enveloping algebra of the current algebra \( U\left(\mathfrak{g}\left[t\right]\right) \) and the coordinate ring of the first congruence subgroup \( \mathcal{O}\left({G}_1\left[\left[{t}^{-1}\right]\right]\right). \) Both of these algebras are obtained from the Yangian by taking the associated graded with respect to an appropriate filtration on Y (𝔤).Bethe subalgebras B(C) in Y (𝔤) form a natural family of commutative subalgebras depending on a group element C of the adjoint group G. The images of these algebras in tensor products of fundamental representations give all integrals of the quantum XXX Heisenberg magnet chain.We describe the associated graded of Bethe subalgebras in the Yangian Y (𝔤) of a simple Lie algebra 𝔤 as subalgebras in \( U\left(\mathfrak{g}\left[t\right]\right) \) and in \( \mathcal{O}\left({G}_1\left[\left[{t}^{-1}\right]\right]\right) \) for all semisimple C ∈ G. In particular, we show that the associated graded in \( U\left(\mathfrak{g}\left[t\right]\right) \) of the Bethe subalgebra B(E) assigned to the unity element of G is the universal Gaudin subalgebra of \( U\left(\mathfrak{g}\left[t\right]\right) \) obtained from the center of the corresponding affine Kac–Moody algebra \( \hat{\mathfrak{g}} \) at the critical level. This generalizes Talalaev’s formula for generators of the universal Gaudin subalgebra to 𝔤 of any type. In particular, this shows that higher Hamiltonians of the Gaudin magnet chain can be quantized without referring to the Feigin–Frenkel center at the critical level.Using our general result on the associated graded of Bethe subalgebras, we compute some limits of Bethe subalgebras corresponding to regular semisimple C ∈ G as C goes to an irregular semisimple group element C0. We show that this limit is the product of the smaller Bethe subalgebra B(C0) and a quantum shift of argument subalgebra in the universal enveloping algebra of the centralizer of C0 in 𝔤. This generalizes the Nazarov–Olshansky solution of Vinberg’s problem on quantization of the (Mishchenko–Fomenko) shift of argument subalgebras.

Khovanov–Rozansky homology for infinite multicolored braids

AbstractWe define a limiting ${\mathfrak {sl}_N}$ Khovanov–Rozansky homology for semi-infinite positive multicolored braids. For a large class of such braids, we show that this limiting homology categorifies a highest-weight projector in the tensor product of fundamental representations determined by the coloring of the braid. This effectively completes the extension of Cautis’ similar result for infinite twist braids, begun in our earlier papers with Islambouli and Abel. We also present several similar results for other families of semi-infinite and bi-infinite multicolored braids.

On category O for affine Grassmannian slices and categorified tensor products

Truncated shifted Yangians are a family of algebras which naturally quantize slices in the affine Grassmannian. These algebras depend on a choice of two weights λ and μ for a Lie algebra g, which we will assume is simply laced. In this paper, we relate the category O over truncated shifted Yangians to categorified tensor products: For a generic integral choice of parameters, category O is equivalent to a weight space in the categorification of a tensor product of fundamental representations defined by the third author using KLRW algebras. We also give a precise description of category O for arbitrary parameters using a new algebra which we call the parity KLRW algebra. In particular, we confirm the conjecture of the authors that the highest weights of category O are in canonical bijection with a product monomial crystal depending on the choice of parameters. This work also has interesting applications to classical representation theory. In particular, it allows us to give a classification of simple Gelfand–Tsetlin modules of U ( gl n ) and its associated W-algebras.

On a variation of the Erdős–Selfridge superelliptic curve

Truncated shifted Yangians are a family of algebras which naturally quantize slices in the affine Grassmannian. These algebras depend on a choice of two weights $\lambda$ and $\mu$ for a Lie algebra $\mathfrak{g}$, which we will assume is simply-laced. In this paper, we relate the category $\mathcal{O}$ over truncated shifted Yangians to categorified tensor products: for a generic integral choice of parameters, category $\mathcal{O}$ is equivalent to a weight space in the categorification of a tensor product of fundamental representations defined by the third author using KLRW algebras. We also give a precise description of category $\mathcal{O}$ for arbitrary parameters using a new algebra which we call the parity KLRW algebra. In particular, we confirm the conjecture of the authors that the highest weights of category $\mathcal{O}$ are in canonical bijection with a product monomial crystal depending on the choice of parameters. This work also has interesting applications to classical representation theory. In particular, it allows us to give a classification of simple Gelfand-Tsetlin modules of $U(\mathfrak{gl}_n)$ and its associated W-algebras.

FUNDAMENTAL REPRESENTATIONS OF QUANTUM AFFINE SUPERALGEBRAS AND R-MATRICES

We study a certain family of finite-dimensional simple representations over quantum affine superalgebras associated to general linear Lie superalgebras, the so-called fundamental representations: the denominators of rational $R$-matrices between two fundamental representations are computed; a cyclicity (and so simplicity) condition on tensor products of fundamental representations is proved.

Gaudin models for 𝔤𝔩(m|n)

We establish the basics of the Bethe ansatz for the Gaudin model associated to the Lie superalgebra 𝔤𝔩(m|n). In particular, we prove the completeness of the Bethe ansatz in the case of tensor products of fundamental representations.

Local Weyl Modules and Cyclicity of Tensor Products for Yangian of Type G 2

Let $\mathfrak{g}$ be the exceptional complex simple Lie algebra of type $G_2$. We provide a concrete cyclicity condition for the tensor product of fundamental representations of the Yangian $Y(\mathfrak{g})$. Using this condition, we show that every local Weyl module is isomorphic to an ordered tensor product of fundamental representations of $Y(\mathfrak{g})$.

Local Weyl modules and cyclicity of tensor products for Yangians

We provide a sufficient condition for the cyclicity of an ordered tensor product L=Va1(ωb1)⊗Va2(ωb2)⊗…⊗Vak(ωbk) of fundamental representations of the Yangian Y(g). When g is a classical simple Lie algebra, we make the cyclicity condition concrete, which leads to an irreducibility criterion for the ordered tensor product L. In the case when g=sll+1, a sufficient and necessary condition for the irreducibility of the ordered tensor product L is obtained. The cyclicity of the ordered tensor product L is closely related to the structure of the local Weyl modules of Y(g). We show that every local Weyl module is isomorphic to an ordered tensor product of fundamental representations of Y(g).

Tensor Products, Characters, and Blocks of Finite-Dimensional Representations of Quantum Affine Algebras at Roots of Unity

We establish several results concerning tensor products, q-characters, and the block decomposition of the category of finite-dimensional representations of quantum affine algebras in the root of unity setting. In the generic case, a Weyl module is isomorphic to a tensor product of fundamental representations and this isomorphism was essential for establishing the block decomposition theorem. This is no longer true in the root of unity setting. We overcome the lack of such a tool by utilizing results on specialization of modules. Furthermore, we establish a sufficient condition for a Weyl module to be a tensor product of fundamental representations and prove that this condition is also necessary when the underlying simple Lie algebra is sl(2). We also study the braid group invariance of q-characters of fundamental representations.

Algebraic Bethe ansatz for the elliptic quantum group Eτ,η(A2(2))

We implement the Bethe ansatz method for the elliptic quantum group Eτ,η(A2(2)). The Bethe creation operators are constructed as polynomials of the Lax matrix elements expressed through a recurrence relation. We also give the eigenvalues of the family of commuting transfer matrices defined in the tensor product of fundamental representations.

Littelmann paths for the basic representation of an affine Lie algebra

A highest-weight representation of an affine Lie algebra g ˆ can be modeled combinatorially in several ways, notably by the semi-infinite paths of the Kyoto school and by Littelmann's finite paths. In this paper, we unify these two models in the case of the basic representation of an untwisted affine algebra, provided the underlying finite-dimensional algebra g possesses a minuscule representation (i.e., g is of classical or E 6 , E 7 type). We apply our “coil model” to prove that the basic representation of g ˆ , when restricted to g , is a semi-infinite tensor product of fundamental representations, and certain of its Demazure modules are finite tensor products.

Finite-dimensional modules of quantum affine algebras

We construct finite-dimensional representations of the quantum affine algebra associated to the affine Lie algebra sl 2 . We define an explicit action of the Drinfeld generators on the vector space tensor product of fundamental representations. The action is defined on a basis of eigenvectors for some of the generators, where the eigenvalues are the coefficients in the Laurent expansion of certain rational functions related to the Drinfeld polynomial corresponding to the module. In these modules we find a maximal submodule and by studying the quotient, we get an explicit description of all finite-dimensional irreducible representations corresponding to Drinfeld polynomials with distinct zeroes. We then describe the associated trigonometric solutions of the quantum Yang–Baxter equation.

Combinatorics of q-Characters of Finite-Dimensional Representations of Quantum Affine Algebras

We study finite-dimensional representations of quantum affine algebras using q-characters. We prove the conjectures from math.QA/9810055 and derive some of their corollaries. In particular, we prove that the tensor product of fundamental representations is reducible if and only if at least one of the corresponding normalized R-matrices has a pole.

Computation of Lickorish's Three Manifold Invariant Using Chern-Simons Theory

It is well known that any three-manifold can be obtained by surgery on a framed link in $S^3$. Lickorish gave an elementary proof for the existence of the three-manifold invariants of Witten using a framed link description of the manifold and the formalisation of the bracket polynomial as the Temperley-Lieb Algebra. Kaul determined three-manifold invariants from link polynomials in SU(2) Chern-Simons theory. Lickorish's formula for the invariant involves computation of bracket polynomials of several cables of the link. We describe an easier way of obtaining the bracket polynomial of a cable using representation theory of composite braiding in SU(2) Chern-Simons theory. We prove that the cabling corresponds to taking tensor products of fundamental representations of SU(2). This enables us to verify that the two apparently distinct three-manifold invariants are equivalent for a specific relation of the polynomial variables.

Algebraic solution of the Hubbard model on the infinite interval

We develop the quantum inverse scattering method for the one-dimensional Hubbard model on the infinite line at zero density. This enables us to diagonalize the Hamiltonian algebraically. The eigenstates can be classified as scattering states of particles, bound pairs of particles and bound states of pairs. We obtain the corresponding creation and annihilation operators and calculate the S-matrix. The Hamiltonian on the infinite line is invariant under the Yangian quantum group Y(su(2)). We show that the n-particle scattering states transform like n-fold tensor products of fundamental representations of Y(su(2) ) and that the bound states are Yangian singlet.

Finite-Dimensional Representations of Quantum Affine Algebras

We present a conjecture on the irreducibility of the tensor products of fundamental representations of quantized affme algebras. This conjecture implies in particular that the irreducibility of the tensor products of fundamental representations is completely described by the poles of R -matrices. The conjecture is proved in the cases of type A_n^{(1)} and C_n^{(1)} .

Yangians, integrable quantum systems and Dorey's rule

We study tensor products of fundamental representations of Yangians and show that the fundamental quotients of such tensor products are given by Dorey's rule.

Fundamental representations of quantum groups at roots of 1

To every finite-dimensional irreducible representation V of the quantum group Ue(g) where e is a primitive lth root of unity (l odd) and g is a finite-dimensional complex simple Lie algebra, de Concini, Kac and Procesi have associated a conjugacy class C V in the adjoint group G of g. We describe explicitly, when g is of type A n , B n , C n , or D n , the representations associated to the conjugacy classes of minimal positive dimension. We call such representations fundamental and prove that, for any conjugacy class, there is an associated representation which is contained in a tensor product of fundamental representations.