arXiv subjects

Sachin Gautam

Publications and source records attributed to Sachin Gautam.

12 recordsLinked to original sources

arXiv2026

The R-matrix formalism for quantized enveloping algebras

Let $U_\hbar\mathfrak{g}$ denote the Drinfeld-Jimbo quantum group associated to a simple Lie algebra $\mathfrak{g}$. We apply a modification of the $R$-matrix construction for quantum groups to the evaluation of the universal $R$-matrix of $U_\hbar\mathfrak{g}$ on any of its finite-dimensional representations. This produces a quantized enveloping algebra $\mathrm{U}_\mathrm{R}(\mathfrak{g})$ whose definition is given in terms of two generating matrices satisfying variants of the well-known $RLL$ relations. We prove that $\mathrm{U}_\mathrm{R}(\mathfrak{g})$ is isomorphic to the tensor product of the quantum double of the Borel subalgebra $U_\hbar\mathfrak{b}\subset U_\hbar\mathfrak{g}$ and a quantized polynomial algebra encoded by the space of $\mathfrak{g}$-invariants associated to the semiclassical limit $V$ of the underlying finite-dimensional representation. Using this description, we characterize $U_\hbar\mathfrak{g}$ and the quantum double of $U_\hbar\mathfrak{b}$ as Hopf quotients of $\mathrm{U}_\mathrm{R}(\mathfrak{g})$ and as fixed-point subalgebras with respect to certain natural automorphisms. As an additional corollary, we deduce that $\mathrm{U}_\mathrm{R}(\mathfrak{g})$ is quasitriangular precisely when $V$ is multiplicity free.

math.QA↗

arXiv2026

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, 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.

math.RT↗

arXiv2025

On the uniqueness of Yangians

Let $\mathfrak{g}$ be a simple Lie algebra over the complex numbers, and let $\mathfrak{g}[u]$ denote its polynomial current algebra. In the mid-1980s, Drinfeld introduced the Yangian of $\mathfrak{g}$ as the unique solution to a quantization problem for a natural Lie bialgebra structure on $\mathfrak{g}[u]$. More precisely, Theorem 2 of [Dokl. Akad. Nauk SSSR 283 (1985), no. 5, 1060-1064] asserts that $\mathfrak{g}[u]$ admits a unique homogeneous quantization, the Yangian of $\mathfrak{g}$, which is described explicitly via generators and relations, starting from a copy of $\mathfrak{g}$ and its adjoint representation. Although the representation theory of Yangians has since undergone substantial development, a complete proof of Drinfeld's theorem has not appeared. In this article, we present a proof of the assertion that $\mathfrak{g}[u]$ admits at most one homogeneous quantization. Our argument combines cohomological and computational methods, and outputs a presentation of any such quantization using Drinfeld's generators and a reduced set of defining relations.

math.QA↗

arXiv2022

Poles of finite-dimensional representations of Yangians

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, and let $Y_{\hbar}(\mathfrak{g})$ be the Yangian of $\mathfrak{g}$. In this paper, we study the sets of poles of the rational currents defining the action of $Y_{\hbar}(\mathfrak{g})$ on an arbitrary finite-dimensional vector space $V$. Using a weak, rational version of Frenkel and Hernandez' Baxter polynomiality, we obtain a uniform description of these sets in terms of the Drinfeld polynomials encoding the composition factors of $V$ and the inverse of the $q$-Cartan matrix of $\mathfrak{g}$. We then apply this description to obtain a concrete set of sufficient conditions for the cyclicity and simplicity of the tensor product of any two irreducible representations, and to classify the finite-dimensional irreducible representations of the Yangian double.

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arXiv2022

On a conjecture of Khoroshkin and Tolstoy

We prove a no-go theorem on the factorization of the lower triangular part in the Gaussian decomposition of the Yangian's universal $R$-matrix, yielding a negative answer to a conjecture of Khoroshkin and Tolstoy from [Lett. Math. Phys. vol. 36 1996].

math.QA↗

arXiv2020

The meromorphic R-matrix of the Yangian

Let g be a complex semisimple Lie algebra and Yg its Yangian. Drinfeld proved that the universal R-matrix of Yg gives rise to rational solutions of the quantum Yang-Baxter equations on irreducible, finite-dimensional representations of Yg. This result was recently extended by Maulik-Okounkov to symmetric Kac-Moody algebras, and representations arising from geometry. We show that this rationality ceases to hold for arbitrary finite-dimensional representations, at least if one requires such solutions to be natural with respect to the representation and compatible with tensor products. Equivalently, the tensor category of finite-dimensional representations of Yg does not admit rational commutativity constraints. We construct instead two meromorphic commutativity constraints, which are related by a unitarity condition. We show that each possesses an asymptotic expansion as s tends to infinity, which has the same formal properties as Drinfeld's R(s), and therefore coincides with the latter by uniqueness. In particular, we give an alternative, constructive proof of the existence of the universal R-matrix of Yg. Our construction relies on the Gauss decomposition R^+(s)R^0(s)R^-(s) of R(s). The divergent abelian term R^0 was resummed on finite-dimensional representations by the first two authors in arXiv:1403.5251. The main ingredient of the present paper is the construction of R^+(s) and R^-(s). We prove that they are rational functions on finite-dimensional representations, and that they intertwine the standard coproduct of Yg and the deformed Drinfeld coproduct introduced in arXiv:1403.5251.

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arXiv2019

An explicit isomorphism between quantum and classical sl(n)

Let g be a complex, semisimple Lie algebra. Drinfeld showed that the quantum group associated to g is isomorphic as an algebra to the trivial deformation of the universal enveloping algebra of g. In this paper we construct explicitly such an isomorphism when g = sl(n), previously known only for n=2.

math.RT↗

arXiv2019

Elliptic quantum groups and their finite-dimensional representations

Let g be a complex semisimple Lie algebra, tau a point in the upper half-plane, and h a complex deformation parameter such that the image of h in the elliptic curve E_tau is of infinite order. In this paper, we give an intrinsic definition of the category of finite-dimensional representations of the elliptic quantum group E_{h,tau}(g) associated to g. The definition is given in terms of Drinfeld half-currents and extends that given by Enriquez-Felder for g=sl_2. When g=sl_n, it reproduces Felder's RLL definition via the Gauss decomposition obtained by Enriquez-Felder for n=2 and by the first author for n greater than 2. We classify the irreducible representations of E_{h,tau} in terms of elliptic Drinfeld polynomials, in close analogy to the case of the Yangian Y_h(g) and quantum loop algebra U_q(Lg) of g. A crucial ingredient in the classification, which circumvents the fact that E_{h,tau} does not appear to admit Verma modules, is a functor from finite-dimensional representations of U_q(Lg) to those of E_{h,tau} which is an elliptic analogue of the monodromy functor constructed in our previous work arXiv:1310.7318. Our classification is new even for g=sl_2, and holds more generally when g is a symmetrisable Kac-Moody algebra, provided finite-dimensionality is replaced by an integrability and category O condition.

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arXiv2017

Meromorphic tensor equivalence for Yangians and quantum loop algebras

Let ${\mathfrak g}$ be a complex semisimple Lie algebra, and $Y_h({\mathfrak g})$, $U_q(L{\mathfrak g})$ the corresponding Yangian and quantum loop algebra, with deformation parameters related by $q=\exp(πi h)$. When $h$ is not a rational number, we constructed in arXiv:1310.7318 a faithful functor $Γ$ from the category of finite-dimensional representations of $Y_h ({\mathfrak g})$ to those of $U_q(L{\mathfrak g})$. The functor $Γ$ is governed by the additive difference equations defined by the commuting fields of the Yangian, and restricts to an equivalence on a subcategory of $Y_h({\mathfrak g})$ defined by choosing a branch of the logarithm. In this paper, we construct a tensor structure on $Γ$ and show that, if $|q|\neq 1$, it yields an equivalence of meromorphic braided tensor categories, when $Y_h({\mathfrak g})$ and $U_q(L{\mathfrak g})$ are endowed with the deformed Drinfeld coproducts and the commutative part of the universal $R$-matrix. This proves in particular the Kohno-Drinfeld theorem for the abelian $q$KZ equations defined by $Y_h({\mathfrak g})$. The tensor structure arises from the abelian $q$KZ equations defined by a appropriate regularisation of the commutative $R$-matrix of $Y_h({\mathfrak g})$.

math.QA↗

arXiv2013

Monodromy of the trigonometric Casimir connection for sl_2

We show that the monodromy of the trigonometric Casimir connection on the tensor product of evaluation modules of the Yangian Ysl_2 is described by the quantum Weyl group operators of the quantum loop algebra U_h(Lsl_2). The proof is patterned on the second author's computation of the monodromy of the rational Casimir connection for sl_n via the dual pair (gl_k,gl_n), and rests ultimately on the Etingof-Geer-Schiffmann computation of the monodromy of the trigonometric KZ connection. It relies on two new ingredients: an affine extension of the duality between the R-matrix of U_h(sl_k) and the quantum Weyl group element of U_h(sl_2), and a formula expressing the quantum Weyl group action of the coroot lattice of SL_2 in terms of the commuting generators of U_h(Lsl_2). Using this formula, we define quantum Weyl group operators for the quantum loop algebra U_h(Lgl_2) and show that they describe the monodromy of the trigonometric Casimir connection on a tensor product of evaluation modules of the Yangian Ygl_2

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arXiv2012

Yangians and quantum loop algebras

Let g be a complex, semisimple Lie algebra. Drinfeld showed that the quantum loop algebra U_h(Lg) of g degenerates to the Yangian Y_h(g). We strengthen this result by constructing an explicit algebra homomorphism Phi defined over Q[[h]] from U_h(Lg) to the completion of Y_h(g) with respect to its grading. We show moreover that Phi becomes an isomorphism when the quantum loop algebra is completed with respect to its its evaluation ideal. We construct a similar homomorphism for g=gl_n and show that it intertwines the geometric actions of U_h(L gl_n) and Y(gl_n) on the equivariant K-theory and cohomology of the variety of n-step flags in C^d constructed by Ginzburg and Vasserot.

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arXiv2008

Cluster algebras and Grassmannians of type G2

We prove a conjecture of Geiss, Leclerc and Schröer, producing cluster algebra structures on multi-homogeneous coordinate ring of partial flag varieties, for the case $G_2$. As a consequence we sharpen the known fact that coordinate ring of the double Bruhat cell $G^{e,w_0}$ is an upper cluster algebra, by proving that it is a cluster algebra.

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