arXiv subjects

Alexander Tsymbaliuk

Publications and source records attributed to Alexander Tsymbaliuk.

At least 19 recordsLinked to original sources

arXiv2026

Scary Wheels (and Super Shrubs)

We prove the shuffle realization for quantum affine superalgebras of types $B,C,D$ in the loop realization, in that we construct an isomorphism between each of these algebras and a suitably defined double shuffle algebra. We explicitly describe the latter shuffle algebra using certain vanishing conditions that generalize the Feigin-Odesskii wheel conditions; a novel feature is the appearance of a so-called scary wheel, which is a particular order $2$ vanishing condition involving $7$ variables. Along the way, we fully develop the theory of shrubs in super types $A$ (affine and toroidal), which are important combinatorial tools in the study of the shuffle algebras associated to toric Calabi-Yau threefolds. Our techniques also allow us to define quantum toroidal superalgebras of types $B,C,D$. More importantly, we provide a general framework to formulate and prove shuffle realizations in the wide generality of quivers with parameters.

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arXiv2026

Chains of affine standard Lyndon words

In this note, we establish the periodicity of chains of affine standard Lyndon words in all types and determine tight bounds on that periodicity, greatly generalizing the $A$-type results of arXiv:2305.16299. Our approach crucially utilizes the convexity and monotonicity of arXiv:2505.15432 together with the new idea to consider the polarization of the root system given by increasing and decreasing chains.

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arXiv2026

Orthosymplectic quantum groups revisited

We present the RLL-realization of extended orthosymplectic quantum supergroups for any parity sequence, with R-matrices evaluated in the earlier work arxiv:2408.16720. Our isomorphism is compatible with the internal structure of generalized doubles. We also relate different sign conventions through 2-cocycle twists. Furthermore, we establish a factorization of the reduced R-matrix within the RLL-realization.

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arXiv2026

Orthosymplectic $R$-matrices

We present a formula for trigonometric orthosymplectic $R$-matrices associated with any parity sequence, and establish their factorization into the ordered product of $q$-exponents parametrized by positive roots in the corresponding reduced root systems. The latter is crucially based on the construction of orthogonal bases of the positive subalgebra through $q$-bracketings and combinatorics of dominant Lyndon words, as developed in [Clark, Hill, Wang, "Quantum shuffles and quantum supergroups of basic type", Quantum Topol. 7 (2016), no.3, 553-638]. We further evaluate the affine orthosymplectic $R$-matrices, establishing their intertwining property as well as matching them with those obtained through the Yang-Baxterization technique of [Ge, Wu, Xue, "Explicit trigonometric Yang-Baxterization", Internat. J. Modern Phys. A 6 (1991), no.21, 3735-3779]. This reproduces the celebrated formulas of [Jimbo, "Quantum $R$ matrix for the generalized Toda system", Comm. Math. Phys. 102 (1986), no.4, 537-547] for classical BCD types and the formula of [Mehta, Dancer, Gould, Links, "Generalized Perk-Schultz models: solutions of the Yang-Baxter equation associated with quantized orthosymplectic superalgebras", J. Phys. A 39 (2006), no.1, 17-26] for the standard parity sequence.

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arXiv2026

On De Concini-Kac forms of quantum groups

Quantum groups of semisimple Lie algebras at roots of unity admit several different forms. Among them is the De Concini-Kac form, which is the easiest to define but, perhaps, hardest to study. In this paper, we propose a suitable modification to the De Concini-Kac form, namely the even part algebra, which has some appealing features. Notably, it behaves uniformly with respect to the order of the roots of unity and admits an adjoint action of the Lusztig form. We revisit several results due to De Concini-Kac-Procesi and Tanisaki for the even part algebra. Namely, we give conceptual definitions of the Frobenius and Harish-Chandra centers and describe the entire center in terms of these two subalgebras getting a complete quantum analog of the Veldkamp theorem on the center of the universal enveloping algebras in positive characteristic. We investigate the Azumaya locus of the even part algebra over its center. We also show that the locally finite part of the even part algebra under the adjoint action of the Lusztig form is isomorphic to the reflection equation algebra, which is the quantized coordinate algebra with the product twisted by $R$-matrix. Some results on Lusztig forms at roots of unity are revisited and proved in greater generality including Kempf vanishing theorem and good filtrations on the quantized coordinate algebra.

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arXiv2025

Fusion and specialization for type ADE shuffle algebras

Root vectors in quantum groups (of finite type) generalize to fused currents in quantum loop groups ([5]). In the present paper, we construct fused currents as duals to specialization maps of the corresponding shuffle algebras ([7,8,9]) in types ADE. Both root vectors and fused currents depend on a convex order of the positive roots, and the choice we make in the present paper is that of the Auslander-Reiten order ([24]) corresponding to an orientation of the type ADE Dynkin diagram.

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arXiv2025

Shuffle algebras and their integral forms: specialization map approach in types $C_n$ and $D_n$

We construct a family of PBWD bases for the positive subalgebras of quantum loop algebras of type $C_n$ and $D_n$, as well as their Lusztig and RTT integral forms, in the new Drinfeld realization. We also establish a shuffle algebra realization of these $\mathbb{Q}(v)$-algebras (proved earlier in arXiv:2102.11269 by completely different tools) and generalize the latter to the above $\mathbb{Z}[v,v^{-1}]$-forms. The rational counterparts provide shuffle algebra realizations of positive subalgebras of type $C_n$ and $D_n$ Yangians and their Drinfeld-Gavarini duals. While this naturally generalizes our earlier treatment of the classical type $B_n$ in arXiv:2305.00810 and $A_n$ in arXiv:1808.09536, the specialization maps in the present setup are more compelling.

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arXiv2025

Orthogonal bases for two-parameter quantum groups

In this note, we construct dual PBW bases of the positive and negative subalgebras of the two-parameter quantum groups $U_{r,s}(\mathfrak{g})$ in classical types, as used in our earlier work arXiv:2407.01450. Following the ideas of Leclerc and Clark-Hill-Wang, we introduce the two-parameter shuffle algebra and relate it to the subalgebras above. We then use the combinatorics of dominant Lyndon words to establish the main results.

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arXiv2025

Bicharacter twists of quantum groups

We apply the general construction of a twist of bigraded Hopf algebras by skew bicharacters to obtain two-parameter quantum groups in the Drinfeld-Jimbo, new Drinfeld (for affine types), and FRT (for both finite and affine) presentations from their standard one-parameter versions. This yields new elementary proofs of the fundamental results on two-parameter quantum groups that appeared in the literature over the last two decades, and also leads to natural generalizations in the super and multiparameter cases.

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arXiv2025

Two-Parameter Quantum Groups and $R$-Matrices: Classical Types

We construct finite $R$-matrices for the first fundamental representation $V$ of two-parameter quantum groups $U_{r,s}(\mathfrak{g})$ for classical $\mathfrak{g}$, both through the decomposition of $V\otimes V$ into irreducibles $U_{r,s}(\mathfrak{g})$-submodules as well as by evaluating the universal $R$-matrix. The latter is crucially based on the construction of dual PBW-type bases of $U^{\pm}_{r,s}(\mathfrak{g})$ consisting of the ordered products of quantum root vectors defined via $(r,s)$-bracketings and combinatorics of standard Lyndon words. We further derive explicit formulas for affine $R$-matrices, both through the Yang-Baxterization technique of [Internat. J. Modern Phys. A 6 (1991), 3735-3779] and as the unique intertwiner between the tensor product of $V(u)$ and $V(v)$, viewed as modules over two-parameter quantum affine algebras $U_{r,s}(\widehat{\mathfrak{g}})$ for classical $\mathfrak{g}$. The latter generalizes the formulas of [Comm. Math. Phys. 102 (1986), 537-547] for one-parametric quantum affine algebras.

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arXiv2025

Affine standard Lyndon words

In this note, we establish the convexity and monotonicity for affine standard Lyndon words in all types, generalizing the $A$-type results of arXiv:2305.16299. We also derive partial results on the structure of imaginary standard Lyndon words and present a conjecture for their general form. Additionally, we provide computer code in Appendix which, in particular, allows to efficiently compute affine standard Lyndon words in exceptional types for all orders.

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arXiv2025

Orthosymplectic Yangians

We study the RTT orthosymplectic super Yangians and present their Drinfeld realizations for any parity sequence, generalizing the results for non-super types BCD, a standard parity sequence, and super A-type.

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arXiv2025

Standard Lyndon loop words: weighted orders

We generalize the study of standard Lyndon loop words from [A.Negut, A.Tsymbaliuk, "Quantum loop groups and shuffle algebras via Lyndon words", Adv. Math. 439 (2024), Paper No. 109482] to a more general class of orders on the underlying alphabet, as suggested in Remark 3.15 of loc.cit. The main new ingredient is the exponent-tightness of these words, which also allows to generalize the construction of PBW bases of the untwisted quantum loop algebra via the combinatorics of loop words.

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arXiv2024

Affine Standard Lyndon words: A-type

We generalize an algorithm of Leclerc describing explicitly the bijection of Lalonde-Ram from finite to affine Lie algebras. In type $A_n^{(1)}$, we compute all affine standard Lyndon words for any order of the simple roots, and establish some properties of the induced orders on the positive affine roots.

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arXiv2024

Quantum loop groups and shuffle algebras via Lyndon words

We study PBW bases of the untwisted quantum loop group $U_q(L\mathfrak{g})$ (in the Drinfeld new presentation) using the combinatorics of loop words, by generalizing the treatment of [29,30,43] in the finite type case. As an application, we prove that Enriquez' homomorphism [11] from the positive half of the quantum loop group to the trigonometric degeneration of Feigin-Odesskii's elliptic algebra [15] associated to $\mathfrak{g}$ is an isomorphism.

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arXiv2024

Orthosymplectic superoscillator Lax matrices

We construct Lax matrices of superoscillator type that are solutions of the RTT-relation for the rational orthosymplectic $R$-matrix, generalizing orthogonal and symplectic oscillator type Lax matrices previously constructed by the authors in arXiv:2001.06825, arXiv:2104.14518, and arXiv:2112.12065. We further establish factorisation formulas among the presented solutions.

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arXiv2024

Shuffle algebras and their integral forms: specialization map approach in types $B_n$ and $G_2$

We construct a family of PBWD bases for the positive subalgebras of quantum loop algebras of type $B_n$ and $G_2$, as well as their Lusztig and RTT (for type $B_n$ only) integral forms, in the new Drinfeld realization. We also establish a shuffle algebra realization of these $\mathbb{Q}(v)$-algebras (proved earlier in arXiv:2102.11269 by completely different tools) and generalize the latter to the above $\mathbb{Z}[v,v^{-1}]$-forms. The rational counterparts provide shuffle algebra realizations of positive subalgebras of type $B_n$ and $G_2$ Yangians and their Drinfeld-Gavarini duals. All of this generalizes the type $A_n$ results of arXiv:1808.09536 by the second author.

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arXiv2023

Shuffle approach towards quantum affine and toroidal algebras

These are detailed lecture notes of the crash-course on shuffle algebras delivered by the author at Tokyo University of Marine Science and Technology during the second week of March 2019. These notes consist of three chapters, providing a separate treatment for: the quantum loop algebras of $\mathfrak{sl}_n$ (as well as their super- and 2-parameter generalizations), the quantum toroidal algebras of $\mathfrak{gl}_1$, and the quantum toroidal algebras of $\mathfrak{sl}_n$. We provide the shuffle realization of the corresponding ``positive'' subalgebras as well as of the commutative subalgebras and some combinatorial representations for the toroidal algebras. One of the key techniques involved is that of ``specialization maps''. Each chapter aims to emphasize a different aspect of the theory: in the first chapter we use shuffle algebras to construct a family of new PBWD bases for type $A$ quantum loop algebras and their integral forms; in the second chapter, we provide a geometric interpretation of the Fock modules and use shuffle description of a commutative subalgebra to construct an action of the Heisenberg algebra on the equivariant $K$-theory of the Hilbert schemes of points; in the last chapter, we relate vertex and combinatorial representations of quantum toroidal algebras of $\mathfrak{sl}_n$ using Miki's isomorphism and use shuffle realization to explicitly compute Bethe commutative subalgebras and their limits. The latter construction is inspired by Enriquez's work relating shuffle algebras to the correlation functions of quantum affinized algebras.

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