Groups and Lie algebras corresponding to the Yang–Baxter equations

Abstract

For a positive integer n we introduce quadratic Lie algebras tr n , qtr n and finitely discrete groups Tr n , QTr n naturally associated with the classical and quantum Yang-Baxter equation, respectively.We prove that the universal enveloping algebras of the Lie algebras tr n , qtr n are Koszul, and compute their Hilbert series. We also compute the cohomology rings for these Lie algebras (which by Koszulity are the quadratic duals of the enveloping algebras). Finally, we construct a basis of U(tr n ).We construct cell complexes which are classifying spaces of the groups Tr n and QTr n , and show that the boundary maps in them are zero, which allows us to compute the integral cohomology of these groups.We show that the Lie algebras tr n , qtr n map onto the associated graded algebras of the Malcev Lie algebras of the groups Tr n , QTr n , respectively. In the case of Tr n , we use quantization theory of 743 Lie bialgebras to show that this map is actually an isomorphism. At the same time, we show that the groups Tr n and QTr n are not formal for n 4.

Braided Drinfeld and Heisenberg doubles

“…(1.2). One application of the results of this paper is to show that given a solution (V, Ψ) to (1.1) and (W, Φ) to (1.2), (V ⊗ W, Ψ (1) ⊗ Φ (1) ⊗ Ψ (2) ⊗ Φ (2) ) again has the structure of a solution to the pentagon equation. For finite-dimensional Hopf algebras, this follows from a twisting result of [22] which is generalized to the context of braided Hopf algebras (so-called braided groups in the braided cocommutative case in [29,28] and other papers), in 3.8.3 and, more generally, monoidal categories in Theorem 2.2.14.…”

On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials

“…1. Let us observe that R w (1, 1) = S w (1), where S w (1) denotes the specialization x i := 1, ∀ i ≥ 1, of the Schubert polynomial S w (X n ) corresponding to permutation w. Therefore, R w (1, 1) is equal to the number of compatible sequences [13] (or pipe dreams, see, e.g., [129]) corresponding to permutation w. Problem 1. 7. Let w ∈ S n be a permutation and l := (w) be its length.…”

“…[7], is a group generated by the set of elements {Q i,j , 1 ≤ i = j ≤ n}, subject to the set of defining relations…”

“…For example, w = [1,4,5,6,8,3,5,7] ∈ S 8 is a grassmannian permutation such that S w (1) = 140, and R w (1, β) = (1, 9, 27, 43, 38, 18, 4).…”

“…25. Consider permutation v = [2,3,5,6,8,9,1,4,7] ∈ S 9 of the length 12, and set x := (1 + βq)(1 + βt). One can check that H v (q, t; β) = x 12 (1 + 2x) 1 + 6x + 19x 2 + 24x 3 + 13x 4 , and F v (β) = (1 + 2β)(1 + 6β + 19β 2 + 24β 3 + 13β 4 ).…”