Quantum groups, Yang-Baxter maps and quasi-determinants

For any quasi-triangular Hopf algebra, there exists the universal R-matrix, which satisfies the Yang-Baxter equation. It is known that the adjoint action of the universal R-matrix on the elements of the tensor square of the algebra constitutes a quantum Yang-Baxter map, which satisfies the set-theoretic Yang-Baxter equation. The map has a zero curvature representation among L-operators defined as images of the universal R-matrix. We find that the zero curvature representation can be solved by the Gauss decomposition of a product of L-operators. Thereby obtained a quasi-determinant expression of the quantum Yang-Baxter map associated with the quantum algebra $U_{q}(gl(n))$. Moreover, the map is identified with products of quasi-Pl\"{u}cker coordinates over a matrix composed of the L-operators. We also consider the quasi-classical limit, where the underlying quantum algebra reduces to a Poisson algebra. The quasi-determinant expression of the quantum Yang-Baxter map reduces to ratios of determinants, which give a new expression of a classical Yang-Baxter map.


Introduction
The Yang-Baxter map [Drinfeld 1990, Veselov 2000 is a map χ defined on a direct product of two sets and satisfies the set-thoretical Yang-Baxter equation (on χ × χ × χ)

Introduction
Introduction Introduction Method For any quantum group U q (g), there exists the universal R-matrix

Additional generators
For i, j ∈ {1, 2, . . . , n} and i ̸ = j, we define The algebra A is isomorphic to U q (sl(n)) under the condition c = 0.

Additional generators
For i, j ∈ {1, 2, . . . , n} and i ̸ = j, we define The algebra A is isomorphic to U q (sl(n)) under the condition c = 0.
We will also use the opposite co-multiplication ∆ ′ , defined by where σ(a ⊗ b) = b ⊗ a for any a, b ∈ A.
Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018 Universal R-matrix The algebra A is a quasi-triangular Hopf algebra. Then there exists an element R ∈ A ⊗ A, which satisfies ∆ ′ (a) R = R ∆(a) for all a ∈ A, (1 ⊗ ∆) R = R 13 R 12 , where R 12 = R ⊗ 1, R 23 = 1 ⊗ R and R 13 = (σ ⊗ 1)R 23 . The quantum Yang-Baxter equation follows from these.
Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018

Universal R-matrix
If we assume that the universal R-matrix has the form where R ∈ N + ⊗ N − : N + and N − are nilpotent sub-algebras generated by {E ij } and {E ji } for i < j, i, j ∈ {1, 2, . . . , n} respectively.

Universal R-matrix
If we assume that the universal R-matrix has the form where R ∈ N + ⊗ N − : N + and N − are nilpotent sub-algebras generated by {E ij } and {E ji } for i < j, i, j ∈ {1, 2, . . . , n} respectively.

Universal R-matrix
The universal is uniquely defined by [Kirillov-Reshetikhin, Rosso, where the product is taken over the reverse lexicographical order on (i, j): (i 1 , j 1 ) ≺ (i 2 , j 2 ) if i 1 > i 2 , or i 1 = i 2 and j 1 > j 2 .

Quantum Yang-Baxter map
Let X = {E ij , q E kk }, i ̸ = j be the set of generators of A and X (a) be the corresponding components in A ⊗ A, Quantum Yang-Baxter map Let X = {E ij , q E kk }, i ̸ = j be the set of generators of A and X (a) be the corresponding components in A ⊗ A, Quantum Yang-Baxter map R : (X (1) , X (2) ) → ( X (1) , X (2) ), Quantum Yang-Baxter map Let X = {E ij , q E kk }, i ̸ = j be the set of generators of A and X (a) be the corresponding components in A ⊗ A, Quantum Yang-Baxter map R : (X (1) , X (2) ) → ( X (1) , X (2) ), Note that any elements of X (1) commute with those of X (2) . In addition, the algebra A a generated by the elements of the set X (a) is isomorphic to the algebra A.

Quantum Yang-Baxter map
Let X = {E ij , q E kk }, i ̸ = j be the set of generators of A and X (a) be the corresponding components in A ⊗ A, Quantum Yang-Baxter map R : (X (1) , X (2) ) → ( X (1) , X (2) ), Note that any elements of X (1) commute with those of X (2) . In addition, the algebra A a generated by the elements of the set X (a) is isomorphic to the algebra A. =⇒ The tensor product structure is preserved under the map.

Another universal R-matrix
One can prove that if R 12 ∈ B + ⊗ B − satisfies definition of the universal R-matrix, then also satisfies the def of the universal R-matrix.
Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018 Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018

R-matrices
Evaluating further the second space (quantum space) of these L-operators in the fundamental representation π, we obtain the block R-matrices

R-matrices
Evaluating further the second space (quantum space) of these L-operators in the fundamental representation π, we obtain the block R-matrices Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018 L and R-operators with a spectral parameter Then we define the spectral parameter dependent L-operator Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018 Zero curvature representation Zero curvature representation Zero curvature representation Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018 Zero curvature representation Evaluating the first space of these (labeled by 0) in the fundamental representation π, we obtain Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018 Zero curvature representation Evaluating the first space of these (labeled by 0) in the fundamental representation π, we obtain Zero curvature representation where L ± 01 = R 12 L ± 01 R −1 12 , L ± 02 = R 12 L ± 02 R −1 12 and we omit the space index 0.

Zero curvature representation
The zero curvature representation gives a rational map among generators for the case U q (sl (2)) [Bazhanov-Sergeev 2015].

Zero curvature representation
The zero curvature representation gives a rational map among generators for the case U q (sl (2)) [Bazhanov-Sergeev 2015].
"In this paper we present detailed considerations of the above scheme on the example of the algebra U q (sl(2)) leading to discrete Liouville equations, however the approach is rather general and can be applied to any quantized Lie algebra. " [Bazhanov-Sergeev 2015] Zero curvature representation The zero curvature representation gives a rational map among generators for the case U q (sl (2)) [Bazhanov-Sergeev 2015].
"In this paper we present detailed considerations of the above scheme on the example of the algebra U q (sl(2)) leading to discrete Liouville equations, however the approach is rather general and can be applied to any quantized Lie algebra. " [Bazhanov-Sergeev 2015] However, this optimistic idea soon run into difficulty if we consider U q (sl(3)) case. Namely, square roots appear in the map for U q (sl(n)), n ≥ 3.

Zero curvature representation
The zero curvature representation gives a rational map among generators for the case U q (sl (2)) [Bazhanov-Sergeev 2015].
"In this paper we present detailed considerations of the above scheme on the example of the algebra U q (sl(2)) leading to discrete Liouville equations, however the approach is rather general and can be applied to any quantized Lie algebra. " [Bazhanov-Sergeev 2015] However, this optimistic idea soon run into difficulty if we consider U q (sl(3)) case. Namely, square roots appear in the map for U q (sl(n)), n ≥ 3. To overcome this difficulty, we will make a change of variables by twisting the universal R-matrix.
Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018 Twisting universal R-matrices Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018 Twisting universal R-matrices then gauge transformed universal R-matrices satisfy the defining relations for the universal R-matrix for the gauge transformed co-multiplication ∆ F (a) = F∆(a)F −1 , a ∈ A.
Twisting universal R-matrices then gauge transformed universal R-matrices satisfy the defining relations for the universal R-matrix for the gauge transformed co-multiplication ∆ F (a) = F∆(a)F −1 , a ∈ A.
R and R * satisfy the same Yang-Baxter relations as R and R * .
Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018 Twisting the universal R-matrices Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018 Twisting L-operators The gauge transformed L-operators are defined by evaluating the gauge transformed universal R-matrices.
Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018 Zero curvature representation for twisted L-operators The zero-curvature representation for the twisted L-operators has the same form as the one for the original L-operators Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018 Zero curvature representation for twisted L-operators The zero-curvature representation for the twisted L-operators has the same form as the one for the original L-operators The set of generators X (a) = {L (u k := q 2ω k = q 2(E 11 +···+E kk ) ), Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018 Zero curvature representation for twisted L-operators The zero-curvature representation for the twisted L-operators has the same form as the one for the original L-operators The set of generators X (a) = {L Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018 Solving the zero curvature representation Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018 Solving the zero curvature representation Solving the zero curvature representation quasi-determinants for the matrix J = L − 1 L + 2 , The other solutions L + 1 , L − 2 can be obtained by substituting L + 2 , L − 1 into the first and the third zero curvature relations.
Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018 The other solutions L + 1 , L − 2 can be obtained by substituting L + 2 , L − 1 into the first and the third zero curvature relations.
quasi-determinants [Gelfand, Retakh] • N × N matrix whose matrix elements a ij are elements of an associative algebra (not necessary commutative algebra): quasi-determinants [Gelfand, Retakh] • N × N matrix whose matrix elements a ij are elements of an associative algebra (not necessary commutative algebra): • m × n sub matrix:

quasi-determinants [Gelfand, Retakh]
In general, there are N 2 quasi-determinants for a N × N matrix A.
In case all the quasi-determinants of A are not zero, the inverse matrix of A can be expressed in terms of them: Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018 quasi-determinants [Gelfand, Retakh] In general, there are N 2 quasi-determinants for a N × N matrix A.
In case all the quasi-determinants of A are not zero, the inverse matrix of A can be expressed in terms of them: If all the matrix elements of A are commutative, then they reduce to |A| ij = (−1) i+j det A/ det A 1,...,î,...N 1,...,ĵ,...N .
Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018 quasi-determinants [Gelfand, Retakh] In general, there are N 2 quasi-determinants for a N × N matrix A.
In case all the quasi-determinants of A are not zero, the inverse matrix of A can be expressed in terms of them: If all the matrix elements of A are commutative, then they reduce to |A| ij = (−1) i+j det A/ det A 1,...,î,...N 1,...,ĵ,...N .
=⇒ Quasi-determinants are non-commutative analogues of ratios of determinants (rather than non-commutative analogues of determinants).
Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018

• Right quasi-Plücker coordinates
Zero-curvature relation and quasi-Plücker coordinates The solution of the zero-curvature relation can be rewritten in terms of quasi-Plücker coordinates of a block matrix: The solution of the zero-curvature relation can be rewritten in terms of quasi-Plücker coordinates of a block matrix: Define a sub-matrix Zero-curvature relation and quasi-Plücker coordinates Zero-curvature relation and quasi-Plücker coordinates Heisenberg-Weyl realization (Minimal representation) The Heisenberg-Weyl algebra W q Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018

Heisenberg-Weyl realization (Minimal representation)
The Heisenberg-Weyl algebra W q Homomorphism from U q (sl(n)) to W q (minimal rep.) where κ ∈ C.
Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018

Heisenberg-Weyl realization (Minimal representation)
The Heisenberg-Weyl algebra W q Homomorphism from U q (sl(n)) to W q (minimal rep.) This realizes a representation which has neither a highest weight nor a lowest weight.
Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018
Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018

Factorization of L-operators
Factorization of L + for minimal rep.
Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018

Factorization of L-operators
Factorization of L + for minimal rep.
Factorization of L − for minimal rep.
Factorization of an R-operator L ±,0 and L ±,∞ give homomorphisms from B ∓ to W q .
Evaluate the universal R-matrix R ∈ B + ⊗ B − by these homomorphisms.
Factorization of the universal R-matrix for minimal rep. Discrete quantum evolution system [cf. Bazhanov-Sergeev 2015] Quantum Yang-Baxter map gives an automorphism Based on this map, we define a discrete quantum evolution system for the algebra of observables Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018 Discrete quantum evolution system [cf. Bazhanov-Sergeev 2015] Quantum Yang-Baxter map gives an automorphism Based on this map, we define a discrete quantum evolution system for the algebra of observables Discrete quantum evolution system [cf. Bazhanov-Sergeev 2015] Quantum Yang-Baxter map gives an automorphism Based on this map, we define a discrete quantum evolution system for the algebra of observables Commuting integrals of motion Transfer matrices are generating function of integrals of motion.

Commuting integrals of motion
Transfer matrices are generating function of integrals of motion.

Commuting integrals of motion
Transfer matrices are generating function of integrals of motion.

Commuting integrals of motion
Transfer matrices are generating function of integrals of motion.
) , Poisson algebra P(gl (n)) Quasi-classical expansion of the universal R-matrix The universal R-matrix is singular in the limit b → 0.
Classical Yang-Baxter map Although the quasi-classical limit of the universal R-matrix becomes singular, its adjoint action ξ ∈ A ⊗ A → RξR −1 ∈ A ⊗ A is well defined. Thus the q → 1 limit of the quantum Yang-Baxter map is well defined. Zero-curvature representation

R = lim
The zero-curvature representation for the classical case has the same as the quantum case.
However, the matrix elements of the L-operators ℓ ± a are commutative.

Solution of the zero-curvature representation
One can obtain the solution by taking the limit q → 1. In particular, the solution is written in terms of ratios of product of minor determinants (Plücker coordinates) of a single matrix.
Example for P(sl (3)) u Quasi-classical limit for the minimal representatioin The Heisenberg-Weyl algebra W q reduces to the classical Heisenberg-Weyl algebra W in the quasi-classical limit.
Minimal representation (homomorphism from P(sl(n)) to W.) Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018 Solution of the zero-curvature relation for classical minimal rep For instance, for n = 3 case, we explicitly obtain 2 ))− 1 , u 2 , w 2 , w 1 , w 2 )

Symplectic form
Under these relations, the following function ) becomes a closed form: Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018

Summary
Quantum Yang-Baxter maps are defined in terms of adjoint action of the univeral R-matrix [Bazhanov-Sergeev 2015].
Solving the zero-curvature representation, we obtained the quantum Yang-Baxter map for U q (sl(n)). It is expressed as a product of quasi-Plücker coordinates over a matrix (written in terms of L-operators, which are image of the universal R-matrix). Twisting of the universal R-matrix was essential for the rationality of the map. Classical Yang-Baxter maps are derived through the quasiclassical limit. Discrete integrable systems (soliton equations) follow from Yang-Baxter maps.
Zengo Tsuboi ( Osaka City University Advanced Mathematical Institute (additional post member) ) Quantum groups, Yang-Baxter maps and quasi-determinants 28 June 2018

Summary
Quantum Yang-Baxter maps are defined in terms of adjoint action of the univeral R-matrix [Bazhanov-Sergeev 2015].
Solving the zero-curvature representation, we obtained the quantum Yang-Baxter map for U q (sl(n)). It is expressed as a product of quasi-Plücker coordinates over a matrix (written in terms of L-operators, which are image of the universal R-matrix). Twisting of the universal R-matrix was essential for the rationality of the map. Classical Yang-Baxter maps are derived through the quasiclassical limit. Discrete integrable systems (soliton equations) follow from Yang-Baxter maps.
Conjecture [Bazhanov-Sergeev 2015]: all the discrete integrable equations could be derived in this way.