Quantum- and quasi-Plücker coordinates

Lauve, Aaron

doi:10.1016/j.jalgebra.2005.12.004

Cited by 8 publications

(19 citation statements)

References 24 publications

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“…There is a rich literature on quantum or noncommutative deformations of grassmannians (see e.g. [32,40,23,21,27]), mostly relying on q-deformations of matrices, so our noncommutative relations are somewhat different and easier to deal with. This is because in our construction the minors of a noncommutative matrix still close to a noncommutative algebra and in §2.4 we have explicitly computed their noncommutativity relations; these will be the noncommutativity relations of the homogeneous coordinate algebra generators of the noncommutative projective space CP N Θ .…”

Section: 3mentioning

confidence: 99%

“…This is because in our construction the minors of a noncommutative matrix still close to a noncommutative algebra and in §2.4 we have explicitly computed their noncommutativity relations; these will be the noncommutativity relations of the homogeneous coordinate algebra generators of the noncommutative projective space CP N Θ . Here we shall follow [32] to define the noncommutative deformation of Plücker equations, or Young symmetry relations, which is an approach to noncommutative grassmannians based on quasideterminants [23].…”

Section: 3mentioning

confidence: 99%

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Algebraic deformations of toric varieties I. General constructions

Cirio

Landi

Szabo

2013

Advances in Mathematics

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We construct and study noncommutative deformations of toric varieties by combining techniques from toric geometry, isospectral deformations, and noncommutative geometry in braided monoidal categories. Our approach utilizes the same fan structure of the variety but deforms the underlying embedded algebraic torus. We develop a sheaf theory using techniques from noncommutative algebraic geometry. The cases of projective varieties are studied in detail, and several explicit examples are worked out, including new noncommutative deformations of Grassmann and flag varieties. Our constructions set up the basic ingredients for thorough study of instantons on noncommutative toric varieties, which will be the subject of the sequel to this paper

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Section: 3mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Algebraic deformations of toric varieties I. General constructions

Cirio

Landi

Szabo

2013

Advances in Mathematics

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“…In this subsection, we summarize formulas on quasi-determinants based on [21,22,23] (see also [30,31,32,33]). Let A = (a ij ) 1≤i,j≤N be a N × N matrix whose matrix elements a ij are elements of an associative algebra.…”

Section: Review On Quasi-determinantsmentioning

confidence: 99%

“…The fact that the quantum Yang-Baxter maps are expressed in terms of quasi-Plücker coordinates over matrices (2.144), (2.151) and (2.160) composed of the Loperators may imply an underlying quantum Grassmannian of the system (cf. [32]). It is known that L-operators are related to Manin matrices [30,31].…”

Section: )mentioning

confidence: 99%

Quantum groups, Yang–Baxter maps and quasi-determinants

Tsuboi

2018

Nuclear Physics B

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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-Plücker 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.2 The notation U h (gl(n)) is often used in literatures for this form of presentation of the quantum algebra. The deformation parameter q is related to h as q = e h . We assume that q is generic.3 The normalization commonly used in literatures can be obtained by the replacement:, L −(2) 21 = (L −(1) L −(2) ) 21 (L −(1) L −(2) ) 22 (L −(1) L −(2) ) 23 J 21 J 22 J 23 J 31 J 32 J 33 , L −(2) 31 = (L −(1) L −(2) ) 31 (L −(1) L −(2) ) 32

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“…Perhaps a theory of non-commutative flag varieties using quasi-Plücker coordinates could help explain the choices for the relations in F q (n). In [12], it is shown that any relation (Y I,J ) (a) has a quasi-Plücker coordinate origin. Section 3 shows that (1) Looking past flag algebras to more general determinantal varieties, the same question is valid.…”

Section: G-proof Of Theoremmentioning

confidence: 99%

QUASI-DETERMINANTS AND q-COMMUTING MINORS

Lauve¹

2010

Glasgow Math. J.

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Abstract. We present two new proofs of the q-commuting property holding among certain pairs of quantum minors of a q-generic matrix. The first uses elementary quasi-determinantal arithmetic; the second involves paths in a directed graph. Together, they indicate a means to build the multi-homogeneous coordinate rings of flag varieties in other non-commutative settings.2010 Mathematics Subject Classification. 20G42, 16T30, 15A15.

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Quantum- and quasi-Plücker coordinates

Cited by 8 publications

References 24 publications

Algebraic deformations of toric varieties I. General constructions

Algebraic deformations of toric varieties I. General constructions

Quantum groups, Yang–Baxter maps and quasi-determinants

QUASI-DETERMINANTS AND q-COMMUTING MINORS

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