Algebraic properties of Manin matrices. I

DOI10.1016/j.aam.2009.02.003zbMath1230.05043arXiv0901.0235OpenAlexW1966411596MaRDI QIDQ840705

Alexander Chervov, Gregorio Falqui, Vladimir Rubtsov

Publication date: 14 September 2009

Published in: Advances in Applied Mathematics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/0901.0235

zbMATH Keywords

quantum groupCayley-Hamilton theoremSchur complementrepresentation theorydeterminantpermanentCapelli identityCramer's rulePlücker coordinatesnoncommutative determinantJacobi ratio theoremManin matrixNewton identitiesCartier-Foata matrixDodgson's condensation algorithmMacMahon master theoremquasidetermininatright-quantum matrixTurnbull identity

Mathematics Subject Classification ID

Combinatorial identities, bijective combinatorics (05A19) Quantum groups and related algebraic methods applied to problems in quantum theory (81R50) Matrices over special rings (quaternions, finite fields, etc.) (15B33)

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Uses Software

• QuantumMACMAHON

Cites Work

• Symmetric Determinants and the Cayley and Capelli Operator
• Quadratic algebras and integrable systems
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Some remarks on Koszul algebras and quantum groups
• Analogue of the identity Log Det = Trace Log for resultants
• A noncommutative version of the matrix inversion formula
• Quantum Berezinian and the classical Capelli identity
• The Capelli identity, the double commutant theorem, and multiplicity-free actions
• A class of left quantum groups modeled after \(\text{SL}_q(n)\).
• Quantum inverse problem method. I
• Determinants of block tridiagonal matrices
• Commutation relations for arbitrary quantum minors.
• Limits of Gaudin systems: classical and quantum cases
• KZ equation, G-opers, quantum Drinfeld-Sokolov reduction, and quantum Cayley-Hamilton identity
• A generalization of Sylvester's identity on determinants and some applications
• A formal inverse to the Cayley-Hamilton theorem
• Quasideterminants. I
• Quantum matrices in two dimensions
• Traces of creation-annihilation operators and Fredholm formulas
• Explicit Newton's formula for \({\mathfrak gl}_n\)
• The quantum Cayley-Hamilton theorem
• Cayley-Hamilton theorem for \(2\times 2\) matrices over the Grassmann algebra
• Quantum symmetry groups of finite spaces
• A theory of noncommutative determinants and characteristic functions of graphs
• Notes on quantum groups and quantum de Rham complexes
• Yangians and Gelfand-Zetlin bases
• Gaudin model, Bethe ansatz and critical level
• Coefficient matrices of a quantum discrete auxiliary linear problem
• The relation between two types of characteristic equations for quantum matrices
• Application of Koszul complex to Wronski relations for \(U(\mathfrak{gl}_n)\)
• Commuting families in skew fields and quantization of Beauville's fibration
• Riemann surfaces, separation of variables and classical and quantum integrability
• Commutation relations among quantum minors in \({\mathcal O}_q(M_n(k))\).
• A left quantum group.
• Quasideterminants
• Noncommutative symmetric functions
• Determinants of matrices over noncommutative rings
• Hitchin systems, higher Gaudin operators and \(R\)-matrices
• Some quantum-like Hopf algebras which remain noncommutative when \(q=1\).
• Representation rings of quantum groups.
• Minor identities for quasi-determinants and quantum determinants
• Quantum immanants and higher Capelli identities
• Noncommutative Sylvester's determinantal identity
• Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities. I: Generalizations of the Capelli and Turnbull identities
• Quasi-commuting families of quantum minors
• Supersymmetric Bethe ansatz and Baxter equations from discrete Hirota dynamics
• A \(q\)-analogue of the centralizer construction and skew representations of the quantum affine algebra
• A basis for the right quantum algebra and the ``\(1= q\) principle
• Quantum- and quasi-Plücker coordinates.
• The quantum Gaudin system
• Skew representations of twisted Yangians
• Combinatorial problems of commutation and rearrangements
• Erratum: “On characteristic equations, trace identities and Casimir operators of simple Lie algebras” [J. Math. Phys. 41, 3192–3225 (2000)]
• Characteristic relations for quantum matrices
• Generalized Gelfand invariants of quantum groups
• Manin matrices and Talalaev's formula
• An algebraic extension of the MacMahon Master Theorem
• SYMMETRIES AND INVARIANTS OF TWISTED QUANTUM ALGEBRAS AND ASSOCIATED POISSON ALGEBRAS
• A Generalization of the Capelli Identity
• Koszul algebras and the quantum MacMahon master theorem
• On Higher-Order Sugawara Operators
• Characteristic identities for semi-simple Lie algebras
• Identities and characters for finite groups
• Generalized Gel’fand invariants and characteristic identities for quantum groups
• Newton’s formula for 𝔤𝔩_{𝔫}
• On representations of the cayley-hamilton equation in the supercase
• On quantum matrix algebras satisfying the Cayley - Hamilton - Newton identities
• The equivalence between the Mandelstam and the Cayley–Hamilton identities
• Yangians and classical Lie algebras
• New determinants and the Cayley-Hamilton Theorem for matrices over Lie nilpotent rings
• Bending Flows for Sums of Rank One Matrices
• A Multiplicative Property of Quantum Flag Minors II
• The spectral theory of a functional-difference operator in conformal field theory
• RING THEORETIC PROPERTIES OF QUANTUM GRASSMANNIANS
• The quantum MacMahon Master Theorem
• Cayley–Hamilton theorem for quantum matrix algebras of $GL(m|n)$ type
• Sylvester's Identity and Multistep Integer-Preserving Gaussian Elimination
• Characteristic Identities for Generators of GL(n), O(n) and Sp(n)
• Les déterminants sur un corps non commutatif