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Isomorphisms of type A affine Hecke algebras and multivariable orthogonal polynomials
"... Introduction In several recent works [10, 11, 9, 29], eigenstates of the rational (type A) CalogeroSutherland model have been investigated from an algebraic point of view. In particular it has been shown that the algebra governing the eigenfunctions of the periodic CalogeroSutherland model (namel ..."
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Introduction In several recent works [10, 11, 9, 29], eigenstates of the rational (type A) CalogeroSutherland model have been investigated from an algebraic point of view. In particular it has been shown that the algebra governing the eigenfunctions of the periodic CalogeroSutherland model (namely the type A degenerate affine Hecke algebra augmented by type A Dunkl operators) is isomorphic to its rational model counterpart. This enables information to be gleaned about the properties of the eigenfunctions in the rational case (the (non)symmetric Hermite polynomials) from the corresponding periodic eigenfunctions (the (non)symmetric Jack polynomials). To summarize the argument, consider the type A Dunkl operators d i := @ @x i p6=i 1 \Gamma s ip x i \Gamma x p which, along with the operators representing multiplication by the variable x i and the elementary transpositions s ij , satisfy the following commutation relations [d i ; x j ] = ( s ij i 6= j 1 + p6=i s ip i
Yangians of Lie Superalgebras
"... This thesis is concerned with extending some wellknown results about the Yangians Y (gl N) and Y (slN) to the case of superYangians. First we produce a new presentation of the Yangian Y (gl mn), using the Gauss decomposition of a matrix with noncommuting entries. Then, by writing the quantum Ber ..."
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This thesis is concerned with extending some wellknown results about the Yangians Y (gl N) and Y (slN) to the case of superYangians. First we produce a new presentation of the Yangian Y (gl mn), using the Gauss decomposition of a matrix with noncommuting entries. Then, by writing the quantum Berezinian in terms of generators from the new presentation we prove that its coefficients generate the centre Zmn of Y (gl mn). We show that the Yangian Y (slmn) is isomorphic to a subalgebra of the Yangian Y (gl mn), and in particular if m ̸ = n, then Y (gl mn) ∼ = Zmn ⊗ Y (slmn). Finally, we show that a Yangian Y (psl nn) associated with the projective special linear Lie superalgebra may be obtained from Y (slnn) by quotienting out the ideal generated by the coefficients of the quantum Berezinian. iii ivAcknowledgements I gratefully acknowledge the help of my supervisor Alex Molev, who provided the original plan for this thesis project and has been helpful and supportive throughout
Symmetry, Integrability and Geometry: Methods and Applications Symmetries of Spin Calogero Models ⋆
, 809
"... Abstract. We investigate the symmetry algebras of integrable spin Calogero systems constructed from Dunkl operators associated to finite Coxeter groups. Based on two explicit examples, we show that the common view of associating one symmetry algebra to a given Coxeter group W is wrong. More precisel ..."
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Abstract. We investigate the symmetry algebras of integrable spin Calogero systems constructed from Dunkl operators associated to finite Coxeter groups. Based on two explicit examples, we show that the common view of associating one symmetry algebra to a given Coxeter group W is wrong. More precisely, the symmetry algebra heavily depends on the representation of W on the spins. We prove this by identifying two different symmetry algebras for a BL spin Calogero model and three for G2 spin Calogero model. They are all related to the halfloop algebra and its twisted versions. Some of the result are extended to any finite Coxeter group. Key words: Calogero models; symmetry algebra; twisted halfloop algebra 2000 Mathematics Subject Classification: 70H06; 81R12; 81R50 1
Symmetries of Spin Calogero Models
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2008
"... We investigate the symmetry algebras of integrable spin Calogero systems constructed from Dunkl operators associated to finite Coxeter groups. Based on two explicit examples, we show that the common view of associating one symmetry algebra to a given Coxeter group W is wrong. More precisely, the sym ..."
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We investigate the symmetry algebras of integrable spin Calogero systems constructed from Dunkl operators associated to finite Coxeter groups. Based on two explicit examples, we show that the common view of associating one symmetry algebra to a given Coxeter group W is wrong. More precisely, the symmetry algebra heavily depends on the representation of W on the spins. We prove this by identifying two different symmetry algebras for a BL spin Calogero model and three for G2 spin Calogero model. They are all related to the halfloop algebra and its twisted versions. Some of the result are extended to any finite Coxeter group.