Traces on the Algebra of Observables of the Rational CalogeroModel Based on the Root System

Konstein, S. E.; Tyutin, I. V.

doi:10.1080/14029251.2013.820403

Cited by 6 publications

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References 11 publications

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“…It is shown in [12,10,9] that the algebra H has a multitude of independent (super)traces. For the list of dimensions of the spaces of the (super)traces on H 1,η (M) for all finite Coxeter groups M, see [8].…”

Section: The (Super)traces On Hmentioning

confidence: 99%

Connection between the ideals generated by traces and by supertraces in the superalgebras of observables of Calogero models

Konstein¹,

Tyutin²

2019

JNMP

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Section: The (Super)traces On Hmentioning

confidence: 99%

Connection between the ideals generated by traces and by supertraces in the superalgebras of observables of Calogero models

Konstein¹,

Tyutin²

2019

JNMP

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Klein operator and the Number of independent Traces and Supertraces on the Superalgebra of Observables of Rational Calogero Model based on the Root System

Konstein¹,

Stekolshchik²

2021

JNMP

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In the Coxeter group W (R) generated by the root system R, let T (R) be the number of conjugacy classes having no eigenvalue +1 and let S(R) be the number of conjugacy classes having no eigenvalue −1. The algebra H W (R) of observables of the rational Calogero model based on the root system R possesses T (R) independent traces; the same algebra, considered as an associative superalgebra with respect to a certain natural parity, possesses S(R) even independent supertraces and no odd trace or supertrace. The numbers T (R) and S(R) are determined for all irreducible root systems (hence for all root systems). It is shown that T (R) ≤ S(R), and T (R) = S(R) if and only if superalgebra H W (R) contains a Klein operator (or, equivalently, W (R) ∋ −1).Keywords: Trace; supertrace; Cherednik algebra; algebra of observables; Calogero model. Mathematics Subject Classification: 17B80, 16W551. Definitions and generalities TracesLet A be an associative superalgebra with parity π. All expressions of linear algebra are given for homogeneous elements only and are supposed to be extended to inhomogeneous elements via linearity.A linear function str on A is called a supertrace ifA linear function tr on A is called a trace if S.E. Konstein and R. StekolshchikLet A 1 and A 2 be associative superalgebras with parities π 1 and π 2 , respectively. Define the tensor product A = A 1 ⊗A 2 as a superalgebra with the product (a 1 ⊗a 2 )(b 1 ⊗b 2 ) = (a 1 b 1 )⊗(a 2 b 2 ) (no sign factors in this formula) and the parity π defined by the formula π(a ⊗ b) = π 1 (a) + π 2 (b).Let T i be a trace on A i . Clearly, the function T such thatis an even supertrace on A . Klein operatorLet A be an associative superalgebra with parity π. Following M.Vasiliev, see, e.g.[21], we say that an element K ∈ A is a Klein operator a if π(K) = 0, K f = (−1) π( f ) f K for any f ∈ A and K 2 = 1. Every Klein operator belongs to the anticenter of the superalgebra A , see [18], p.41. b Any Klein operator, if exists, establishes an isomorphism between the space of even traces and the space of even supertraces on A . Namely, if f → T ( f ) is an even trace, then f → T ( f K) is a supertrace, and if f → S( f ) is an even supertrace, then f → S( f K) is a trace. Group algebraLet V = R n and G ⊂ End(V ) be a finite group. The group algebra C[G] of G consists of all linear combinations ∑ g∈W (R) α gḡ , where α g ∈ C. We distinguish g considered as an element of the group G ⊂ End(V ) from the same elementḡ ∈ C[G] considered as an element of the group algebra. The addition in C[G] is defined as follows:and the multiplication is defined by setting g 1 g 2 = g 1 g 2 .Note that the additions in C[G] and in End(V ) differ. For example, if I ∈ G is unity and the matrix J = −I from End(V ) belongs to G, then I + J = 0 in End(V ) while I + J = 0 in C[G]. Root systemsLet V = R N be endowed with a non-degenerate symmetric bilinear form (·, ·) and the vectors a i constitute an orthonormal basis in V , i.e. Co-published by Atlantis Press and Taylor & Francis Copyright: the authors 29...

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The number of independent traces and supertraces on symplectic reflection algebras

Konstein¹,

Tyutin²

2021

JNMP

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It is shown that A := H 1, η (G), the sympectic reflection algebra, has T G independent traces, where T G is the number of conjugacy classes of elements without eigenvalue 1 belonging to the finite group G ⊂ Sp(2N ) ⊂ End(C 2N ) generated by the system of symplectic reflections.Simultaneously, we show that the algebra A, considered as a superalgebra with a natural parity, has S G independent supertraces, where S G is the number of conjugacy classes of elements without eigenvalue −1 belonging to G.We consider also A as a Lie algebra A L and as a Lie superalgebra A S . It is shown that if A is a simple associative algebra, then the supercommutant [A S , A S ] is a simple Lie superalgebra having at least S G independent supersymmetric invariant non-degenerate bilinear forms, and the quotient [A L , A L ]/([A L , A L ] ∩ C) is a simple Lie algebra having at least T G independent symmetric invariant non-degenerate bilinear forms. *

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Traces on the Algebra of Observables of the Rational CalogeroModel Based on the Root System

Cited by 6 publications

References 11 publications

Connection between the ideals generated by traces and by supertraces in the superalgebras of observables of Calogero models

Connection between the ideals generated by traces and by supertraces in the superalgebras of observables of Calogero models

Klein operator and the Number of independent Traces and Supertraces on the Superalgebra of Observables of Rational Calogero Model based on the Root System

The number of independent traces and supertraces on symplectic reflection algebras

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