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

Abstract

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...

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

“…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]. In particular, there is an m-dimensional space of traces and an (m + 1)dimensional space of supertraces on H 1,η (I 2 (2m + 1)).…”

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

“…e The dimension of the space of supertraces on H W (A n−1 ) (η) is the number of the partition of n 1 into the sum of different positive integers, see [6], and the space of the traces on H W (A n−1 ) (η) is one-dimensional for n 2 due to Theorem 2.3, see also [9].…”

“…Astonishingly, the proof differs from the one in [6] and [8] in several signs only, and we provide it here indicating change of signs by means of a parameter κ with κ = −1 for the supertraces and κ = +1 for the traces. As a result, some parts of this text are almost copypasted from [6] and [8], especially Subsection 4.3 and Appendix.…”

“…It is proved in [9] that if H W (R) (η) has isomorphic spaces of the traces and supertraces, then H W (R) (η) contains a Klein operator.…”

“…It was shown in [6] and [8] that, for every associative superalgebra H W (R) (η) of observables of the rational Calogero model based on the root system R, the space of supertraces is nonzero. The dimensions of these spaces for every root system are listed in [9].…”