TWISTED YANGIANS AND FOLDED ${\mathcal W}$ -ALGEBRAS

Abstract

We show that the truncation of twisted Yangians are isomorphic to finite Walgebras based on orthogonal or symplectic algebras. This isomorphism allows us to classify all the finite dimensional irreducible representations of the quoted W-algebras. We also give an R-matrix for these W-algebras, and determine their center.

Representations of Reflection Algebras

“…The Yangian symmetries have been found in various areas of physics including the theory of integrable models in statistical mechanics, conformal field theory, quantum gravity. We note a surprising connection of the Yangian and twisted Yangians with the finite W-algebras recently discovered in [26,27,25]; see also [2].…”

-superalgebras as truncations of super-Yangians

“…For conciseness, we will just sketch the results. In particular, one gets the following theorems Proof: Same proof as for twisted Yangians, see [10], using the results given in [17] for Y (M|2n) + . Indeed, as for the gl(M|N) case, one needs to have S (r) = 0 for r ≥ p to get a representation of the W-superalgebra.…”

“…Using the representations classification of twisted super-Yangians given in [17], one can then deduce the classification of irreducible finite-dimensional representations for truncated twisted super-Yangians in the same way it has been done in [10] for ordinary twisted Yangians. For conciseness, we will just sketch the results.…”

“…Then [10], this connection was extended to a class of W-algebras, namely the algebras of type W[so(2mp), m.sl(p)], W[so((2m + 1)p), m.sl(p) + so(p)] and W[sp(2np), n.sl(p)], which where related to truncation of twisted Yangian Y ± (N). Note that although Yangians based on orthogonal and symplectic algebras exist [2], and admit an FRT presentation [11], it is the twisted Yangians introduced by Olshanski [12,13] which enter into the game.…”

“…Nevertheless, this relation allows to give an R-matrix presentation of the W-algebras under consideration, with however the slight change that it is an "RSRS" relation which occurs, not an "RTT" one. The classification of finite-dimensional irreducible representations of the W-algebras then follows [10].…”