Poisson algebras and Yang-Baxter equations

We connect generalizations of Poisson algebras with the classical and associative Yang-Baxter equations. In particular, we prove that solutions of the classical Yang-Baxter equation on a vector space V are equivalent to “twisted” Poisson algebra structures on the tensor algebra TV . Here, “twisted” refers to working in the category of graded vector spaces equipped with Sn actions in degree n. We show that the associative Yang-Baxter equation is similarly related to the double Poisson algebras of Van den Bergh. We generalize to L∞-algebras and define “infinity” versions of Yang-Baxter equations and double Poisson algebras. The proofs are based on the observation that Lie is essentially unique among quadratic operads having a certain distributivity property over the commutative operad; we also give an L∞ generalization. In the appendix, we prove a generalized version of Schur-Weyl duality, which is related to the use of nonstandard Sn-module structures on V . 1. Twisted Poisson algebras and the CYBE Throughout, we will work over a characteristic-zero field k. The tensor algebra TV = TkV satisfies the following twisted-commutativity property: each graded component V ⊗m is equipped with an Sm-module structure by permutation of components, and given homogeneous elements v, w ∈ TV of degrees |v|, |w|, we have (1.1) w ⊗ v = (21)(v ⊗ w), where (21) ∈ S|v|+|w| is the permutation of the two blocks {1, . . . , |v|}, {|v|+ 1, . . . , |v|+ |w|}. We thus say that TV is a twisted commutative algebra. Similarly, we may define twisted Lie algebras. Again let A = ⊕ m≥0 Am together with an Sm action on Am for all m. A twisted Lie algebra is A together with a graded bracket { , } : A⊗A → A satisfying {w, v} = (21){v, w}, (1.2) {u, {v, w}}+ (231){v, {w, u}}+ (312){w, {u, v}} = 0, (1.3) where σ = (i1i2 . . . in) ∈ Sn denotes the element σ(j) = ij, and given τ ∈ S3, τ a,b,c ∈ Sa+b+c denotes the permutation acting by permuting the blocks {1, . . . , a}, {a+1, . . . , a+ b}, {a+ b+1, . . . , a+ b+ c}. (We will not use cycle notation in this paper.) The motivating observation of this paper is as follows: If A = TV is endowed with a twisted Lie algebra structure satisfying the Leibniz rule, (1.4) {u⊗ v, w} = u⊗ {v, w} + (213)(v ⊗ {u,w}), then the Jacobi identity restricted to degree one, V ⊗ V ⊗ V → T V , says that the bracket yields a skew solution of the well-known classical Yang-Baxter equation (CYBE): interpreted as a map r : V ⊗V → 1The notion of twisted algebras is an old notion from topology dating to at least the 1950’s; see, e.g., [Bar78, Joy86, Fre04, LP06]. They are related to superalgebras and color Lie algebras [RW78, Sch79]. 2The CYBE is a central equation in physics and the study of quantum groups. 1 V ⊗ V , we have r = −r, (1.5) [r, r]− [r, r] + [r, r] = 0, (1.6) where here r denotes r acting in the i-th and j-th components (e.g., r = IdV ⊗ r). The starting point for this paper is then Theorem 1.7. Let V be any vector space. Skew solutions r ∈ End(V ⊗ V ) of the CYBE are equivalent to twisted Poisson algebra structures on TV , equipped with its usual twisted commutative multiplication ⊗. Here, a twisted Poisson structure on TV is the same as a twisted Lie algebra structure satisfying (1.4). The proof is based on the twisted generalization of the following well-known fact: a Poisson algebra structure on SymV is the same as a Lie algebra structure on V (Proposition 1.10). Precisely, recall that an S-module is a graded vector space V = ⊕ m≥0 Vm together with Sm-actions on each Vm. S-modules form a symmetric monoidal category, and the notion of SymV (the free commutative monoid in the category of S-modules) makes sense, and yields a twisted commutative algebra. In the case that V is concentrated in degree zero, the twisted commutative algebra SymV , viewed as an ordinary vector space with the induced multiplication map, is the usual symmetric algebra SymV0. In the case V is concentrated in degree one, SymV , viewed as an ordinary vector space with an associative multiplication, is the usual tensor algebra TV1. Then, as explained in §1.1 below, a standard proof that Poisson structures on SymV are the same as Lie algebra structures on V carries over to the twisted setting, and yields Theorem 1.7. Remark 1.8. P. Etingof pointed out to the author a connection with the Lie algebra tr from [BEER05] (which is generated by rij subject to the universal relations satisfied by r ij for any skew solution r of the CYBE). More precisely, the universal enveloping algebra of tr contains the space of all possible operations V ⊗m → V ⊗m obtainable from the twisted Lie structure on TV (in terms of an indeterminate r). 1.1. Proof of Theorem 1.7. We recall first the definition of the symmetric monoidal structure on the category of S-modules: Given S-modules V = ⊕