Theoretical and Mathematical Physics, 192(1): 939–957 (2017)
A GENERALIZATION OF LIE H-PSEUDOBIALGEBRAS
and Fang Li
We investigate Hom–Lie H-pseudobialgebras. We present some examples and a theorem that allows
constructing these new algebraic structures. We consider coboundary Hom–Lie H-pseudobialgebras and
the corresponding classical Hom–Yang–Baxter equations.
Keywords: Hom–Lie bialgebra, Hom–Lie H-pseudobialgebra, classical Hom–Yang–Baxter equation,
The recently introduced notion of a Lie H-pseudoalgebra over a cocommutative Hopf algebra  is a
multivariable generalization of the concept of a Lie conformal algebra , introduced by Kac in connection
with vertex algebras . The algebraic properties of each element in a Lie H-pseudoalgebra can be refor-
mulated in terms of its Fourier coeﬃcients, which in the physical terminology are sometimes called creation
and annihilation operators. The space of all annihilation operators is a (typically inﬁnite-dimensional) Lie
algebra, and the Lie bracket is continuous in a linearly compact topology. The studies of linearly compact
inﬁnite-dimensional Lie algebras by Cartan  and Guillemin ,  can then be useful in investigating
The (Lie) H-pseudoalgebras are closely related to the diﬀerential Lie algebras of the Ritt and Hamilto-
nian formalism in the theory of nonlinear evolution equations –. But because they are new algebraic
structures, their role in many ﬁelds of mathematics is not yet completely understood. Experts in this ﬁeld
expect that under certain conditions, there is a similar relation of “multidimensional” Lie pseudoalgebras
to “multidimensional” vertex algebras introduced in . In the case of a commutative Lie algebra g,the
Lie H-pseudoalgebras encode the operator product expansion between ultralocal ﬁelds (and also the linear
Poisson brackets) for H = U(g).
The Hom–algebra structures have been recently investigated by many authors. The main feature of
these algebras is that the identities deﬁning the structures are twisted by homomorphisms. Such algebras
appeared in the 1990s in examples of q-deformations of Witt and Virasoro algebras –. These examples
and their generalizations motivated studies of the classes of Hom–associative algebras, Hom–Lie algebras and
Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou, China,
Department of Mathematics, Zhejiang University, Hangzhou, China, e-mail: firstname.lastname@example.org.
This research is supported by the Natural Science Foundation of Zhejiang Province of China (Grant
No. LQ13A010018) and the National Natural Science Foundation of China (Grant Nos. 11226069 and 11401530).
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i
Matematicheskaya Fizika, Vol. 192, No. 1, pp. 3–22, July, 2017. Original article submitted March 8, 2016; revised
November 11, 2016.
2017 Pleiades Publishing, Ltd.