Doubles of Associative Algebras and Their Applications
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HYSICS OF ELEMENTARY PARTICLES AND ATOMIC NUCLEI. THEORY
Doubles of Associative Algebras and Their Applications D. Gurevicha, b, * and P. Saponovc, d, ** a
LMI, UPHF, Valenciennes, 59313 France Interdisciplinary Scientific Center J.-V. Poncelet, Moscow, 119002 Russia c National Research University Higher School of Economics, Moscow, 101000 Russia dInstitute for High Energy Physics, NRC “Kurchatov Institute”, Protvino, 142281 Russia *e-mail: [email protected] **e-mail: [email protected] b
Received November 15, 2019; revised January 15, 2020; accepted February 28, 2020
Abstract—For a couple of associative algebras we define the notion of their double and give a set of examples. Also, we discuss applications of such doubles to representation theory of certain quantum algebras and to a new type of Noncommutative Geometry. DOI: 10.1134/S1547477120050167
1. INTRODUCTION In this letter by a double of associative algebras we mean an ordered couple ( A, B ) of associative unital algebras A and B such that their tensor products B ⊗ A can be also endowed with an associative product by means of a permutation map σ : A ⊗ B → B ⊗ A . If the algebra A is equipped with a counit (an algebra homomorphism) ε : A → , then under some natural conditions on σ and ε the algebra A can be represented in the algebra B . The simplest example of such a double is a Heisenberg–Weyl (HW) algebra. The smash-product of a bialgebra A and an A -module M is another example of a double. In this case the role of the algebra B can be played by the free tensor algebra T (M ) = ⊕k M ⊗k or by some of its quotient algebras. We are mainly interested in doubles related to braidings. Let V be a finite dimensional complex vector space, dimV = N . An invertible operator R : V ⊗2 → V ⊗2 is called a braiding, if it is subject to the braid relation
R12 R23R12 = R23R12 R23, R12 = R ⊗ I , R23 = I ⊗ R. Hereafter, I stands for the identity operator or its matrix. A braiding R is called respectively an involutive or a Hecke symmetry if it is subject to a supplementary condition
R 2 = I or (qI − R)(q −1I + R) = 0 q ∈ / {0, ±1}. The best known examples of Hecke symmetries come from the Drinfeld–Jimbo Quantum Groups (QG) U q (sl(N )) . They are deformations of the usual flips.
Nevertheless, there exist involutive and Hecke symmetries, which are neither deformations of the flips nor super-flips. All symmetries, we are dealing with, are assumed to be skew-invertible (see [1]). We mainly deal with Hecke symmetries R = R(q) at a generic value of the parameter q . To any such a Hecke symmetry R we associate R -analogs of the symmetric and skew-symmetric algebras of the space V by respectively setting
Sym R (V ) = T (V ) Im(qI − R) , −1
Λ R (V ) = T (V ) Im(q I + R) . Besides, we consider the so-called RTT and Reflection Equation (RE) algebras defined respectively by
R12TT 1 2 − TT 1 2 R12 = 0,
(1.1)
R12 L1R12 L1 − L1R12 L1R12 = 0,
(1.2)
where T = ti j
1≤i , j ≤ N
, L = li j
1≤ i, j ≤ N
, T1 = T ⊗ I and
T2 = I ⊗ T etc. In Section 3 we exhibit examples of dou
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