Graded Lie Algebras Defined by Jordan Algebras and Their Representations

In this talk we introduce the notion of a generalized representation of a Jordan algebra with unit which has the following properties: 1) Usual representations and Jacobson representations correspond to special cases of generalized representations. 2) Every simple Jordan algebra has infinitely many nonequivalent generalized representations. 3) There is a one-to-one correspondence between irreducible generalized representations of a Jordan algebra A and irreducible representations of a graded Lie algebra L(A) = U −1⊕U0⊕U1 corresponding to A (the Lie algebra L(A) coincides with the TKK construction when A has a unit). The latter correspondence allows to use the theory of representations of Lie algebras to study generalized representations of Jordan algebras. In particular, one can classify irreducible generalized representations of semisimple Jordan algebras and also obtain classical results about usual representations and Jacobson representations in a simple way. Introduction Jordan algebras were introduced by P. Jordan, J. von Neumann and E. Wigner (see [1]) in the connection with some problems of quantummechanics. Already there it was found that simple Jordan algebras have only a finite number of nonequivalent irreducible representations (homomorphisms into a space of linear operators with operation X ∗ Y = XY + Y X). Moreover (A. Albert [2]), the exceptional Jordan algebra E3 has no such representations at all. This situation demonstrates a big difference between Lie and Jordan algebras. (As is known a Lie algebra has infinitely many nonequivalent irreducible representations.) To improve the situation, N. Jacobson introduced [3] another notion of a representation of a Jordan algebra (see below the definition of the Jacobson representation) and showed that every simple Jordan algebra has at least one nontrivial Jacobson representation. But still the number of Jacobson representations is also finite. In this talk we will introduce a notion of a generalized representation of a Jordan algebra with unit and will describe the irreducible generalized representations. The irreducible generalized representations are in a one-to-one correspondence with the irreducible representations of the 3-graded Lie algebra L(A) corresponding to A. In particular, every simple Jordan algebra has infinitely many nonequivalent irreducible generalized representations. This work was supported by the RFBR grant 03–01–00056 and Swedish Academy of Sciences. 1 2 ISSAI KANTOR AND GREGORY SHPIZ The usual and Jacobson representations correspond to special cases of generalized representations. Moreover, this correspondence preserves the irreducibility and the equivalence of representations. In particular, it allows to classify irreducible usual and Jacobson representations. The authors are very grateful to Bruce Allison, Kevin McCrimmon and Ivan Shestakov for useful discussions. §1 A graded Lie algebra L(A) defined by a Jordan algebra A We need a construction of a 3-graded Lie algebra L(A) defined by a Jordan algebra A. The construction of L(A) is presented as it was originally given in [4], [5] (see also [6]). This construction coincides with what is called the TKK construction when Jordan algebras A has a unit, but does not coincide with it in general (for example dimU−1 is not equal in general to dimU1 ). The Lie algebra L(A) has the following two important properties: 1) There is an element Ā ∈ U1 such that [[Ā, x], y] = x ∗ y ∀x, y ∈ U−1, where ∗ is the multiplication in the given algebra A. (The space U−1 is identified with the space of the algebra A.) 2) The Lie algebra L(A) is generated by the space U−1 and the element Ā ∈ U1. To construct the Lie algebra L(A) let us denote by U the space of the algebra A and by Ā(x, y) = x ∗ y the multiplication in A. We denote also La(x) = a ∗ x, Aa(x, y) = (x ∗ a) ∗ y + (y ∗ a) ∗ x− a ∗ (x ∗ y), (0.1) S = {La, [La, Lb] | ∀a, b ∈ U}, Ū = {Ā, Aa | ∀a ∈ U}, (0.2) where {...} is the linear span of elements in the braces. Consider a direct sum U ⊕ S ⊕ Ū . (0.3) Let a, b ∈ U , S, S1, S2 ∈ S. (0.4) Define [a, b] = 0, [Ā, Ab] = 0, [Aa, Ab] = 0, [S, a] = S(a), [S1, S2] = S1S2 − S2S1, (0.5) [Ā, La] = Aa, [Aa, Lb] = Aa∗b. (3.6) Theorem 1. The space (0.3) with the commutation relations (0.5) and (0.6) is a graded Lie algebra L(A) = U−1 ⊕ U0 ⊕ U1, (0.7) where U−1 = U , U0 = S, U1 = Ū . Remark 1. The multiplication in the Jordan algebra A can be restored as a double commutator [[Ā, x], y] ∀x, y ∈ U−1. Example 1. Let An be a Jordan algebra of matrices of order n with operation B ∗ C = BC + CB. GRADED LIE ALGEBRAS DEFINED BY JORDAN ALGEBRAS AND THEIR REPRESENTATIONS3 Then the Lie algebra L(An) = A2n−1, i.e. linear Lie algebra of matrices of order 2n A2n−1 = (