Simple Jordan Superalgebras Research Articles

Representations of simple noncommutative Jordan superalgebras II

In this article we continue the study of representations of simple finite-dimensional noncommutative Jordan superalgebras. We prove that an irreducible unital bimodule over a simple noncommutative Jordan superalgebra U of degree ≥2 is either an irreducible bimodule over its symmetrized superalgebra U ( + ) or is equal to one of its Peirce components. Applying this result, we describe irreducible finite-dimensional representations of simple noncommutative Jordan superalgebras U ( V , f , ⋆ ) and K ( Γ n , A ) , completing the description of simple finite-dimensional unital bimodules over simple noncommutative Jordan superalgebras over an algebraically closed field of characteristic 0.

Codimension growth of simple Jordan superalgebras

Codimension growth of simple Jordan superalgebras

Second cohomology group of the finite-dimensional simple Jordan superalgebra 𝒟t, t≠0

The second cohomology group (SCG) of the Jordan superalgebra [Formula: see text], [Formula: see text], over an algebraically closed field [Formula: see text] of characteristic zero is calculated by using the coefficients which appear in the regular superbimodule [Formula: see text]. Contrary to the case of algebras, this group is nontrivial thanks to the non-splitting caused by the Wedderburn Decomposition Theorem [F. A. Gómez-González, Wedderburn principal theorem for Jordan superalgebras I, J. Algebra 505 (2018) 1–32]. First, to calculate the SCG of a Jordan superalgebra we use split-null extension of the Jordan superalgebra and the Jordan superalgebra representation. We prove conditions that satisfy the bilinear forms [Formula: see text] that determine the SCG in Jordan superalgebras. We use these to calculate the SCG for the Jordan superalgebra [Formula: see text], [Formula: see text]. Finally, we prove that [Formula: see text], [Formula: see text].

The Superalgebras of Jordan Brackets Defined by the $ n $-Dimensional Sphere

We study the generalized Leibniz brackets on the coordinate algebra of the $ n $ -dimensional sphere. In the case of the one-dimensional sphere, we show that each of these is a bracket of vector type. Each Jordan bracket on the coordinate algebra of the two-dimensional sphere is a generalized Poisson bracket. We equip the coordinate algebra of a sphere of odd dimension with a Jordan bracket whose Kantor double is a simple Jordan superalgebra. Using such superalgebras, we provide some examples of the simple abelian Jordan superalgebras whose odd part is a finitely generated projective module of rank 1 in an arbitrary number of generators. An analogous result holds for the Cartesian product of the sphere of even dimension and the affine line. In particular, in the case of the 2-dimensional sphere we obtain the exceptional Jordan superalgebra. The superalgebras we constructed give new examples of simple Jordan superalgebras.

Representations of simple Jordan superalgebras

This paper completes description of categories of representations of finite-dimensional simple unital Jordan superalgebras over algebraically closed field of characteristic zero.

Representations of simple noncommutative Jordan superalgebras I

In this article we begin the study of representations of simple finite-dimensional noncommutative Jordan superalgebras. In the case of degree ≥3 we show that any finite-dimensional representation is completely reducible and, depending on the superalgebra, quasiassociative or Jordan. Then we study representations of superalgebras Dt(α,β,γ) and K3(α,β,γ) and prove the Kronecker factorization theorem for superalgebras Dt(α,β,γ). In the last section we use a new approach to study noncommutative Jordan representations of simple Jordan superalgebras.

The Capelli eigenvalue problem for Lie superalgebras

For a finite dimensional unital complex simple Jordan superalgebra J, the Tits–Kantor–Koecher construction yields a 3-graded Lie superalgebra $$\mathfrak {g}^\flat \cong \mathfrak {g}^\flat (-1)\oplus \mathfrak {g}^\flat (0)\oplus \mathfrak {g}^\flat (1)$$, such that $$\mathfrak {g}^\flat (-1)\cong J$$. Set $$V:=\mathfrak {g}^\flat (-1)^*$$ and $$\mathfrak {g}:=\mathfrak {g}^\flat (0)$$. In most cases, the space $$\mathscr {P}(V)$$ of superpolynomials on V is a completely reducible and multiplicity-free representation of $$\mathfrak {g}$$, and there exists a direct sum decomposition $$\mathscr {P}(V):=\bigoplus _{\lambda \in \Omega }V_\lambda $$, where $$\left( V_\lambda \right) _{\lambda \in \Omega }$$ is a family of irreducible $$\mathfrak {g}$$-modules parametrized by a set of partitions $$\Omega $$. In these cases, one can define a natural basis $$\left( D_\lambda \right) _{\lambda \in \Omega }$$ of “Capelli operators” for the algebra $$\mathscr {PD}(V)^{\mathfrak {g}}$$ of $$\mathfrak {g}$$-invariant superpolynomial differential operators on V. In this paper we complete the solution to the Capelli eigenvalue problem, which asks for the determination of the scalar $$c_\mu (\lambda )$$ by which $$D_\mu $$ acts on $$V_\lambda $$. We associate a restricted root system $$\varSigma $$ to the symmetric pair $$(\mathfrak {g},\mathfrak {k})$$ that corresponds to J, which is either a deformed root system of type $$\mathsf {A}(m,n)$$ or a root system of type $$\mathsf {Q}(n)$$. We prove a necessary and sufficient condition on the structure of $$\varSigma $$ for $$\mathscr {P}(V)$$ to be completely reducible and multiplicity-free. When $$\varSigma $$ satisfies the latter condition we obtain an explicit formula for the eigenvalue $$c_\mu (\lambda )$$, in terms of Sergeev–Veselov’s shifted super Jack polynomials when $$\varSigma $$ is of type $$\mathsf {A}(m,n)$$, and Okounkov-Ivanov’s factorial Schur Q-polynomials when $$\varSigma $$ is of type $$\mathsf {Q}(n)$$. Along the way, we prove that the natural map from the centre of the enveloping algebra of $$\mathfrak {g}$$ into $$\mathscr {PD}(V)^{\mathfrak {g}}$$ is surjective in all cases except when $$J\cong F $$, where $$ F $$ is the 10-dimensional exceptional Jordan superalgebra.

Structure of some Unital Simple Jordan Superalgebras with Associative Even Part

Studying the unital simple Jordan superalgebras with associative even part, we describe the unital simple Jordan superalgebras such that every pair of even elements induces the zero derivation and every pair of two odd elements induces the zero derivation of the even part. We show that such a superalgebra is either a superalgebra of nondegenerate bilinear form over a field or a four-dimensional simple Jordan superalgebra.

Simple finite-dimensional modular noncommutative Jordan superalgebras

We classify the central simple finite-dimensional noncommutative Jordan superalgebras over an algebraically closed field of characteristic p>2. The case of characteristic 0 was considered by the authors in the previous paper [21]. In particular, we describe Leibniz brackets on all finite dimensional central simple Jordan superalgebras except mixed (nor vector neither Poisson) Kantor doubles of the supercommutative superalgebra B(m,n).

Wedderburn principal theorem for Jordan superalgebras I

We consider finite dimensional Jordan superalgebras J over an algebraically closed field of characteristic 0, with solvable radical N such that N=20 and J/N is a simple Jordan superalgebra of one of the following types: Kac K10, Kaplansky K3, superform or Dt.We prove that an analogue of the Wedderburn Principal Theorem (WPT) holds if certain restrictions on the types of irreducible subbimodules of N are imposed, where N is considered as a J/N-bimodule. Using counterexamples, it is shown that the imposed restrictions are essential.

The Structure of Simple Noncommutative Jordan Superalgebras

In this paper we describe all subalgebras and automorphisms of simple noncommutative Jordan superalgebras $K_3(\alpha,\beta,\gamma)$ and $D_t(\alpha,\beta,\gamma)$ and compute the derivations of the nontrivial simple finite-dimensional noncommutative Jordan superalgebras.

Simple Jordan superalgebras with associative nil-semisimple even part

Under study are the simple infinite-dimensional abelian Jordan superalgebras not isomorphic to the superalgebra of a bilinear form. We prove that the even part of such superalgebra is a differentially simple associative commutative algebra, and the odd part is a finitely generated projective module of rank 1. We describe unital simple Jordan superalgebras with associative nil-semisimple even part possessing two even elements which induce a nonzero derivation.

Group gradings on the Lie and Jordan superalgebras Q(n)

ABSTRACTWe classify gradings by arbitrary abelian groups on the classical simple Lie and Jordan superalgebras Q(n), n ≥ 2, over an algebraically closed field of characteristic different from 2 (and not dividing n + 1 in the Lie case): Fine gradings up to equivalence and G-gradings, for a fixed group G, up to isomorphism.

(−1, 1)-Superalgebras of vector type: Jordan superalgebras of vector type and their universal envelopings

We establish a connection between the integral domains, the projective finitely generated modules over them, the derivations of an integral domain and the (−1, 1)-superalgebras of vector type. We study the properties of the universal associative envelopings of the simple Jordan superalgebras of vector type.

Ternary Derivations of Jordan Superalgebras

We give a description of ternary and generalized derivations of finite-dimensional semisimple Jordan superalgebras over an algebraically closed field of characteristic 0, and also of simple Jordan superalgebras with semisimple even part over an algebraically closed field in arbitrary characteristic other than 2.

Novikov–Poisson Algebras and Superalgebras of Jordan Brackets

We prove that for a generalized Novikov–Poisson algebra ⟨ A, ·, ○ ⟩ the bracket is a Jordan bracket, and therefore, the Kantor superalgebra J(A, {,}) is a Jordan superalgebra. Also we prove that ⟨ A, ○ ⟩ is nontrivial simple Novikov algebra and A = AA if and only if J(A, {,}) is a simple Jordan superalgebra.

New examples of simple Jordan superalgebras over an arbitrary field of characteristic 0

An new example of a unital simple special Jordan superalgebra over the field of real numbers was constructed in (10). It turned out to be a subsuperalgebra of the Jordan superalgebra of vector type J( ,D), but not isomorphic to a superalgebra of this type. Moreover, its superalgebra of fractions is isomorphic to a Jordan superalgebra of vector type. A similar example of a Jordan superalgebra over a field of characteristic 0 in which the equation t 2 + 1 = 0 has no solutions was constructed in (12). In this article we present an example of a Jordan superalgebra with the same properties over an arbitrary field of characteristic 0. A similar example of a superalgebra is found in the Cheng-Kac superalgebra.

Simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0

We classify the central simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0. As a corollary, we describe the Poisson brackets on the simple finite-dimensional Jordan superalgebras of characteristic 0.

On $𝛿$-superderivations of simple superalgebras of Jordan brackets

We described $\delta$-derivations and $\delta$-superderivations of simple Jordan superalgebra <<KKM Double>> (also known as superalgebra of Jordan brackets) and unital simple finite-dimensional Jordan superalgebras over algebraic closed fields with characteristic $p\neq2$. As a consequence, we received relationship between non-trivial $\delta$-derivations and specialty of simple superalgebra <<KKM Double>>. We constructed new examples of non-trivial 1/2-derivations of Jordan superalgebras of vector type.

δ-Superderivations of semisimple finite-dimensional Jordan superalgebras

We described $\delta$-derivations and $\delta$-superderivations of simple and semisimple finite-dimensional Jordan superalgebras over algebraic closed fields with characteristic $p\neq2$. We constructed new examples of 1/2-derivations and 1/2-superderivations of simple Zelmanov's superalgebra $V_{1/2}(Z,D).$