We derive certain structural results concerning centroids of Lie supertriple systems. Centroids of the tensor product of a Lie supertriple system and a unital commutative associative algebra are studied. Furthermore, the centroid of a tensor product of a simple Lie supertriple system and a polynomial ring is partly determined.

The notion of Lie triple systems already appeared in Cartan’s work on Riemannian geometry, albeit the formal concept was not defined until 1949 by Jacobson in his study of associative algebras that are closed with respect to triple commutators (cf. [

As a natural generalization of Lie triple systems, the concept of Lie supertriple systems was introduced in the study of Yang-Baxter equations (cf. [

The centroid

In this paper we present new results concerning the centroids of Lie supertriple systems and give some conclusions of the tensor product of a Lie supertriple system and a unital commutative associative algebra. Furthermore, we completely determine the centroid of the tensor product of a simple Lie supertriple system and a polynomial ring. The paper is organized as follows. In Section

Throughout this paper, the base field

An integral domain is a ring with no left or right zero divisors.

An algebraically closed field

An idempotent of a ring is an element

A Lie supertriple system

Let

(1) Let

(2) Any Lie supertriple system

A derivation of a Lie supertriple system

An ideal of a Lie supertriple system

Let

Let

By the definition of Lie supertriple system we conclude that if

It is clear that the scalar maps will always be in the centroid.

Let

It is easily seen that

For any

If

If

Clearly,

If

Let

When

Let

if

Consider the following:

(1) If

On the other hand, suppose

(2) Let

Since

Let

For any

Let

Let

The map

By restriction, there is an algebra homomorphism

If

Suppose

is injective.

If

Consider the following:

(1) It is easy to see that

(2) If

(3) We can see that

If the characteristic of

If

To show the inverse inclusion, let

Let

if

(1) For all

We now prove

Let

If

Let

(1) For any

(2) From Lemma

(3) It follows from

Now, we study the relationship between the centroid of a decomposable Lie supertriple system and the centroid of its factors.

Suppose that

(1) Letting

Next, we prove

Conversely, for

(2) Clearly,

A generalized version of the above theorem is stated below without proof.

Suppose that

Benkart and Neher investigated the centroid of the tensor product of associative algebras and Lie algebras in [

Let

Let

For

It is easy to show that if

The transformation

It is easy to see that

Let

For

Let

if

if the map

(1) Set

Now, suppose

(2) In the discussion above, we get

Next we will determine the centroid of the tensor product of a simple Lie supertriple system and a polynomial ring.

Let

Now we write out a basis of

One has

Let

For any

Letting

From Lemma

Let

Thus, now we may write

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the referee for valuable comments and suggestions on this paper. This research was supported by NNSF of China (nos. 11171055 and 11471090) and the Fundamental Research Funds for the Central University (no. 14ZZ2221).