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Segre products and Segre morphisms in a class of Yang–Baxter algebras
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AbstractLet $$(X,r_X)$$
(
X
,
r
X
)
and $$(Y,r_Y)$$
(
Y
,
r
Y
)
be finite nondegenerate involutive set-theoretic solutions of the Yang–Baxter equation, and let $$A_X = \mathcal {A}({{\textbf {k}}}, X, r_X)$$
A
X
=
A
(
k
,
X
,
r
X
)
and $$A_Y= \mathcal {A}({{\textbf {k}}}, Y, r_Y)$$
A
Y
=
A
(
k
,
Y
,
r
Y
)
be their quadratic Yang–Baxter algebras over a field $${{\textbf {k}}}$$
k
. We find an explicit presentation of the Segre product $$A_X\circ A_Y$$
A
X
∘
A
Y
in terms of one-generators and quadratic relations. We introduce analogues of Segre maps in the class of Yang–Baxter algebras and find their images and their kernels. The results agree with their classical analogues in the commutative case.
Title: Segre products and Segre morphisms in a class of Yang–Baxter algebras
Description:
AbstractLet $$(X,r_X)$$
(
X
,
r
X
)
and $$(Y,r_Y)$$
(
Y
,
r
Y
)
be finite nondegenerate involutive set-theoretic solutions of the Yang–Baxter equation, and let $$A_X = \mathcal {A}({{\textbf {k}}}, X, r_X)$$
A
X
=
A
(
k
,
X
,
r
X
)
and $$A_Y= \mathcal {A}({{\textbf {k}}}, Y, r_Y)$$
A
Y
=
A
(
k
,
Y
,
r
Y
)
be their quadratic Yang–Baxter algebras over a field $${{\textbf {k}}}$$
k
.
We find an explicit presentation of the Segre product $$A_X\circ A_Y$$
A
X
∘
A
Y
in terms of one-generators and quadratic relations.
We introduce analogues of Segre maps in the class of Yang–Baxter algebras and find their images and their kernels.
The results agree with their classical analogues in the commutative case.
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