# Frobenius polycyclic ring

Frobenius multi-component ring(Frobenius, Arrowhead: Frobenius algebra), orFrobenius algebra Is a thing with special in giving a good dual theory among the studies.

Frobenius multiple circle began to be studied as a generalization by and in the 1930's, and it was named after. () And especially () For the first time discovered a rich dual theory. Using this, we characterize Frobenius multi-component ring in (), and this property of Frobenius multi-component ring is perfect I called it duality. Frobenius multi-way ring was generalized to. This is right. Recently, the interest in Frobenius multi-circle has been rising from the connection with.

⇧ Frobenius multiple circle ⊃ symmetric multiple ring ⊃ ⊃ ⊃

Definition

A on * k isFrobenius multiple circleis σ : _ A _ x _ A _ _ _ _ _

σ _ ( ab , _ c ) = σ _ (_a , bc )

It means that there are things that satisfy. This bilinear format is calledFrobenius format(Frobenius form).

Equivalent characterization refers to linear mapping λ ​​: _ A _ → _ k _ and that does not include left non-zero (λ ).

When Frobenius type σ is Frobenius polygon, or when the same condition λ _ ( ab ) = _ λ _ ( ba _) is satisfied, It is called symmetric multi-dimensional ring.

There are also different concepts that are almost irrelevant.

An example

1. Frobenius multivariate with Frobenius form σ _ ( a , _ b ) = ( ab ) on the body k. 2. Arbitrary Finite Dimension Unit Coupled Multi-rings A _ has its own homomorphism to its own homomorphic ring End (_A ). The bilinear form can be defined on A as in Example 1. If this bilinear form is non-degenerate, A _ has the structure of Frobenius multi-circle. 3. All on the body is the Frobenius multi-dimensional ring with the following Frobenius form _σ , ie σ _ ( a , _ b ) is the factor of the identity element in _ a _ _ . This is the special case of Example 2. 4. For body _k, 4-dimensional k - algebra _ k _ [ x , _ y ] / ( x _ 2, _ y _ 2) are Frobenius multi-component rings. This follows the characterization of the exchange Frobenius ring described below. This ring is a local ring with the ideal generated with x _ and _ y _ as the local idea because it has the only minimal ideal generated by _ x _ . 5. For _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ is not a Frobenius multi - dimensional**. From xA _ _ derived from x ↦ y {\ displaystyle x \ mapsto y}! [x \ mapsto y] (https://wikimedia.org/api/rest_v1/media/math/render/svg/eb2452f2d32e5424f3db361de033fd49a73f9dcc) Of _A homomorphism can not be extended to A homomorphism from A _ _ _, hence the ring is not self - primitive and not Frobenius.

nature

• Frobenius multi-dimensional ring and / or are Frobenius multi-way ring.
• It is the same that having a finite dimensional multiple ring on the body is Frobenius, that the right is introductory and that the plural ring has unique.
• Convertible locality Frobenius polygon is just like being local and including finite dimensions on the remainder.
• Frobenius multiple ring is (), in particular, it is left (right) and left (right).
• For body k, it is equivalent that the finite dimensional unitary coupled multi-way ring is Frobenius and that the imported right_A_ - group Hom _ k _ ( A , _ k ) is isomorphic to the right of _ A .
• For infinite field k, the finite dimensional unitary coupled multi-way ring is Frobenius unless there are only a finite number of local minima. If _ _ is a finite order of _ _, finite dimensions _ F _ - multiple rings are naturally finite dimensions _ k _ - multiple rings by, it is equivalent to Frobenius _ F _ - multicircular ring and Frobenius _k _ - multi-ring . In other words, the Frobenius nature does not depend on the body unless the multi-component ring is a finite dimensional multiple ring.
• Similarly, if F is a finite order extension field of k, all k - multiple rings _ A _ naturally generate F _ - multiply ring _ F _ ⊗ _ k _ A _ _ is Frobenius k _ - multicircular ring and _ F _ ⊗ It is equivalent that _k _ A _ is Frobenius _ F _ - multi-component ring.
• In the unitary coupling multiple rings of right finite dimension expression, the Frobenius plural circle A _ is exactly like a multinary circle whose _M _ has the same dimensions as its _ A _ dual Hom _ A _ (_M , _ A ) is there. Among these multiple rings, the _A dual of simple additions is always simple.

footnote

1. ****, Definition 4.2.5.

References

; (1937),, Proc. Nat. Acad. Sci. USA 23(4): 236-240,:,,, DeMeyer, F., Ingraham, E. (1971), Separable Algebras over Commutative Rings , Lect. Notes Math 181, Springer (1958), "Remarks on quasi-Frobenius rings", Illinois Journal of Mathematics 2: 346-354,, (1903), "Theorie der hyperkomplexen Größen I" (German), Sitzungsberichte der Preussischen Akademie der Wissenschaften -: 504-537, Kock, Joachim (2003), Frobenius Algebras and 2D Topological Quantum Field Theories, London Mathematical Society student texts, Cambridge: Cambridge University Press, Lam, T. Y. (1999),, Graduate Texts in Mathematics No. 189, Berlin, New York: Lurie, Jacob,, (1939),, (Annals of Mathematics)40(3): 611-633,::,,, (1941),, __ (Annals of Mathematics)42(1): 1-21,:,,, (1938),, 39(3): 634-658,::,,,, Onodera, T. (1964), "Some studies on projective Frobenius extensions", __18: 89-107 Weibel, Charles A. (1994).. Cambridge University Press.

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Post Date : 2018-02-17 19:00